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Minimum-variance sampling schemes for the scaling of logs by weight Nokoe, Sagary 1976

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MINIMUM-VARIANCE SAMPLING SCHEMES FOR THE SCALING OF LOGS BY WEIGHT by SAGARY NOKOE B.Sc. (For.) Hons., University of Ibadan, 1972 M.F., University of B r i t i s h Columbia, 1974 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY (Forestry) in THE FACULTY OF GRADUATE STUDIES We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1976 ® Sagary Nokoe, 1976 In presenting th is thes is in pa r t i a l fu l f i lment o f the requirements for an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f ree ly ava i lab le for reference and study. I fur ther agree that permission for extensive copying of th is thes is for scho la r ly purposes may be granted by the Head of my Department or by h is representa t ives . It is understood that copying or p u b l i c a t i o n of th is thes is for f i n a n c i a l gain sha l l not be allowed without my writ ten permission. Depa rtment The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 f) Supervisor: Professor A. Kozak i i # ABSTRACT T r a d i t i o n a l l y , logs have been measured i n d i v i d u a l l y (scaled) to estimate t h e i r t o t a l cubic foot content of wood, inside bark. Recently, procedures have been developed for estimating firm wood volumes from r a t i o of volume of logs to t h e i r weight. The objectives of the study were to examine minimum-variance sampling schemes for r a t i o estimation, and the selection of the most appropriate sampling procedure for weight s c a l i n g . Minor objectives included the use of double-sampling and p o s t - s t r a t i f i c a t i o n procedures for greater e f f i c i e n c y . After considering expected v a r i a t i o n , appropriate sampling procedures were devised, and then used to draw samples from generated populations and weight scale data respectively, without replacement. The sampling schemes included the "completely random", "representative (or r e s t r i c t -ed) random", "modified (or uniform) random", frequency^-dependent, and size-dependent sampling schemes. The d i f f e r e n t populations were generated according to hypothetical r a t i o estimates, and the exponent (p) by which the a u x i l i a r y variates were re l a t e d to the variance of the corresponding variates of the population of i n t e r e s t . I n i t i a l r e s u l t s indicated that for a part i c u l a r sample size , a scheme r e s u l t i n g in the highest sample c o r r e l a t i o n c o e f f i c i e n t did not necessarily give the smallest variance. Among a l l modifications of a particular sampling scheme, that modification r e s u l t i n g in the highest sample c o r r e l a t i o n c o e f f i c i e n t also gave the smallest variance of the population of i n t e r e s t . For populations with positive "p" values, including the weight scaling data, i t was found that sampling for r a t i o estimation with p r o b a b i l i t y proportional (or inversely proportional) to the magnitude of the a u x i l i a r y variable, and for only f i v e frequency classes, led to the "best" minimum-variance estimate of the mean of population of in t e r e s t . These schemes gave co n s i s t e n t l y large mean devia-tions from the mean of the population of in t e r e s t . With increasing number of classes, the magnitude of the mean deviations was reduced but the property of minimum-variance was not necessarily retained. For populations with negative values of "p", the eUfoerdtSichemest pef.fiorixied^ b'efe'.t'e^ -.v.tH'a^ -' r.-thesietJ-bu-ta s'tdtllstha'dl magc&rmfd&fdoiefrci'esin i e fa . On the basis of reasonably small variances and devia-tions, the "modified random sampling" scheme was selected as the best. Examination of the scaling data showed i v . almost constant variances for sub-divisions based on a r r i v a l times or periods. For the modified random scheme, equal numbers of observations were taken within each group of a r r i v a l of truck loads. Its ultimate selection as the best of a l l the minimum-variance schemes for weight scal i n g , in p a r t i c u l a r , could therefore be linked with the homogeneity of the group variances. P o s t - s t r a t i f i c a t i o n resulted in greater e f f i c i e n c y , e s p e c i a l l y for large numbers of groups. Double-sampling procedures did not lead to any improvement i n the r e s u l t s . I t was suggested that "modified (or uniform) random" sampling procedures, used j o i n t l y with p o s t - s t r a t i f i c a t i o n by a r r i v a l of truck loads, would be ideal for the s c a l i n g of logs by weight. v. TABLE OF CONTENTS Page 1.0 INTRODUCTION 1 2.0 BACKGROUND INFORMATION AND REVIEW 3 2.1 Ratio estimation - theory, biased and unbiased type estimators 3 2.2 Sampling schemes for r a t i o estimation 15 2.3 Double-sampling procedure for r a t i o estimation -23 2.4 Log scaling by weight 29 2.5 Generated versus c o l l e c t e d data 33 3.0 BASIC FORMULATION AND PROCEDURE 35 3.1 Ratio estimate, bias and variance 35 3.I . i Ratio estimate, r 35 P 3 . 1 . i i Bias of r 36 P 3 . 1 . i i i Variance of rp 38 3.1.iv An i l l u s t r a t i o n of the formulae 39 for s i m p l i f i e d weight scaling 3.2 Data generation procedure for cases 1-21 42 3.3 Determination of sample size 44 3.4 Description of sampling schemes 46 4.0 DATA SUMMARIES AND DESCRIPTION 51 4.1 Generated data - case studies 1-21 and A1-A3 51 4.2 IFS Log Scaling and weight data 66 v i . Page 5.0 RESULTS FOR CASE STUDIES 1-21 AND A1-A3 72 5.1 General introduction 72 5.2 Case studies 1-21 and A1-A3 72 5.2.i Relationship between RHO and standard errors 72 5 . 2 . i i Estimates of Rp versus R^ for same samples 5 . 2 . i i i Screening for 'best' sampling schemes 80 5.2.iv Mean deviation as function of p and sample siz e , n 102 6.0 RESULTS OF APPLICATIONS TO IFS SCALE DATA AND SUMMARY OF RESULTS 107 6.1 IFS scale data 107 6.2 Summary of re s u l t s 126 7.0 STRATIFICATION, DOUBLE-SAMPLING AND OTHER CONSIDERATIONS 131 7.1 The data and additional notations 131 7.2 Numerical i l l u s t r a t i o n of t h e o r e t i c a l bias 134 7.3 S t r a t i f i c a t i o n for greater e f f i c i e n c y 138 7.3.i Sampling from respective populations 140 7 . 3 . i i Assuming sample loads from large population sizes 142 7.4 Double-sampling procedures for companies A, D,F,J,K,L, M 149 7.5 Summary of r e s u l t s 152 8.0 CONCLUSIONS AND RECOMMENDATIONS 153 LITERATURE CITED 158 APPENDICES 165 LIST OF TABLES v i i . Table Page 1 POPULATION PARAMETERS AND GENERATED DATA (CASES 1-21) . 54 2 CASE 1 - FREQUENCY DISTRIBUTION FOR 10 CLASSES 55 3 CASE 2 - FREQUENCY DISTRIBUTION FOR 10 CLASSES 55 4 CASE 3 - FREQUENCY DISTRIBUTION FOR 10 CLASSES 55 5 CASE 4 - FREQUENCY DISTRIBUTION FOR 10 CLASSES 56 6 CASE 5 - FREQUENCY DISTRIBUTION FOR 10 CLASSES 56 7 CASE 6 - FREQUENCY DISTRIBUTION FOR 10 CLASSES 56 8 CASE 7 - FREQUENCY DISTRIBUTION FOR 10 CLASSES 57 9 CASE 8 - FREQUENCY DISTRIBUTION FOR 10 CLASSES 57 10 CASE 9 - FREQUENCY DISTRIBUTION FOR 10 CLASSES 57 11 CASE 10 - FREQUENCY DISTRIBUTION FOR 10 CLASSES 58 12 CASE 11 - FREQUENCY DISTRIBUTION FOR 10 CLASSES 5 8 13 CASE 12 - FREQUENCY DISTRIBUTION FOR 10 CLASSES 5 8 14 CASE 13 - FREQUENCY DISTRIBUTION FOR 10 CLASSES 59 15 CASE 14 - FREQUENCY DISTRIBUTION FOR 10 CLASSES 59 16 CASE 15 - FREQUENCY DISTRIBUTION FOR 10 CLASSES 59 17 CASE 16 - FREQUENCY DISTRIBUTION FOR 10 CLASSES 60 18 CASE 17 - FREQUENCY DISTRIBUTION FOR 10 CLASSES 60 19 CASE 18 - FREQUENCY DISTRIBUTION FOR 10 CLASSES 60 20 CASE 19 - FREQUENCY DISTRIBUTION FOR 10 CLASSES 61 21 CASE 20 - FREQUENCY DISTRIBUTION FOR 10 CLASSES 61 22 CASE 21 - FREQUENCY DISTRIBUTION FOR 10 CLASSES 61 23 INITIAL SUMMARY FOR CASES A1-A3 62 24 CASE A l - FREQUENCY DISTRIBUTION FOR 10 CLASSES 64 2 5 CASE A2 - FREQUENCY DISTRIBUTION FOR 10 CLASSES 64 2 6 CASE A3 - FREQUENCY DISTRIBUTION FOR 10 CLASSES 64 2 7A POPULATION PARAMETERS FOR DATA OBTAINED FROM IFS - GROUPED IN 2O'S 68 27B POPULATION PARAMETERS FOR DATA OBTAINED FROM IFS - GROUPED IN 40'S 69 27C POPULATION PARAMETERS FOR DATA OBTAINED s FROM IFS - GROUPED IN 80'S 69 2 8 FREQUENCY DISTRIBUTION (5 CLASSES) FOR IFS SCALE DATA 71 29 FREQUENCY DISTRIBUTION (10 CLASSES) FOR IFS 71 SCALE DATA 30 FREQUENCY DISTRIBUTION (15 CLASSES) FOR IFS SCALE DATA 71 31 CASE 1 - EFFICIENCY OF SAMPLING SCHEMES (SS), P = -1.0 (SAMPLE OUTPUT) 7 3 32 CASE 11 - EFFICIENCY OF SAMPLING SCHEMES (SS), P = 0.5 (SAMPLE OUTPUT) 74 v i i i . LIST OF TABLES (Continued) Table P a 9 e 33 CASE 15 - EFFICIENCY OF SAMPLING SCHEMES (SS); P = -1.0 (SAMPLE OUTPUT) 75 34 CASE A l - EFFICIENCY OF SAMPLING SCHEMES (SS); EST. P = 1.8808 (SAMPLE OUTPUT) 7 6 35A CASE 1: RANKS BASED ON STANDARD ERRORS FOR ALL SCHEMES APPLIED 86 36A CASE 2: RANKS BASED ON STANDARD ERRORS FOR ALL SCHEMES APPLIED 86 37A CASE 3: RANKS BASED ON STANDARD ERRORS FOR ALL SCHEMES APPLIED 86 35B CASE 1: DEVIATION FOR BEST SCHEMES (P = -1.0) 87 36B CASE 2: DEVIATION FOR BEST SCHEMES (P = 0.5) 87 37B CASE 3: DEVIATION FOR BEST SCHEMES (P = 0.0) 87 38A CASE 4: RANKS BASED ON STANDARD ERRORS FOR ALL SCHEMES APPLIED 88 39A CASE 5: RANKS BASED ON STANDARD ERRORS FOR ALL SCHEMES APPLIED 88 40A CASE 6: RANKS BASED ON STANDARD ERRORS FOR ALL SCHEMES APPLIED 88 3 8B CASE 4: DEVIATION FOR BEST SCHEMES (P = 0.5) 89 39B CASE 5: DEVIATION FOR BEST SCHEMES (P = 1.0) 89 40B CASE 6: DEVIATION FOR BEST SCHEMES (P = 1.5) 89 41A CASE 7: RANKS BASED ON STANDARD ERRORS FOR ALL SCHEMES APPLIED 90 42A CASE 8: RANKS BASED ON STANDARD ERRORS FOR ALL SCHEMES APPLIED 90 43A CASE 9: RANKS BASED ON STANDARD ERRORS FOR ALL SCHEMES APPLIED . 90 41B CASE 7: DEVIATION FOR BEST SCHEMES (P = 2.0) 91 42B CASE 8: DEVIATION FOR BEST SCHEMES (P = 1.0) 91 43B CASE 9: DEVIATION FOR BEST SCHEMES (P = -0.5) 91 44A CASE 10: RANKS BASED ON STANDARD ERRORS FOR ALL SCHEMES APPLIED 92 45A CASE 11: RANKS BASED ON STANDARD ERRORS FOR 92 ALL SCHEMES APPLIED 46A CASE 12: RANKS BASED ON STANDARD ERRORS FOR ALL SCHEMES APPLIED 92 44B CASE 10: DEVIATION FOR BEST SCHEMES (P = 0.0) 93 45B CASE 11: DEVIATION FOR BEST SCHEMES (P = 0.5) 93 46B CASE 12: DEVIATION FOR BEST SCHEMES (P = 1.0) 93 47A CASE 13: RANKS BASED ON STANDARD ERRORS FOR ALL SCHEMES APPLIED 94 48A CASE 14: RANKS BASED ON STANDARD ERRORS FOR ALL SCHEMES APPLIED 94 LIST OF TABLES (continued) i x . Table Page 49A CASE 15: RANKS BASED ON STANDARD ERRORS FOR ALL SCHEMES APPLIED 9 4 47B CASE 13: DEVIATION FOR BEST SCHEMES (P = 1.5) 95 48B CASE 14: DEVIATION FOR BEST SCHEMES (P = 2.0) 95 49B CASE 15: DEVIATION FOR BEST SCHEMES (P = 1.0) 95 50A CASE 16: RANKS BASED ON STANDARD ERRORS FOR ALL SCHEMES APPLIED 96 51A CASE 17: RANKS BASED ON STANDARD ERRORS FOR ALL SCHEMES APPLIED 96 52A CASE 18: RANKS BASED ON STANDARD ERRORS FOR ALL SCHEMES APPLIED 96 5OB CASE 16: DEVIATION FOR BEST SCHEMES (P = -0.5) 97 5IB CASE 17: DEVIATION FOR BEST SCHEMES (P = 0.0) 97 52B CASE 18: DEVIATION FOR BEST SCHEMES (P = 0.5) 97 53A CASE 19: RANKS BASED ON STANDARD ERRORS FOR ALL SCHEMES APPLIED 98 54A CASE 20: RANKS BASED ON STANDARD ERRORS FOR ALL SCHEMES APPLIED 98 55A CASE 21: RANKS BASED ON STANDARD ERRORS FOR ALL SCHEMES APPLIED 98 53B CASE 19: DEVIATION FOR BEST SCHEMES (P = 1.0) 99 54B CASE 20: DEVIATION FOR BEST SCHEMES (P = 1.5) 99 55B CASE 21: DEVIATION FOR BEST SCHEMES (P = 2.0) 99 56A CASE A l : RANKED VALUES BASED ON STANDARD ERRORS 100 57A CASE A2: RANKED VALUES BASED ON STANDARD ERRORS 100 58A CASE A3: RANKED VALUES BASED ON STANDARD ERRORS 100 •56B MEAN DEV. OF BEST SIX SCHEMES FROM TABLE 56A 101 57B MEAN DEV. OF BEST SIX SCHEMES FROM TABLE 57A 101 58B MEAN DEV. OF BEST SIX SCHEMES FROM TABLE 5 8A 101 59 REGRESSION COEFFICIENTS FOR DEV., R, P AND n RELATIONS 106 60 SAMPLE SIZE DETERMINATION FOR ± A% OF YBAR PRECISION REQUIREMENT 109 61 SAMPLE SIZE DETERMINATION FOR ± A CUBIC FEET PRECISION REQUIREMENT 109 62A RANKED VALUES BASED ON STANDARD ERRORS FOR 14 SCHEMES HO 62B RANKED VALUES BASED ON STANDARD ERRORS FOR 14 SCHEMES H I 63 NUMBER OF +VE & -VE DEV. FOR BEST SIX SCHEMES H2 64 MEAN DEVIATION FOR THE BEST SCHEMES SELECTED 112 65 MEAN OF ABSOLUTE DEVIATIONS FOR ABOVE SCHEMES 112 66 EFFICIENCY OF SAMPLING SCHEMES - POP YBAR = 1329.38; SAMPLE SIZE = 31 (SAMPLE OUTPUT) 119 67 EFFICIENCY OF SAMPLING SCHEMES - POP YBAR = 1329.38; SAMPLE SIZE = 36 (SAMPLE OUTPUT) 120 X. LIST OF TABLES (continued) Table Page 68 EFFICIENCY OF SAMPLING SCHEMES - POP YBAR = 132 9.3 8; SAMPLE SIZE - 36 (SAMPLE OUTPUT) 121 69 EFFICIENCY OF SAMPLING SCHEMES - POP YBAR = 1329.38; SAMPLE SIZE = 45 (SAMPLE OUTPUT) 122 70 EFFICIENCY OF SAMPLING SCHEMES - POP YBAR = 1329.38; SAMPLE SIZE = 61 (SAMPLE OUTPUT) 123 71 EFFICIENCY OF SAMPLING SCHEMES - POP YBAR = 1329.38; SAMPLE SIZE =164 (SAMPLE OUTPUT) 124 72 EFFICIENCY OF SAMPLING SCHEMES - POP YBAR = 1329.38; SAMPLE SIZE = 180 125 7 3 SUMMARY OF LEAST-VARIANCE SCHEMES USED IN TABLES 35B-58B 129 74 SUMMARY FOR COMPANIES A-M 133 75 RESULTS FOR COMP.G: SAMPLE SIZE = 70 (SAMPLE) 135 76 RESULTS FOR COMP.K: SAMPLE SIZE = 150 (SAMPLE) 136 77 RESULTS FOR COMP.A: SAMPLE SIZE = 30 (SAMPLE) 137 7 8 STRATIFICATION BY SIZE OF X FOR COMPANIES A,D, 143 F,J,K,L,M & T 7 9 STRATIFICATION BY SIZE OF X FOR COMPANIES B,C, E,G,H,I 144 80 STRATIFICATION BY ARRIVALS FOR COMPANIES A,D, F , J,K,L,M 145 81 STRATIFICATION BY ARRIVALS FOR COMPANIES B,C, E,G,H,I 146 LIST OF FIGURES F igure Page 1 Type 5 for 5 classes 50 2 Type 6 for 5 classes 50 3 Histograms for A l , A2 and A3-5 and 15 classes 65 4 Rangexof&deyiat'i6n's».fo r K o.tiviatior. f p and n respectively 103 ACKNOWLEDGEMENT Dr. Antal Kozak suggested the problem and supervised me on the study. The members of my committee—Drs. J . Demaerschalk, A. Kozak, D. D. Munro and S. W. Nash—reviewed the rough d r a f t . Dr. S. W. Nash checked the derivations and gave useful suggestions. Dr. J. H. G. Smith of the Faculty of Forestry, and Mr. A. Fraser of the B r i t i s h Columbia Forest Service (B.C.F.S.) Research D i v i s i o n provided valuable comments. I am most gra t e f u l to a l l . I am also g r a t e f u l to Dean J. A. F. Gardner, the Faculty of Forestry and the Biometrics Group, for sponsoring my p a r t i c i p a t i o n at the ,!Log Scaling by Weight' short course at Oregon State University, A p r i l 26-28, 1976. Data for the study were made available by the In d u s t r i a l Forest Service Limited, and the Research D i v i s i o n of the B.C.F.S. I am p a r t i c u l a r l y g r a t e f u l to Mr. D. E r r i c o of the B.C.F.S. Research D i v i s i o n for the i n i t i a l data screening and preparation. The University of B r i t i s h Columbia Computing Centre i s also acknowledged for the provision of computing f a c i l i t i e s . F i n a n c i a l assistantships were given in the form of Faculty of Forestry Research/Academic Grants, the MacMillan-Bloedel Fellowship in Mensuration, and the University of B r i t i s h Columbia Graduate Fellowships. I am g r a t e f u l to a l l those who made these possible. F i n a l l y , I am g r a t e f u l to Theresa, my wife, for her love and encouragement throughout the period of my studies at the University of B r i t i s h Columbia. < > n l n2 x. 1 X I V , EXPLANATION OF SYMBOLS Symbol Page f i r s t used Explanation 27 Less than ^ 2 5 Less than or equal to 7 8 Greater than ^ 6 Greater than or equal to ^ 7 8 Far greater than = 11 Approximately equal to OC 21 Proportional to % 2 9 percentage n 3 Sample size 25 First-phase sample size 2 5 Second-phase sample size N 5 population size k 4 constant of proportion-a l i t y 2 V(...), s ... 4,7 Variance of ... ^ 4 Summation sign ^ 8 Summation for a l l possible ways /NN \ n) 8 Number of combinations of n from N x 3 A u x i l i a r y variable (weight) .th 3 I variate of x 3 Variable of interest (volume) Symbol Page f i r s t used Explanation y, 3 i variate of y x, y 4 sample means X, Y 3 Population means rB l 4 ratio-of-means estimator (P = 1) 4 any r a t i o estimator 7 Beale's estimator r T 8 'True' r a t i o estimate rg 6 Quenouille's estimate of R r ^ r ^ 11 Product estimate of R p 4 a factor (variable) i n the r e l a t i o n s h i p of V(Y.) and x_., e.g. -1 V ( y j ) = kx? b 6 a factor (variable) e.g. n b 9 48 proportion for extremes of population «K 44 Type 1 error (1 - ^  = confidence prob.) Pr. 20 p r o b a b i l i t y of s e l e c t -ing the I*-*1 sample Pr 12 Pr o b a b i l i t y P' population proportion 1 2 5 for l^1 stratum Q. 26 1 - P' 1 i # 46 number Symbol Page f i r s t used Explanation E(...) 6 Expectation of ... D [ . . . j 104 Deviation for scheme 25 a l l i d e n t i c a l functions that give estimates of p.'Y. f 11 the f r a c t i o n n/N B 181? number of blocks, defined by N/n (to the nearest integer) t 44 student-t s t a t i s t i c r 7 Konijn's estimate of R k r 4 Mean-of-ratios estimate 2 r 9 Hartley and Ross r a t i o HR estimator r 11 Mickey's estimator MI S(...) 6 Standard deviation of.. g 6 number of groups m 6 sample size for each g s 31 number of species or species groups m. 2 5 number of samples for i 1 - " stratum st 2 5 number of strata c 47 number of classes S^  26 standard deviation of l - i : n stratum 46 number of classes or groups, e.g. MZ th f^ 47 frequency of i class Symbol Page f i r s t used Explanation x v i i . a random variate with 12 expectation Z 0~ a population standard 1 2 deviation G Fisher's measure of 13 kurtosis ^ Population c o r r e l a t i o n x ^ 11 c o e f f i c i e n t of x, y k„ 12 Fisher's 4 t h cummulant 4 Cov(...), S , etc. 7 Covariance of.... x and y e tc. C x, C sample c o e f f i c i e n t of 7 v a r i a t i o n of x, y respectively C,,„ sample c o e f f i c i e n t of 7 v a r i a t i o n of x and y ^ c o e f f i c i e n t of independent 21 v a r i a b l e . In r a t i o estimation. G>= R V P c 21 error term b (weighted) estimate w 21 of <?(=?>) estimate, e.g. Y, 6 estimate of Y E' required precision of 44 ¥ in cubic feet or i n any volume measurement A required precision of 44 Y in percent 1 q r cubic feet CV 44 a c o e f f i c i e n t of v a r i a t i o n Symbol Page f i r s t used E x p l a n a t i o n YCV%, XCV% 51 c o e f f i c i e n t o f d e t e r -m i n a t i o n o f y, x i n perce n t a g e r e s p . RHO 52 c o r r e l a t i o n c o e f f i c i e n t between x and y XTOT, YTOT 53 T o t a l s f o r x, y r e s p -e c t i v e l y XVAR. YVAR XMID. 51 53 v a r i a n c e s f o r x, y r e s p e c t i v e l y t h e m i d - p o i n t o f j x - c l a s s t h y |x h ( y j x ) f ( * , y ) u, v 42 42 42 42 42 n o t a t i o n f o r a b s o l u t e v a l u e y g i v e n x c o n d i t i o n a l d e n s i t y o f y g i v e n x j o i n t d i s t r i b u t i o n o f x and y s t a n d a r d normal v a r i a b l e s 35 sum o f w e i g h t e d square d e v i a t i o n s dg. ^ P 0 35 38 36 d e r i v a t i v e o f q w i t h r e s p e c t t o 1 x n m a t r i c e s o f x, y r e s p e c t i v e l y F u n c t i o n o f x, .Y r e p r e s e n t i n g r d0 \ < d x j / o 36 E v a l u a t i o n o f d0 dx j a t p o i n t o f e x p e c t e d v a l u e s Symbol F i r s t page used Explanation CR 46 RS 46 MZ 46 1, z 47 2, Z 47 3, Z 47 4, Z 48 5, Z 48 6, Z 49 IXZ 49 DXZ 49 SYBA.R 72 SYBA., SYBAR 72 Completely random sampling scheme Representative sampling scheme Modified random sampling schemes for Z groups Type 1 frequency sampling, Z classes Type 2 frequency sampling, Z. classes Type 3 frequency sampling, Z classes Type 4 frequency sampling, Z classes Type 5 frequency sampling, Z classes Type 6 frequency sampling Z classes Frequency sampling with p r o b a b i l i t y increas-ing with magnitude of x. Frequency sampling with p r o b a b i l i t y decreasing with magnitude of x. Standard error of YBAR involving r a t i o estima-t i o n (that i s , use of an a u x i l i a r y variable) Standard error of YBAR (no use of r a t i o estima-tion) . SS 72 Sampling schemes Symbol Page f i r s t used Explanation CS MIN MAX TBIAS P-SYB.R S-SYB.R Upper-case Y: 51 132 132 132 132 132 l e t t e r s , e.g. Case study (or data set) Minimum Maximum Theoretical bias of the estimate of Y P o s t - s t r a t i f i c a t i o n SYBA.R P r e - s t r a t i f i c a t i o n SYBA.R Parameters, i n general Lower-case l e t t e r s , e.g. y: S t a t i s t i c s (estimates of parameters), i n general. 1. 1.0 INTRODUCTION Log sc a l i n g by weight, simply r e f e r r e d to as weight scaling, i s a technique that r e l i e s on the p r i n c i p l e of r a t i o estimation or the use of a u x i l i a r y information (load weight). It i s being used, as an alternative to one hundred per cent scaling, by the B r i t i s h Columbia Forest Service (B.C.F.S.) for stumpage assessments. The Forest Services of the States of C a l i f o r n i a and Texas, respectively, are also using the technique, for the same objective ( B l a i r , 1965 and Burns, 1970). The usual process of selection of samples and that required in B. C. i s "the representative sampling scheme" (B.C.F.S. Scaling Manual, 1975). This scheme i s geared towards equal p r o b a b i l i t y sampling and also ensures that the population i s adequately covered over time. It does not, however, completely avoid the p o s s i b i l i t y of sampling only from pa r t i c u l a r groups of the a u x i l i a r y population, unless s t r a t i f i c a -t i o n procedures are used. Moreover, the r e l a t i v e e f f i c i e n c y or i n e f f i c i e n c y y o f t h i s scheme i s not known i n r e l a t i o n to other schemes hitherto not used for weight s c a l i n g . The main objectives of t h i s study are the examination of minimum-variance sampling schemes for r a t i o estimation, and the s e l e c t i o n of the most appropriate sampling scheme(s) for weight s c a l i n g . A scheme w i l l be regarded as the "most approp-r i a t e " i f i t r e s u l t s in an estimate with a reasonably small 2 . variance and minimum deviation from the parameter of i n t e r e s t . Minor objectives include the use of p o s t - s t r a t i f i c a t i o n procedures for greater e f f i c i e n c y , and the use of double-sampling procedures for the estimation of the p r o b a b i l i t y d i s t r i b u t i o n s of the a u x i l i a r y population. The study was performed on representative generated data, as well as actual sample scale data c o l l e c t e d for particular period of operations. It was not intended to examine the r e l a t i v e e f f i c i e n c i e s of several ratio-type estimators, even though some discussion i s provided in some chapters to provide a j u s t i f i c a t i o n of a par t i c u l a r estimate. B i o l o g i c a l and economic considerations did not form e s s e n t i a l parts of the study. I t should be possible, however, to extend some of the r e s u l t s to cover these considerations. For instance, a scheme r e s u l t i n g i n small variances would imply small sample sizes would be required for the attainment of s p e c i f i e d precisions, and less sampling costs. Chapter 2 provides background information on r a t i o estimation, weight scaling and sampling procedures. Chapter 3 considers t h e o r e t i c a l formulation of the r a t i o formulae, and the procedure adopted in t h i s study. The remaining chapters are devoted to data description, application of sampling and the discussion of the r e s u l t s . Wherever necessary, summaries are provided for quick reference. 3. 2.0 BACKGROUND INFORMATION AND REVIEW 2<1 Ratio estimation - theory, biased and unbiased-type estimators. Throughout t h i s and subsequent discussion, upper-case l e t t e r s w i l l be used to designate parameters and lower-case l e t t e r s (or capped l e t t e r s ) as s t a t i s t i c s (parameter estimates). The r a t i o estimation procedure of interest in t h i s study involves paired measurements—auxiliary va r i a t e , x^ ( i = l , . . . n ) , from a population in which the mean, X i s known or can be estimated and a correspond-ing y^ ( i = l , . . . , n ) , as the variate of the population of i n t e r e s t . I t i s intended to estimate Y using a r a t i o involving x and y, and X. The other case of r a t i o estimation which involves independent samples of x and y, each of size n (Hansen, et_ al__. 1953), w i l l not be covered because of i t s i n a p p l i c a b i l i t y to weight s c a l i n g . th In weight scaling, the variate x_. i s the j truck load weight and yj i s the j load volume or scale. Conditions required for the effectiveness of r a t i o estimates as discussed by Cochran (1963), Hartley and Ross (1954), Hendricks (1956), Rao (1966), Robson and Vithyasai (1961); Singh (1965, 1967), Sukhatme (1954) and with Sukhatme (1970), Konijn (1973) and others, include a) x, y should be p o s i t i v e l y correlated; b) regression of y on x should be a perfect straight l i n e through the o r i g i n ; and c) the c o e f f i c i e n t of v a r i a t i o n (as a percentage) for both the sample means x and y should be less than one per cent. Furthermore, the choice of the type of r a t i o to use for a par t i c u l a r s i t u a t i o n i s dependent on the r e l a t i o n -ship between the variance of yj (from a group of same j ' s ) , V(y_j) and X j . The r e l a t i o n s h i p i s of the form V ( Y j ) = kxP (2.1.1) where k i s a constant of proportionality. The usual r a t i o of means estimate of the r a t i o , n , n 21 y. / 1 i = l i i = l is based on p = 1, hence the subscript 1 on the est-r, =  21 x. = y/x, (2.1.2) 1 i=i ' i i=i i pnafcof. r . Unsubscripted r or R should be interpreted as any general or appropriate r a t i o , i n t h i s thesis. For p = 2, n fn \ r_ = Z. (y./x,)/ n = 121 r ; /n = r (2.1 .3) 2 i = l 1 l \l=l i / which i s the mean of the r a t i o s r ^ . It i s thus easy to observe that the functional r e l a t i o n s h i p between the 5 . variance of and x^ i s very important. Usually t h i s r e l a t i o n s h i p i s unknown but g i v e n a w i d e range*o.f data i t > c a n b , e , e s t i m a t e d , i f , o n e i s injt - e c e s^tedr; 1- i t S a ^ n c e r 2 i s .usually s u b j e c t-rt ; Q j ; m R G ' h largerate:ainp<iLing- ,f l o e t u ait i o n s than the r a t i o £p0d:J§©.odma.n -and. H a r t l e y E( 195-8'.)'<r-su'ggfes'4?edfc that rdne should be: r(e,l\ucfca*nit t o u s e a:^ wiitihO'iiit (cl<e'a>r ( e v i d e n c e o f ±*trs> p r e f ' e r a b m l i t y / . . Much of the general theory of r a t i o estimation has been documented i n basic textbooks on sampling techniques such as Yates (1960), Cochran (196 3), Sukhatme (1954), Hansen et_ a l . (1954), Murthy (1967), Konijn (1973) and others, for one stratum and several s t r a t a . I t i s therefore intended to provide a discussion on only the sections of interest in t h i s study. Let us assume that the population size i s N -known or unknown - and a required sample s i z e , for a s p e c i f i e d precision, n. Suppose that V(y_j) i s proportion-a l to XJ (implying p = 1), and that the n observations are drawn randomly with x^ and y^ measured on each occasion, then an estimate of the r a t i o R-^  i s given by equation (2.1.2) with a variance of N 2 i Z ( 7 i - RlXd.) V ( r n ) = — (2.1.4) nX N This could be estimated by v ( r l > = ^ ~ N ( 2 - 1 - 5 ) where the l a s t m u l t i p l i e r i s the f i n i t e population correction factor for sampling without replacement. The estimated standard errortisr*given.*by:i Ly sfr\) ={v? 1)] 3 S (2.1.6) The estimation of equation (2.1.4) i s based on •'4 ;the f i r s t two .terms-in the.eexpans.ionriof fthe> ;b"ias-.of'' the r a t i o estimate [ E ( r 1 ) - R J . This variance has been shown by several authors to be biased in the order 1/n. Using more terms i n the expansion instead of two, i t was shown that t h i s bias could be reduced to 1/n where b ^ 1. Quenouille (1956) suggested a procedure 2 which was shown to reduce the bias to order ,o,f,"l/n , using the previous variance. The procedure involved the breakdown of the samples into g groups each of size m (n - gm), and then the computation of the r a t i o estimates for each group using equations (2.1.2) or (2.1.3), whichever was appropriate. The o v e r a l l r a t i o estimate was obtained as r Q =gr - i a _ z _ l l [ r Q l + + ... + r Q g j . . . (2.1.7) with r n (j = 1, g) indicating various group r a t i o s obtained in the same manner as r . However, the optimum number of groups required for the estimation has not been documented. Durbin (1959) commented favourably on Quenouille's estimator, but T i n (1965), using expansions for the bias and variance of these estimators, obtained r e s u l t s that led to h i s r e j e c t i o n of Quenouille's estimator with g = 2, i n favour of two estimators due to Konijn and Beale respectively. Konijn's estimator i s of the form r k = r [ l + |(N-n)/NnJ | r S y x / y x - £2 J*}} . . (2.1.8) which may be s i m p l i f i e d by replacing the sample variance 2 . . s x and covariance s y x divided by t h e i r respective means into 2ll r k = r [ l + {(N-n)/Nnj.(c y x - 6*) . (2.1.9) Q y x i s the sample c o e f f i c i e n t of v a r i a t i o n of x and y 2 (not i n percentage) and c i s the square of the c o e f f i c i e n t of v a r i a t i o n of x. The above estimator was obtained by subtracting from r , an estimator of the f i r s t term in the bias expansion. Beale's estimator, r B # was given as rB = r j l + c y x (N-n) /Nnj/ j l + 5 2 x (N-n)/NnJ . . . . (2.1.10) Tin further obtained conclusions, which have been said 8. to be non-deducible from his derivations. The con-clusions were that up to the order 1/n, r ^ and r g were free from bias, while that of r ^ / r was - ^ C 2 X - c X y } / / N ' With ten sets of 1000 observations, Konijn concluded that Tin's r e s u l t s which were based on one set of sample of size 1000 were unreliable for some biya r i a t e normal d i s t r i b u t i o n s . Konijn further i l l u s t r a t e d the d i f f i c u l t i e s in obtaining a desired standard error using sample sizes of as high as 1000 even when there was a high c o r r e l a t i o n c o e f f i c i e n t . Koop (1968) i l l u s t r a t e d i n a short a r t i c l e that the use of the approximate formula for the r a t i o estimate qpuld.^reafcLy* underestMa'td-lts tpue r a t i o , r , when the sample size i s small. Using empirical i l l u s t r a t i o n s , he showed that i n comparison with the true variance of r T , (equation 2.1.11 below), the approximation could lead to misleading r e s u l t s and as high as 40.7% i n e f f i c i e n c y for a d i s t r i b u t i o n with a r a t i o estimate of about 0.087976: of y/x using a simple random sample of size n from N, without replacement. V(r ) T (2.1.11) where X' i s the summation over the(^jpossible values Smith (1969) made i t clear that Koop (1968) had compared the true variance only with "the f i r s t order approximations based on the supposition that deviations (r-R) may be adequately described by a l i n e a r function of the deviations of numerators and denominators." \ Using the assumption of b i v a r i a t e normal population and higher order moments, some formulae were presented which indicated considerable improvement in the variance formula. However, as Smith (1969) pointed out, except when the present population has widely fl u c t u a t i n g y^ at small x^, the usual variance approximation formula should be appropriate. For equation (2.1.3) which has an estimated variance o f ^ r £ 2 ] V ( r 2 ) =|2.(r£ - r 2 ) j /(n - 1) , (2.1.12) Hartley and Ross (1954) provided an unbiased estimate of the r a t i o based on the application of a correction factor to r 2 : rHR = r2 + n (N - 1) fy - r x I (2.1.13) (n - 1) NX I Z J Adopting an abbreviated notation used by Tukey (1956), Robson (1957) obtained an exact variance for r H R which was shown as N approached i n f i n i t y to reduce to the form derived and given independently by Goodman and Hartley 1 (1958). 1 At the time of Robson's (1957) publication, Goodman and Hartley's (1958) publication had not been published but was c i t e d as a 1956 unpublished report. No d i s t r i b u t i o n a l assumptions were made. 1 0 . lim V(r„ RX) = - Is 2 + R 2S 2 - 2R S + ^ T - \ s 2 S 2 HR n\_y 2 x 2 xy n-1 \_ r2 x + S 2 . x \t ( 2 . 1 . 1 4 ) 2. or 1 2 2 2 ^ v , 1 V(r X) = - 4 s + R„S - 2R„S 1 + — K BR n \ y 2 x 2 xy n n - 1 { S 2 S 2 + S 2 I (2.1.15) r 2 x r 2 x j 2 where S i s the covariance between r 0 and x, and r 2 x 2 a l l other symbols are as previously defined. Furthermore, for large sample s i z e , the second term i n equation (2.. 1.15) can be ignored and an approximate unbiased estimate of V(r 2x) can be obtained. It was the contention of Goodman and Hartley, supported later by Cochran, that the r a t i o estimate proposed by Hartley and Ross (equation 2.1.13) would be (Cochran, 1963) "more precise in large samples i f the l i n e Y +-R| (x^ - X) f i t s the value y^ more c l o s e l y than the li n e R-^  x^". The estimator was based on the fact that (Hartley and Ross, 1954) " i f the p r o b a b i l i t y for any x^ to vanish i s zero, an unbiased r a t i o estimator may be construct-ed as a function of the sample average of the r a t i o s rj_ = Yi_/x^. Furthermore, t h i s estimator may require modification in the method of sampling to ensure that for no y^ in the sample, x^ vanishes". Murthy (1962) described a technique for estimating any non-linear function by correcting for the bias employ-ing usual estimates based on samples drawn by the same process of s e l e c t i o n . It was also shown later by Murthy (1964) that when x and y are negatively correlated, one should use the product estimator. The product e s t ima t or , r , cwasa jg ii ve nUb.y ibhiei lr:e±atd on s: M " r M = Nyx/X (2.1.16) This was r e a d i l y found to have a r e l a t i v e bias (1 - f)C v, r/n , and a variance given by V<rM> * <^H { S \ + ^ M ^ x + 2 * ?xy Sx Sy}'- ( 2 ' 1 - 1 7 ) where f = n/N and ^ i s the population c o r r e l a t i o n c o e f f i c i e n t of x, y. Subrahmanya (1967) commented favourably on equation (2.1.17). Murthy and Nanjamna (1959) published on 'almost unbiased estimates', but the publication of Mickey (1959), on 'some f i n i t e population unbiased r a t i o and regression estimators' had been found to be less objectionable by other (Konijn, 1973). Mickey's estimator involved averaging r a t i o s over a l l permutations in the sample, which, on further s i m p l i f i c a t i o n , yielded, - - g r = w | r + (1 - w £ ) Z r /g (2.1.18) MI x Q x j=l g j where w = g j l - (n-m ) / N J and a l l other symbols are the same as for the Quenouille's estimator (2.1.7). For further discussion on r a t i o estimators, the reader i s r e f e r r e d to the works of Durbin (1959), Nieto De Pascual (1961), Rao (1966), Robson and Vithyasai (1961) and others. On other classes of unbiased estimators, not necessarily involving d i r e c t use of r a t i o estimation, suitable references may be obtained from Durbin (1953), Hansen and Hurwitz (1943, 1949), Horvitz and Thomson (1952), L a h i r i (1951), Midzuno (1952), Raj (1954, 1956) and Yates and Grundy (1953). L a h i r i and Midzuno, respectively, provided estimates which are of special interest since they involved only a modification in the sampling procedure. Whereas L a h i r i ' s was based on the drawing of a sample with p r o b a b i l i t y proportional to the sum of X£'s, Midzuno used the p r o b a b i l i t y Pr =2-x^ / I })-X of drawing a c e r t a i n sample. Estimates from such samples have been shown to be unbiased but lacked exact expres-sions for t h e i r variances. Generally, there i s a tremendous e f f e c t of non-normality of the observations on the estimated variance irrespective of the type of estimate or sampling procedure used. Using Fisher's cummulant [Fisher (1932)] given by k^ = E (z^ - Z ) 4 - 3cr4 (= 0 for normal population, and i s a random variate with expectation Z ) , 13. we have (Cochran, 196 3), n-1 2 <r 1 + n-1 2n a (2.1.19) or n-1 2 & (2.1.20) where o~is the population standard deviation, and G i s Fisher's measure of k u r t o s i s . I t i s thus easy to observe that for a leptokuntie-type.of population 2 (positive G) , V(s •) i s l a r g e l y i n f l a t e d and according to Cochran, such positive G's "are common in most sampling". Deviations from normality should therefore be minimal or corrected before using the r a t i o estimation technique. In general, where there i s a s i g n i f i c a n t l y large positive c o r r e l a t i o n , an ordinary r a t i o estimate w i l l give a more r e l i a b l e and precise estimate than an estimate of the same class without the use of an a u x i l i a r y v a r i a b l e . Hajek (1957), has for instance discussed some of the issues and also compared two confidence i n t e r v a l estimations for r a t i o j ^ F i e l l e r (1932, 1940) and Paulson (1942) procedures versus the usual method of large-sample theory approximationj. Hajek concluded that F i e l l e r ' s confidence i n t e r v a l i n particular was never shorter than the usual large-sample approxima-t i o n . The a r t i c l e of Sen et_ al_. (1975) on the use of the r a t i o estimation technique in the estimation of a w i l d l i f e population based on successive sampling may also be of i n t e r e s t . For extensions of t h i s review to cover s t r a t i f i c a t i o n , the reader may refer to some of the basic texts mentioned e a r l i e r . 15. 2.2 SAMPLING SCHEMES FOR RATIO ESTIMATION Schemes i n use for r a t i o estimation, and not necessarily for weight scaling include: a) Simple random sampling with or without replacement. The general idea behind simple random sampling i s that the p r o b a b i l i t y of drawing a sample of size n from N i s the same for a l l [ ^ ) p o s s i b l e ways. A common variance formula ( i d e n t i c a l to equation (2.1.4) or (2.1.5)) for estimate, r , i s given as: V(r,) = - J - ( S 2 u - 2R S w + R V x ) (2.2.1) 1 nx y 1 xy i x for sampling with replacement or i n f i n i t e N, and V(r,) = -\(S2 r - 2R,S + R 2S 2 )N-n . . . . (2.2.2) 1 n_2 y 1 xy N for sampling without replacement. Murthy (1967) had observed that the r a t i o of the square of the bias to the above variances decreased with increasing n. Ajgaonkar (1967) demonstrated that i f an estimator i s an average function, i t s precision increased with increasing sample size when the elements were drawn with varying p r o b a b i l i t i e s at each draw. Hanurav (1968) contended, however, that the r e s u l t i n g increase in precision as a r e s u l t of increasing sample size was generally untrue except 16. " f o r some ca s e s l i k e s a m p l i n g from an i n f i n i t e p o p u l a t i o n and (or) when t h e s a m p l i n g i s done w i t h r e p l a c e m e n t from a f i n i t e p o p u l a t i o n , w i t h the same s e t o f p r o b a b i l i t i e s o f s e l e c t i o n used f o r t h e d i f f e r e n t draws t o e s t i m a t e the p o p u l a t i o n mean i n each c a s e . " S a k u n t a l a (1969) used s i m u l a t i o n s t u d i e s t o compare d i f f e r e n t e s t i m a t o r s commonly used i n s u r v e y w o r k — the mean, r a t i o , r e g r e s s i o n and p r o d u c t e s t i m a t o r s — u s i n g s i m p l e random s a m p l i n g p r o c e d u r e s . G e n e r a l l y , t h e r e was p r e f e r e n c e f o r t h e r e g r e s s i o n e s t i m a t o r but f o r p o s i t i v e c o r r e l a t i o n , the r a t e o f de c r e a s e o f the e s t i m a t e d s a m p l i n g v a r i a n c e was more f o r the r a t i o t h a n f o r t h e r e g r e s s i o n e s t i m a t o r . As n was i n c r e a s e d , t h e e s t i m a t e d s a m p l i n g v a r i a n c e a l s o was d e c r e a s e d , as was t h e r a t e . The r a t e o f d e c r e a s e was ,hhowever f of.ound t o be most r a p i d f o r the p r o d u c t e s t i m a t o r and l e a s t r a p i d f o r the r e g r e s s i o n e s t i m a t o r , w i t h the mean and r a t i o e s t i m a t o r s i n between. F o r n e g a t i v e c o r r e l a t i o n , t h e r e was a r e v e r s a l o f the b e h a v i o r of th e p r o d u c t and r a t i o e s t i m a t o r s . F o r the case o f s a m p l i n g w i t h or w i t h o u t r e p l a c e -ment, i t has been recommended t h a t e s t i m a t o r r 2 s h o u l d be a v o i d e d u n l e s s t h e r e was r e a s o n f o r i t s p r e f e r a b i l -i t y . Moreover, s i n c e i t s b i a s does n ot depend on n, the Bias^need/notdchangehw.ith i n c r e a s i n g - m ,. "uniLike.fr 17 b) Systematic sampling Under t h i s technique, sample locations are pre-select-ed on the basis of an i n i t i a l randomly selected point. Using i n t r a - c l a s s c o r r e l a t i o n c o e f f i c i e n t s (measure of c o r r e l a t i o n between pairs of sampling u n i t s ) , f c , f c y , ( J c x , Swain (1964), Murthy (1967) expressed the variance as V(r) = ^ [ s 2 y { l + (n-D? c y J + R 2 S 2 X ' nX { l + ( n - l ) f c x } - 2RS x y | { l + ( n - l ) f c x } . { l + (n-Dfcy}^] (2.2.3) which under the condition that ?cx _ fey ~ ^ c' gives an estimate which i s 11 + (n-1) times an i d e n t i c a l variance from simple random sampling. Thus, for t h i s scheme to be more e f f i c i e n t than that of simple random sampling, o should be negative. Singh (1966) provided an inter e s t i n g discussion on t h i s . Generally, the design should be such that the v a r i a t i o n between the r a t i o estimates from each class i s minimum. For more complicated designs, an exact variance based on the same p r i n c i p l e as above has not been documented. 18. c) R e p r e s e n t a t i v e (or r e s t r i c t e d ) random s a m p l i n g T h i s i s one o f t h e most common schemes used i n w e i g h t s c a l i n g . The procedure i n v o l v e s t h e d i v i s i o n o f the a n t i c i p a t e d p o p u l a t i o n s i z e , Ni, by t h e r e q u i r e d sample s i z e f o r d e s i r e d p r e c i s i o n , n, i n t o B b l o c k s where B = N / t i ( 2 . 2 . 4 ) F o r each b l o c k , as d e t e r m i n e d b y the o r d e r of a r r i v a l s o f t r u c k l o a d s ( i n w e i g h t s c a l i n g ) , one random sample i s drawn. U n l e s s a d e f i n i t e N i s i n mind, o r ^ t h e - s a m p l i n g f r a c t i o n , ' i s jh;ejLd^ico.nstant;pjit i s p o s s i b l e to- under-sample e s p e c i a l l y , , i n . j , t h e . n c a s e ^ o f unexpectedly) -larg.e, N... f T h i s , j i a y make i t i m p o s s i b l e t o s a t i s f y t h e p r e c i s i o n r e q u i r e m e n t . S i i r : i n tc- r. -.-r t.ho whole , u«-r c t -.rrxv-t I a >. .. E s t i m a t i o n o f the average t r u c k l o a d t h r o u g h r a t i o e s t i -'^r, c i ' the. rm'.:• - T ;••'!** Lvos r tha <•' inoirva„ " i _:V'\tirvi m a t ion i s u s u a l l y made u s i n g r-^ and i t s c o r r e s p o n d i n g v a r i a n c e . t hn a rviqe t r u c k Load -<ic*n x'-*Lio •'•t^ nM*.!«..•• i t . • I t has 'been s p e c u l a t e d t h a t t h e r e s t r i c t e d random s a m p l i n g scheme may -have some b e a r i n g w i t h o r d e r e d e s t i -mation 1" prlfcedu-fes'v -esp'ecia^lry • f o r w e i g h t 'sca^l^rrgv-* Ordered e-s|timateso!.a"r.er^esti.mates5 which'. t a k e r intoitaceio'uht*"the o r d e r i n which.,4-the, units.rare„drawn., , Thej .device^ is,.-constructed such t h a t the e x p e c t a t i o n a s s o c i a t e d , w i t h , each new v a r i a t e e q u a l s Or mates ••« < - _ . - • HSC<* on u o. ••<•:?> the p o p u l a t i o n v a l u e o f t h e o r i g i n a l v a r i a t e (Das (1951), Sukhatme (1953), Des Raj (1956), Murthy (1957) and Sukhatme t >at it - nypct.- '.ti^n - . '•.» ^••HI t.u t h e pop>«.. wr ' v va.l »>u and Sukhatme (1970)). F o r e v e r y unordered random sample o f s i z e n, s e l e c t e d w i t h o u t r e p l a c e m e n t , t h e r e i s n: o r d e r e d samples". The* 'sampling- p r o c e d u r e s ' gi^vdh by L a h i r i (1951) and Midizunoa (jl.9;52j)^lar;e;iisuch 'examples. d) Varying Probabi l i t y sampling Most of the varying p r o b a b i l i t y sampling schemes have been geared towards obtaining unbiased or almost unbiased class of r a t i o estimates. Midzuno (1952)'s procedure, for instance, involved the selection of one unit with p r o b a b i l i t y proportional to x, and the r e s t by a simple random sampling procedure without replacement. Hansen and Hurwitz (1943) suggested that the use of varying p r o b a b i l i t i e s in drawing samples from a large or i n f i n i t e population may reduce the sampling variance of the estimate considerably. Similar conclusions have been drawn by Sukhatme and Sukhatme (1970). A general-i z a t i o n of the r e s u l t was also given by Horvitz and Thomson (1952). Raj (1956) used the a r t i f i c i a l population of Yates and Grundy (1953), and the technique of linear programming to show that " i f the r e l a t i o n between y and x can be assumed to be l i n e a r , the optimum assignment of the p r o b a b i l i t i e s (say, proportional to a u x i l i a r y X) can bring about marked reduction i n variance". Goodman and Kish (1950, Das (1951), Narain (1957) provided suitable references. For the variance of the r a t i o estimator, Murthy (1967) gave an expression of the form 20. V(r) = - T - (y ± " R x ± ) 2 (2.2.5) nX i where Pr^ ref e r s to the p r o b a b i l i t y of selecting the t i l i sample. Some authors including Sukhatme and Sukhatme (1970) are, however, of the opinion that i f p r o b a b i l i t y sampling i s done with replacement, there could be j u s t i f i c a t i o n for the use of simple random sampling formulae. e) Regression sampling This scheme has not been very widely used for r a t i o estimation due to the unfixed x's. A modification that might be useful could involve the d i v i s i o n of the popula-ti o n into x-classes such that the c o r r e l a t i o n c o e f f i c i e n t , within each class i s minimum, to make use of a single class value less debatable. If the x's could be fixed, say by the above modification, a scheme similar to that recommended by Demaerschalk and Kozak (1974) may be applicable. They suggested the use of uniform sampling procedures to cover as wide a range as possible. Since both x and y'are variables i n r a t i o estimation, a study of the seriousness of the errors in independent variables might be necessary. Williams (1972) observed that v i o l a t i o n of the assumption of known independent variables without e r r o r , r e s u l t e d i n tremendous u n d e r - e s t i m a t i o n o f the r e g r e s s i o n c o e f f i c i e n t s f o r i n c r e a s i n g v a r i a n c e o f the e r r o r of the independent v a r i a b l e s . K e r r i c h (1966) d i s c u s s e d a proc e d u r e f o r t h e f i t t i n g o f the l i n e y = ax when e r r o r s o f o b s e r v a t i o n were p r e s e n t i n b o t h v a r i a b l e s . However, s i n c e h i s procedure was based on t h e ass u m p t i o n t h a t the v a r i a b l e s were l o g - n o r m a l l y d i s t r i b u t e d w i t h the same v a r i a n c e , the t e c h n i q u e might not be a p p l i c -a b l e t o our s i t u a t i o n . Draper and Smith (1966) i l l u s t r a t e d an example of w e i g h t i n g w i t h t h e f i t t i n g o f the model y± = < ? x i + £ i (2.2.6) where the v a r i a n c e o f t h e e r r o r t e r m E^, V(£^) or o f Y^/ V ( y ^ ) v a r i e d w i t h x^ by a w e i g h t w^. L e a s t squares e s t i m a t e gave b w = ( I w ^ ) / Sw ix i 2 (2.2.7) I f we r e p r e s e n t V[Y^) = V(E^) by <T^, th e n f o r t h e case where er^ o£ or o l = k * i , W i = cr 2 / K x i . (2.2.8) and b w = Z y i / Z x i = y/x (2.2.9) w h i c h i s i d e n t i c a l t o r]_ e x c e p t f o r t h e d i f f e r e n c e i n s a m p l i n g . The v a r i a n c e o f b w i s g i v e n as v ( b w ) = <s2 /2.W±K±2 , ( 2 . 2 . 1 0 ) c ^ 2 = c f 2 b e i n g a p p r o x i m a t e d or e s t i m a t e d by t h e mean squares r e s i d u a l . F o r unweighted r e g r e s s i o n and as c o n d i t i o n i n g o f the i n t e r c e p t as f o r e q u a t i o n ( 2 . 2 . 6 ) , 2 2 the V ( e s t i m a t e d r e g r e s s i o n s l o p e ) e q u a l s cr / 2 L ( x . - x) . . . . (2.2.11) F o r an o r d i n a r y s i m p l e r e g r e s s i o n model w i t h the i n t e r c e p t c o n d i t i o n e d t o z e r o , an e s t i m a t e o f b i s o b t a i n e d as ^ = b = Z x ^ / ^ X i 2 (2.2.12) w h i c h i s i d e n t i c a l t o b w e x c e p t t h a t Wj_ e q u a l s a c o n s t a n t , and V(b) = <r2 / ^ X i 2 (2.2.13) (|S» " r e s u l t s -from v a r i o u s ass;Mip''t1i6,rfs"' r e l a t i o n s h i p - • • between <5j.? ,and--x^~ from' equation.;" (2 . 2,. 7()i;o'«^o.uruT, «r- i_c F i n a l l y ,,.a^  t h orough knowledge of t h e ex p e c t e d p o p u l a t i o n w i l l h e l p :a g r e a t d e a l i f r e g r e s s i o n p r o c e d u r e s a re t o be a p p l i e d 2.3 DOUBLE-SAMPLING PROCEDURE FOR RATIO ESTIMATION Neyman (1938) used a two-phase sampling (double-sampling) procedure to solve a problem on Human Population Sampling I t was done witltthe knowledge that the cost of obtaining information on the population of inter e s t was higher than that of an a u x i l i a r y variable that was correlated with i t The procedure involved a large first-phase random sample of size n^, selected without replacement. From t h i s sample, the various strata proportions of the population were estimated. With these estimated proportions, the second-phase sample of size was drawn with proportion-a l a l l o c a t i o n . The estimate of the parameter of the population of interest was then obtained from the l a t t e r phase. Neyman's procedure, commonly referred to as "double-sampling for s t r a t i f i c a t i o n " , could be applied to regression, r a t i o or difference < estimators as long as t h i s w i l l r e s u l t in desirable and r e l i a b l e estimates based on the second phase samples. Das (1951), Cochran (1963), Hansen et a l . (1953), Bose (1941), Olkin (1958), Sukhatme and Koshal (1959), Hanurav (1967), Murthy (1967), Sen (1971), Konijn (1973), Sukhatme (1954) and others, have • 24. discussed optimum conditions for use of double-sampling procedures for r a t i o estimation. In h i s discussion, Bose (1942) emphasized that a knowledge of x w i l l improve the estimation "provided there i s a s i g n i f i c a n t c o r r e l a t i o n between the two characters x and y." Sen (1971) discussed the application of successive sampling to the case of two a u x i l i a r y variables in obtain-ing a combined estimate based on a multivariate double-sample r a t i o estimate (Olkin, 1958). Sen concluded that there was a considerable gain in precision over a simple a u x i l i a r y variable when the two a u x i l i a r y variables were uncorrelated with each other, but 'moderately' to highly correlated with the variable of i n t e r e s t . For two cor-r e l a t e d a u x i l i a r y variables, no advantage i n precision over a simple a u x i l i a r y variable was found. Weight scaling involves, at present, only one a u x i l i a r y v a r i a b l e . We w i l l therefore r e s t r i c t ourselves from hereon to the case of one a u x i l i a r y variable despite Sen's (1971) r e s u l t s . There are two common cases of double-sampling procedures that are of relevance in t h i s study. a) Neyman's procedure (Neyman, 193 8) This involves the use of s t r a t i f i e d sampling procedures for the estimation of the sub-sample c h a r a c t e r i s t i c . The f i r s t phase gives an idea of the size of the various s t r a t a in the population. Let P 1 be the proportion of the stratum i in the i population of size N. Select n^ f i r s t phase random observations without replacement and determine p', an i estimate of P^. For a second-phase sample of size n , (n 2^n^) c o l l e c t m^  sub-samples by proportional a l l o c a -t i o n so that, mi = p: n 2 . (2.3.1) st and n = Z m. (2.3.2) 2 i = l 1 _ Estimate the mean of the population of intere s t Y by the unbiased estimate, ^ st Y = 2 PlY, (2.3.3) where p^ i s as defined, and y^ i s the sample mean of the i stratum (of several st s t r a t a ) . i f , as indicated by Neyman (1938), we l e t F| represent a l l i d e n t i c a l functions that give estimates of T B'' Y. , then the expectation of a. t h e F ^ L w h i c h w i l l - g i v e -"-the ^ s m a l l e s t v a r i a n c e s h o u l d g i v e Y . The v a r i a n c e o f »FJ>', V/(E:£)vcouldrfchen be g i v e n as or 2 V (F ) = E (Ft Y)~ o  st V(F t).= E [ Z (p!y. - P ^ ) ] (2.3.4) On further reduction and other assumptions of independence _ i among the y^'s, and between y^ and p^, equation (2.3.4), y i e l d -ed V(F L) = J 7 2 ( i . S i ^ + P i Q i n 1 " i f ) ' ^ m. 1 st + - Z P H (Y- - Y ) 2 (2.3.5) n *~ 1 1 where = 1 - P^ and i s i stratum standard deviation. It follows from equation (2.3.5) that once n 2 and n^ are fi x e d in some way, say by the use of cost functions, V(F^) w i l l depend only on the m^'s and the value they give to the second term on the r i g h t of equation (2.3.5). Thus the variance w i l l be minimum whenever m i = " ^ i ^ i + K^i'1}2/ S s i + wr 1}'-'•• " ' 1 ^ >..l t. n ; - v. •.-..•..'.•..(-.2.3.6) . A major disadvantage with the use of Neyman's procedure i s that the f i r s t phase w i l l be too expensive and time consuming i n the case of weight scaling since that w i l l involve scaling from the f i r s t phase. Perhaps, we could use the general idea of a f i r s t phase sample to s t r a t i f y based on x, rather than y. b) Double-sampling f o r e s t i m a t i n g X and R The procedure''giv'entibelow has been more commonly used than the one p r e v i o u s l y -described. "The m a i n r o b j-ectlve' i's-fcHe o b t a i n i n g of a r e l i a b l e estimate of the a u x i l i a r y p a r a -meter X. There are g e n e r a l l y two a l t e r n a t i v e schemes i n v o l v e d : a. Second phase sample as a subsample of f i r s t b. Both phase samples drawn independently K .> I t has been shown t h a t i f we c o n s i d e r the v a l u e s of (Y - Y) and (Y - Y) t o the f i r s t order of approximations, an unbiased estimate i s o b t a i n e d and the mean square e r r o r i s the v a r i a n c e of the f i r s t order of a p p r o x i -mation g i v e n by V(Y) = V(Y~) - 2RCov(Y ,X ) + R 2V (X ) + R 2V(X,) 2 2 2 1 + 2RCov(Y X ) - 2 R 2 C o v ( X 1 X „ ) . . . (2.3.7) 2 ' 1 1 2 and V(Y) = V ( Y ) / N 2 (2.3.8) In t h i s procedure, o n l y the a u x i l i a r y v a r i a t e s are measured f o r the mean, X i n the f i r s t phase. The p a i r e d measurements x^, y^ are taken d u r i n g the second phase. I t i s , however, c o n c e i v a b l e that i f V ( X 2 ) i s r e p l a c e d by V(X.), assuming n 9< n,, a more r e l i a b l e estimate may be obtainable. Freese (1960) suggested some minor modifications in equation (2.3.7), but these are not considered very e s s e n t i a l . 2.4 LOG SCALING BY WEIGHT The usual basis for timber transactions has been one hundred per cent (100%) s c a l i n g . The introduction of any sampling method should therefore involve the willingness on the part of the buyer and/or s e l l e r to r i s k a possible monetary loss rather than spend addition-a l funds for a more accurate or 100% scale. Burns (1970) wrote that "the use of weight scaling may reduce your log handling costs. It has on Jackson State Forest i n young growth redwood and Douglas-fir." Dilworth (1976) r e c a l l e d that the use of weight scaling dates as far back as a quarter of a century ago. It was being used by a company on salvage trees in the Mount Hood area. During the same period, the Grown Zellerbach Company used the technique for pulp logs at i t s operations in the Olympic Peninsula. Dilworth (197 6) also indicated that the most immediate application might be for small logs, fiber logs, and logs of low value. Hinthorne (1976) indicated that h i s company, which operates in B r i t i s h Columbia, uses weight scaling for 70% of firmwood cubic and 100% s t i c k scaling for the remaining 30% to s a t i s f y the B r i t i s h Columbia Forest 3.0. Service sc a l i n g requirement. To s a t i s f y internal corporate scaling requirements, weight scaling i s used only 10% of a l l firmwood cubic, but Fraser (personal communication -1976) pointed out that t h i s was applied to "small l o t s , weekly volume production, log grades and values other than stumpage". Gains in scaling e f f i c i e n c y or precision could be brought about by s t r a t i f i c a t i o n . C r i t e r i a for s t r a t i f i c a t i o n could include l o c a l i t y , species groups, l o c a l and seasonal weather, method of logging, kind and duration of storage. Others are the s i z e , age, position of logs i n tree and log grades. However, with increased s t r a t i f i c a t i o n and sampling int e n s i t y , an increase i n cost to a l e v e l sometimes higher than for 100% s t i c k s c a l i n g might occur. P r e - s t r a t i f i c a t i o n poses serious p r a c t i c a l problems of sample size determination a r i s i n g mainly from the d i f f i c u l t y of lack of knowledge of each stratum. Wensel (1973b) discussed the use of p o s t - s t r a t i f i c a t i o n procedures. He indicated that without p o s t - s t r a t i f i c a t i o n , "using weight scaling and the r a t i o estimator only increased the precision 3% over simple random sampling".... However, with the use of p o s t - s t r a t i f i c a -t i o n , "even without weights caused an increase i n precision of 24%", with the largest gain obtained from that with combination of p o s t - s t r a t i f i c a t i o n and r a t i o estimation. Such gains were usually associated with reduced sample si z e s . Old growth forests might render use of weight scaling or i n fact any other type of sampling i n e f f i c i e n t as a r e s u l t of the high v a r i a b i l i t y in th e i r weight. The type of loads also c a r r i e d by the trucks irrespective of the source of the loads might also create another problem. A mixture of small and large volume/weight r a t i o s may lead to unreliable r e s u l t s . S t r a t i f i c a t i o n , before or after sampling, might be necessary to iso l a t e extreme r a t i o s , but the cost of f i n d -ing such may be overwhelming. The general set-up of weight scaling i s i l l u s t r a t e d as follows. For a population of size N, a l l truck net weights x^, i = 1, N are taken to give the t o t a l X. For a sample (subsample of N) determined by a required precision with 95% probability, and selected by an approp-r i a t e sampling scheme (usually representative sampling scheme), the scale (or volume) y^ ( i = 1, n) and the corresponding weight x^ ( i = 1, n) are noted. I f possible, the d i f f e r e n t species are recognised and th e i r scale volume Y j ^ are noted: YL= X Y-ii (2.4.1) j=l 3 1 for s d i f f e r e n t or recognisable species or species groups. P e r i o d i c a l l y , the data may be checked to f i n d out whether an adjustment i n sample size i s necessary. Should an 32. adjustment become a necessity, the previous data may be treated as for a separate stratum (Wensel, 1976-personal communication). Using the r a t i o of means estimator, the o v e r a l l n n r a t i o i s determined as ^_Yj_/ >^_x- # a n c ^ f ° r the separate n n 1 • species y^ .. / Jj> x. , (BCESf scaling manual, 1975). M u l t i -plying these r a t i o s by X, the t o t a l o v e r a l l weight, gives the expected t o t a l scales (or volumes) for the period. Which of the two procedures may be used depends on whether a single stumpage value i s applicable for a l l species or not. . Of course a l l inventory depletion and sale accounting records require an estimate of volume by species groups which i s obtainable from the check s c a l i n g . Further discussion on weight scaling may be obtained from Freeman (1962), Lange (1962), Johnson et_ al_. (1963), B l a i r (1965), Turnbull , et_ a_l. (1965), Fraser and Highsted (1966), Moss (1966), Huey (1967), . Hamilton (19 7 5 T - a n d t h e i r r e f -erences . F o r discussion on how weight scaling i s presently done, the reader may refer to the B r i t i s h Columbia Forest Service Scaling manual (1976). Other manuals e s s e n t i a l l y explain the same pro-cedure. Hinthorne's (1976) paper provides inte r e s t i n g discussion on some of the constraints posed by management needs. Also, see Appendices XXV-XXVII. 2.5 GENERATED VERSUS COLLECTED DATA Generated or simulated data give us the opportun-i t y to monitor the behavior of sampling schemes and other c h a r a c t e r i s t i c s . In a recent a r t i c l e on 'numbers and data', Finney (1975) commented i n his summary that "numbers chosen to i l l u s t r a t e a s t a t i s t i c a l method can have any magnitudes that do not contradict the mathematical model. Observa-t i o n a l data are subject to many requirements of i n t e r n a l consistency and of p l a u s i b i l i t y i n r e l a t i o n to previous information. Though such requirements are not necessarily absolute constraints, departures from them should be detected and examined thoroughly before a d e f i n i t i v e s t a t i s t i c a l analysis begins." Kahn (1956) discussed i n a working paper the general p r i n c i p l e s of Monte Carlo calculations with pa r t i c u l a r emphasis on reduction of the amount of work involved. On c o l l e c t e d data, i f we assume that either x or y or both come from populations that are normally d i s t r i b u t -ed, the domination of the extremes, say, i n the o v e r a l l data c o l l e c t e d could seri o u s l y increase the variance of the estimate and thus decrease the p r e c i s i o n . Cochran (1963) advised that even though at times the presence of these extremes could lead to a better estimate of the mean, " i t might be wise to segregate them and make 34. separate plans for coping with them, perhaps by taking a complete enumeration of them i f they are not numerous. This removal of the extremes from the main body of the population reduces the skewness and improves the normal approximation." In t h i s study, data w i l l be generated under s p e c i f i e d conditions and used i n the i n i t i a l segments of the study to examine the e f f i c i e n c y of the basic sampling schemes referre d to i n section 2 . 2 and in later chapters. The process w i l l then be repeated on c o l l e c t e d weight scal i n g data. Further extensions of the r e s u l t s w i l l be r e s t r i c t e d to some new sets of c o l l e c t e d data not used previously so as to study other conditions that may not have been detected i n the i n i t i a l portions of the study. 3.0 BASIC FORMULATION AND PROCEDURE 3.1 Ratio estimate, bias and variance 3.I.i Ratio estimate, r hr We wish to evolve the formula for the r a t i o estimate that includes the order of the r e l a t i o n s h i p of the variance of y^ and p, that i s , V(yi) a X i P (3 .1 .1) Let us consider the general no-intercept model, Yi = R p x i + £ i (3.1.2) where y i , x± are paired variates as previously defined and R^ i s the appropriate p-dependent r a t i o or the c o e f f i c i e n t of x^ i n equation (3 .1.2). We can then write the sum of the weighted squared deviations, q, as n 2 ^ -P o q = Z (Yi " Rp xi) = Z x ± £ i . . . . (3.1.3) i=l i=l which on d i f f e r e n t i a t i o n with respect to R gives, P 1 d2_ 2 dR ^ x i ~ p ^ i - vi' (3-1-4) P Setting equation (3.1.4) to zero, yields Rp = r p = ^ . p (3.1.5) Z * i whereZ_ = Z^ ' unless otherwise s p e c i f i e d . If we set i=l p = 1 or 2 in equation (3 .1.5), we obtain estimators 3.6. (equation 2.1.2) or (equation 2.1.3) res p e c t i v e l y . For f i x e d x's, as i n regression sampling, r ^ i s unbiased since t h i s i s a least squares estimate. For 2. constant variances, cr, as m ordinary regression, the expectation of "q given the x's" i s E(q| x's) = X x i P (x iPcT 2) = ncj2 (3.1.6) 3 . 1 . i i Bias of r P Let us write r = 0 (x,, ...,x n, y , y ) = 0 , and p n l e t the expectations of the x^'s be X^'s, and for y i ' s , YL'a. (E (xL) = X± , E(y ±) = Y ± ) . We can then write, using Taylor's expansion 0 (x , x n, y , y n) = 0 (X 1 # X^, + terms involving higher order derivatives, (3.1.7) where /d0 \ , / d 2 0 \ , a n d so on, are t h e d e r i v a t i v e s \ d X j J o \ d X j d x k j o e v a l u a t e d a t t h e i r r e s p e c t i v e e x p e c t e d v a l u e s o f x^, X j & x k and so on. Thus, the e x p e c t a t i o n of r , E -I 0(x , x , y , , y )' i s E 0(x , x n, y , y n) = 0 ( X , X R 1 Y ) + — T lij-A v (x.) + - y (l-V 2.2. I—.2 J> 2 V j = l X % ° 3=1 V ( y . ) + X 7 ~ ) C o v ( x x ) -j<K^n * j 2 _. \ K / + X 2 1 ( M ^ C o v ( y , , y v ) + ^ Z l«j<k$n^ a y j a V o J j=i k = l V 3 k'o Cov(x j,y k) (3.1.8) where C O V ( X J , x k ) , and so on, are the covariances. One can say that for j ^ k, Cov(Xj, x k) = 0, Cov(yj, y k)= 0 and Cov ( X J , y k) =0. I f we assume independence above, and not just zero c o r r e l a t i o n s , then terms l i k e E -j (x_. - Xj) (x^ . - Xj^) J . , and so on, could j u s t i f i a b l y be excluded from equation (3.1.8). More terms w i l l also vanish since (d 0 \ and /d 0 \ equal zero. If a l l the above assumptions are r i g h t , then for r = 0 ( x , — , x , y , . . . , y ) , one has P n n n E(r } = U 2-p +r 2_ U 2 1 p - r X. / j=l \ 3 1° + n I 2 d 0 j = l \ D X J D Y J >° C o v ( x j > y j ) (3.1.9) so that the bias of the r a t i o estimator could be approx-imated as Bias (r ) P n 2 ZL dx V(x n -2 j , , Cov(x.,y.) dx.dy./ 3 i j = l \ 3 • 3/o J (3.1.10) 3 . 1 . i i i Variance of r P Let x, y_, X, Y be the respective (1 x n) matrices of x, 2 y, X and Y. Then, i f we expand jjZf(x,y_) - 0(X,Y)J| i n a Taylor expansion about the point (X, Y) as far as quartic terms and then take t h e i r expectations, the variance of rp, V(rp) can be shown to be: V ( r p ) =E^(2f(x 1, .. ., x n, y x, y n ) - 0 ( X 1 , *»' Y l (I, [ % r l v ( x ^ n z - V(Yj) + 2 l _ d x ^ - J C o v ^ y . ) 3 /o n def 2 d ef 3=1 \d*j-'/o \ d x j 2 / o L<x. - X.) (continued) 3.1.iv An i l l u s t r a t i o n of the formulae for s i m p l i -f i e d weight scaling The s i m p l i f i e d weight scaling i s defined by a l l or some combinations of the following conditions: a) Similar or i d e n t i c a l species from a given b i o -geographical area. b) Population i s made up of data which are deter-mined by the order of a r r i v a l s of truck loads. c) For each stratum j , E(x_j) = X and E(yj) = Y. Stratum variances of x, y are the same. d) X i s known; Y estimatedt^si-hgTiratids*. e S T TT-:UI.:> ^ . -40. e) Sample size, f o r a l l s t r a t a together i s n from a p o p u l a t i o n s i z e of N. f) N and X become known o n l y at the end of the pr oduction per i o d . C o n d i t i o n s b) and e) imply t h a t o n l y one observa-t i o n i s taken from each stratum, or -could -have1 •••li/2' strata-and"sample Wo t r u c k s from^each stratum. cHowever,' c f r o m con-d i t i o n s - ua.)i^ndt © K .we eoWd^eens^tler trie whWle°n o b s e r v a t i o n s as c o n s t i t u t i n g . a;ns;ample)nfrom one major stratum f o r the f o l l o w i n g i l l u s t r a t i o n . I f ih', -'Hi i\ • «. . i a u t f " •.• t i r ; -I f the d e r i v a t i v e s or r p are e v a l u a t e d a t the p o i n t of t h e i r expected v a l u e s , P Q: (X,Y), and the r e s u l t s s u b s t i t u t e d i n the f i r s t t h ree terms of the r i g h t hand s i d e e x p r e s s i o n i n e q u a t i o n (3.1.11), we o b t a i n -" 2 2 2 V ( r p ) = X ( R p S x + S y - 2 R p S x y ) . n T h i s e x p r e s s i o n i s the same as equation (2.2.1) and the f i r s t term on the r i g h t of e q u a t i o n (2.1.15). T h i s i n d i c a t e s the s i m p l i c i t y i n the assumption of weight s c a l i n g . Under c o n d i t i o n s a) and c ) , Idx,jo - _ 1 / \ -- 1 - RpX and / d0_) = x 1. _ n \dy^ Io n l The e x p r e s s i o n i n v o l v e d T a y l o r ' s expansion up to the q u a d r a t i c terms. E v a l u a t i o n of higher order terms and 4 1 . the addition of these to the above expression can be expected to r e s u l t in i n s i g n i f i c a n t improvement. The bias of r from equation (3.1.10) could be shown to reduce to _2 R X 2 Bias L(r ) = J? (np - n + 2 - p)S v p _______ x n _-2 _p-2 + X (p-2 + (l-p)X )S (3.1.12) n X Y i f we substitute the following expressions 2 _-2 fd 0 \ = 2R pX (np-n+2-p) and . \ d x ± 2 J 0 ~ ? / 2 x _-2 _p-2 d 0 \ = X (p-2 + (l-p)X ) . ^dx-jdy-Jo n 2 S p e c i f i c a l l y , for p = 0, 1 and 2 respectively, _-2 -2 -2 Bias ( r Q ) = R QX (2-n)S x 2 + X (X - 2 ) s x y . (3.1.13) n x n _-2 __2 Bias ( r x ) =____- S x 2 -2L__ s x y (3.1.14) n n _-2 2 _-2 and Bias ( r 9 ) = R 9X S v - X S (3.1.15) 2 _ x xy n In general, however, i t can be observed from equations (3,1.. 1.3) and' (3.hU 1.4) .ithat cdhe^b^FsRs i s . .o&rolrder^ti/ri-, tand that Mtifehe sca.twdQSicS^/X /and S^/X^ralrei small bifjtfe Wi^s ^ i 11 be negl i g i b l e ffror l~a^gre rn,. Similar expressions have been mentioned by other workers (Cochran, 1963; Konijn, 1973; Sukhatme and Sukhatme, 1970 and Wensel, 1974). 3.2 Data g e n e r a t i o n procedure f o r cases 1-21 Data f o r 21 d i f f e r e n t combinations of p and R (cases 1-21) were generated f o r the i n i t i a l study on the performance of the v a r i o u s sampling schemes wi t h regards t o d i f f e r e n t p o p u l a t i o n s . The cases w i t h p o s i t i v e p v a l u e s r e p r e s e n t e d s l i g h t m o d i f i c a t i o n s c o f a a c t u a l weight s c a l i n g d a ta. The model used i s e x p l a i n e d as f o l l o w s : Let us w r i t e the e x p e c t a t i o n of y gi v e n x as i E ( y j x ) and suppose t h a t j^y - E ( y | x) J i s independent of x. Assume the c o n d i t i o n a l d e n s i t y of y given x, h (y x) i s t c k.x.p My|x) = 1 e ' 2 kl x» (y-R x) p 2 hp f 2 T k |x| f ( x , y ) = g ( x ) h ( y j x ) , where the d e n s i t y g(x) can be any s u i t a b l e d e n s i t y . f ( x , y ) i s the j o i n t d i s t r i b u t i o n o f y and x. Thus, E(y|x) = R px and V ( y j x ) = k | x | P ; E(y) = R pE(x) = R pX say, and V(y) = k E [ |x| "] One can get standa r d normal u and v randomly and independ-e n t l y , and then s e t h p x = X .+'T<3xuaaridVyy7 R X + k | x | V Twenty-one s e t s of p a i r e d x,y data c o n s t i t u t i n g cases 1-21, of s i z e N = 1000 each were then generated by s e t t i n g R = R*, k = V (y')/X and the a r b i t r a r y s p e c i f i e d P * values: V(y') = 250, C T 2 = 900, X = 150 x p = -1.0, -0.5, 0.0, 0.5, 1,0,1.5, 2.0, and R' = 0.10, 0.25, 0.40 Cases 1-21 are represented r e s p e c t i v e l y by the twenty-one combinations of p and R' (Table 1 i n Chapter 4). The se l e c t i o n of the seven d i f f e r e n t p values was made to include possible values of p for actual weight scaling data under various conditions. Probable sources of v a r i a -t i o n in r a t i o s of volume to weight include the s p e c i f i c g r a v i t y of the species, the nature and extent of decay and voids present, season of logging, and so on. The t h e o r e t i c a l R values used in the generation were higher than those for weight s c a l i n g . It was f e l t , however, that the use of 21 d i f f e r e n t t h e o r e t i c a l populations and 14 sets of weight scal i n g data from d i f f e r e n t areas of B. C., was enough to unearth the best minimum-variance scheme and most e f f i c i e n t scheme for weight scaling sampling. Three additional cases A1-A3 were obtained by combining a l l previous cases with the same R. Since p values were unknown for these new cases, they were obtained using a weighted non-linear least squares estimation procedure (University of B r i t i s h Columbia programme UBC:BMD X85). 44. 3.3 Determination of sample size Conforming with t r a d i t i o n , the usual method of large -sample theory approximation for confidence i n t e r v a l estimation was used i n t h i s study to determine the size of the sample required for a given p r e c i s i o n . The approx-imation required a minimum sample size of t h i r t y (30), and so a size less than 30 f i n a l l y obtained may not necessarily y i e l d the desired precision. The l i m i t s for the estimated mean of the population of interest (obtained through r a t i o estimation), Y are R given by P r ( V t V 2V($ R..) 3 5<Y<$ R + tK/2V {2R)hy 1 - c e . . (3.3.1) In equation (3.3.1), l-'<*-is the l e v e l of confidence, t ^ ^ is the value of the s t u d e n t — t (or normal deviate for n>30) corresponding to the chosen confidence, and Pr (...) means the p r o b a b i l i t y that . . . . We can write equation (3.3.1) as Pr (Y R - E < Y < Y R + E') = 1 - °< (3.3.2) Thus, i f we specify E by a certa i n quantity or as a percentage of Y R and set = 1.96 for °c = 0.05 (n?30) , _ _2 and notice that V(Y R) = X V(R p), we could obtain the r e q u i r -ed sample s i z e . Usually, use i s made of the c o e f f i c i e n t of v a r i a t i o n of y, CV, giving n = t l / 2 ( C V ) 2 ( 3 _ 3 ^ 3 ) A 2+ t 2 ? / 2 ( C V ) 2 / N where A i s the required precision in percentage of Y, and a l l other symbols as previously defined. The algorithm used, i n most cases, involved the drawing of an a r b i t r a r y sample of size 20 from the generat-2 2 ed data to estimate S v, S.,, R and so on, for the purposes x y of estimating the r a t i o variance and then V(Y R) or the coef f i c i e n t of v a r i a t i o n , for substitution i n equation ( 3 . 3 . 3 ) or i t s modification. I f r e l i a b l e prior information e x i s t s , the variances could be estimated from such. However, i f the samples are determined from the i n i t i a l segments of the population, as i s done in practice for weight scaling, i t should be recognised that in the process of sampling i t might be necessary to re-estimate the sample size in order to meet the o v e r a l l precision requirement. Extra samples, precision unknown, were also taken according to a p a r t i c u l a r pattern of i n t e r e s t , and were a r b i t r a r i l y set not to exceed twenty percent of the respective population s i z e s . 3.4 Description of sampling schemes The following schemes were used i n the study: a) Completely random procedure - (CR) n random numbers associated with Xj_, (i=l, ...,n) were randomly selected between 1 and N without replacement, such that the p r o b a b i l i t y of drawing such n was the same as that of the otherjj^j - ij possible ways. b) Representative (or r e s t r i c t e d ) random sampling - (RS) As described i n Chapter 2, section 2, subsection c. c) Modified random sampling - (MZ, Z = # of groups) This may appropriately be described as 'uniform sampling within ordered groups'. The population to be sampled was divided into a s p e c i f i e d number of groups, say Z = 5, 10, 15 and so on. The required number of samples was then spread equally (with adjustment for rounding-up) for a l l groups. Random samples of the approximately n/Z size were then taken from each group. The greater the number of groups, the closer t h i s scheme i s to RS. If 'groups' do vary i n variance, and there are enough group samples, they could be treated as s t r a t a . Otherwise, the whole aggregate of the samples could be treated s t a t i s t i c a l l y as from one population without major consequences. d) Type 1 freguency sampling - £(1,Z), Z = # of classes] If the frequency d i s t r i b u t i o n of the X's was known (which was true of our case studies), or can be estimated by an appropriate procedure, the samples could be drawn randomly within each class by proportional a l l o c a t i o n . Thus the class with the greatest number of observations would get the most samples. In view of the normal nature of the generated data, t h i s type of sampling may be dubbed 'Gaussian-like sampling procedure'. For c classes, and for any i cl a s s , m ± = ( f i / ^ f ^ n (3.4.1) til where f i i s the x-frequency (or proportion) of the i c l a s s . e) Type 2 frequency sampling - j j 2 , z ) , z = # of classesj EgOa;!- sized samples were drawn,T1from the-frequency classes,' except for the two extremes where no samples were taken. For some normally d i s t r i b u t e d populations and for large number of classes, more than the two extreme classes were l e f t unsampled. f) Type 3 frequency sampling - £(3,z), z = # of classesj Same as for Type 2 above, except that the extremes were not excluded from the sampling. Serious problems arose when there were not enough extreme data to sample from. In some cases the scheme was abandoned. 48. g) Type 4 frequency sampling - £(4,z), z = # of classesj. This i s a s l i g h t modification of Type 2 or 3. A few equal size samples were taken from the extremes, and equal a l l o c a t i o n procedures applied to the mid-classes with the remaining samples. Extreme class sample sizes were usually less than the mid-class s i z e s . The set up could be as: and m = 9n (3.4.2) c 1 ~ , c-2 ,*"1n (3.4.3) i f 1 or c -where 9 was a given proportion for each of the two extremes such that 9 < (l-29)/(c-2) . Only the case of c = 5 and 9 = .05 was considered under t h i s scheme for the a r t i f i c i a l data. h) Type 5 frequency sampling - £(5,z), z = # of c l a s s e s j Ignoring the extreme classes, the samples were drawn randomly within each of the remaining classes according to an a l l o c a t i o n based on the inner numbered classes. For 5 classes, say, the middle classes were numbered 1, 2 and 3 respectively, so that the proportion for the o r i g i n a l second class was 1/6 = 0.1667 ( F i g . l ) . For c e r t a i n types of population where there were enough extreme classes, they were included in the process. i ) Type 6 frequency sampling - ^ ( 6 , z ) , z = # of c l a s s e s j . T h i s was s i m i l a r t o type 5 except t h a t the order of numbering or assignment of p r o b a b i l i t i e s was r e v e r s e d . ( F i g . 2 ) . As i n d i c a t e d i n F i g u r e s 1 and 2, these schemes were designed t o take care o f p o s i t i v e or negative skewness j) Frequency sampling i n c r e a s i n g with magnitude of x - [_ i x z , ' ,- z = # of cl a s s e s j . S i m i l a r t o type 5 except t h a t the c l a s s mid-point v a l u e s of X j , XMIDj, were used i n s t e a d of the a r b i t r a r y a s s i g n e d numbers, so that m - XMID-; m j " c ^ I - 3 - .n ( 3 . 4 . 4 ) y XMITK i=2 f o r j - 2 , . . . , c-1. k) Frequency sampling d e c r e a s i n g w i t h magnitude of x - ^ DXZ, Z = # of classes]. A r e v e r s a l of IXZ, and i d e n t i c a l t o type 6 . For i re d u c -i n g from c-1 t o 2 (c>2), and j i n c r e a s i n g from 2 t o c-1 by increments o f 1, XMID; m . = i — . n J c " ! ; - ( 3 . 4 . 5 ) j=2 3 The e l e v e n schemes d e s c r i b e d and .their m o d i f i c a t i o n s a c t u a l l y belong t o 5 main d i v i s i o n s , namely: i ) Random sampling - CR, RS, MZ 50. i i ) P r o b a b i l i t y proportional to frequency of x - Type 1 i i i ) Equal p r o b a b i l i t y for x-classes - Types 2,3,4 iv) P r o b a b i l i t y proportional to size of x - Type 5 and IXZ v) P r o b a b i l i t y inversely proportional to size of x - Type 6 and DXZ It should be noted that a l l the schemes may not be a p p l i c -able to a l l types of population. The computer programme that was written and used in t h i s study, ensured that where there were not enough samples i n a pa r t i c u l a r group or cl a s s , the remainder were equally spread to the others. I f t h i s remainder was excessive, say more than 25% of the expected sample size for that class or r e s u l t -ed i n less than 75% of the t o t a l sample size required, the whole scheme was abandoned. Sampling was done without replacement. 5000 .5000 3333 1667 0 1 2 3 0 0 1 2 3 0 numbered classes numbered classes F i g . 1 Type 5 for 5 classes F i g . 2 Type 6 for 5 classes 4.0 DATA SUMMARIES AND DESCRIPTION 4.1 Generated data - case studies 1-21 and A1-A3 Table 1 summarizes the parameters for the case studies (CS) 1-21 with the indicated p and R (=R') values. The respective CS represents a sample of a population with the t h e o r e t i c a l set of p and R values as used i n the data generation. However, we s h a l l assume that the size of 1000 was large enough to constitute a population from which we intend to sample. YBAR and XBAR represent the population means Y and X respectively, and were computed by N Y = T. y ±/N (4.1.1) N X = Jx±/S (4.1.2) • The population variances of Y, YVAR and X, XVAR were comput ed by N 2 YVAR = V(Y) =2! (Y_-Y)/N (4.1.3) N _2 YVAR = V(X) = £ (x i-X)/N (4.1.4) For sample estimates, the denominator N was replaced by n-1, and summation was up to n terms. YCV%, XCV% are the respective c o e f f i c i e n t of variations expressed in percentages, YCV% = (YVAR2/Y) . 100 (4.1.5) XCV% = (X V A R V X) . 100 (4.1.6) The c o r r e l a t i o n c o e f f i c i e n t s for x, y were determined as RHO = Cov (x #y) ^ (WAR . XVAR) n = 1 (*i - x)(yi-Y) m ± ( 4 a l _ 7 ) , (YVAR . XVAR) n-1 As could be observed from Table 1, the variance and the mean of x were kept almost constant, hence the i d e n t i c a l coeff icientso©'fvva&iitai4i>on. The mean of y was dependent on X and R, whereas the variance was dependent on X and p. Standard tests of normality performed confirmed the normal nature of the x-data generated. Correlation c o e f f i c i e n t s for the cases ranged from 0.296218 for case 5 to 0.956412 for case 15. In general, the f i r s t seven cases (R = 0.10) had the smallest c o e f f i c -ients, followed by the next seven (R = 0.25) and then the l a s t cases (R = 0.40). This was a r b i t r a r i l y done. p The r e l a t i o n s h i p between the variance of y j and X j may be obtained'' . from some of the frequency tables presented as Tables 2-22. The s t a t i s t i c s were computed with i d e n t i c a l r e l a t i o n s h i p s given in the e a r l i e r paragraphs of t h i s section. In p a r t i c u l a r , N was replaced by nj, where n^ represented the number of observations available i n the j 1 " " c l a s s . Frequency d i s t r i b u t i o n s for classes 5, 15 and 20 were also examined but were excluded from the tables . Class c o r r e l a t i o n c o e f f i c i e n t s should be noted care-f u l l y since a large value would suggest the need to question the use of an a r b i t r a r y x-value for x j . Undefined c o r r e l a t i o n c o e f f i c i e n t s , as a consequence of near or zero variance for either x or y, or both, were indicated with several asterisks. Moreover, cor-r e l a t i o n c o e f f i c i e n t s for only two samples were ignored and i n place " ;asterisks -were written. XTOT and YTOT represent the respective class t o t a l s for X and Y, and were computed by summing up a l l the elements within each c l a s s . For case studies A1-A3, CSl-21 were combined in the following manner: Al = CS1-7 R' = 0.10, A2 = CS8-14 R' = 0.25, and A3 = CS15-21 R' = 0.40. The R' values were not determined but were only obtained by v i r t u e of the c l a s s i f i c a t i o n (see Table 1) . Using the frequency class variances of yj (for n-j^3) and the class x-mid points, XMID^, the values of p were estimated 54v TA6LE 1 2 POPULATION PARAMETERS FOR GENERATED DATA (CASES 1-21) CASE YBAR WAR YCVS XBAR XVAR XCV2 RHO P R • 1 34.78 193.73 40. 01 3 50.56 8115.09 25.70 .715519 - l o 0 o 10 2 36.47 188.12 37. 60 349. 85 7877. 59 25„ 3 7 o 43 7412 -0.5 .10 3 35. 57 242.50 43.78 343.73 8346.35 26.58 .361C29 C O . 10 4 38.G0 309.51 46. 2 9 340, 81 8499.98 27.05 .301836 0.5 .10 5 39.31 454.10 54.20 341.31 7 943.32 26.11 .296218 1.0 . 10 6 43. A3 649.12 58.66 340.76 8300.49 26.74 .265849 1.5 . 10 7 47. 83 982.76 65o54 335. 68 7666* 31 26. 08 .417291 2.0 .10 8 90. 03 5 06.7 2 25.00 358.11 7583.C3 24.32 .892252 -1.0 .25 9 88. 77 595.60 27. 49 352. 2C 8170.26 25.66 .849279 -0.5 .25 10 88.31 615.40 28. 09 352. 93 7860.74 25.12 .772650 0.0 . 25 11 88. 90 749.81 30. 8 0 356.46 7993.16 25.08 .710757 0.5 .25 12 87.29 895.60 34. 2 8 350.94 8163.82 25. 75 o 656592 1.0 .25 13 90.36 127 0.94 39.4 5 351.19 3168.29 25.73 .5792C6 1.5 .25 14 87.00 159G.95 45. 84 347.37 3381.92 26.36 .455369 2.0 .25 15 141.77 1283.48 25.27 353*46 8095.44 25.46 .956412 -1.0 .40 16 14C.92 1263.38 25.27 353.29 7454.01 24.44 „ 9 2 9 8 7 9 -0.5 . 40 17 140.89 1355.34 26. 13 351. 72 8009.66 25.45 .900850 0.0 .40 18 138.46 1397.14 27.00 346.15 8072.21 25.96 .870913 C. 5 .40 19 140.61 1595.34 28.41 352. 3 7 7425. 52 24.42 .793471 1.0 .40 20 138.56 1972.97 32.06 350. 26 8371. £3 26. 12 .729199 1.5 .40 21 137.50 2527.55 36. 56 346.4 0 8662.81 2 6.87 .622431 2.0 .40 For t h i s and subsequent t a b l e s (unless otherwise s p e c i f i e d ) , YTOT, YBAR and XTOT, XBAR, X-CELL LIMITS are i n any a p p r o p r i a t e u n i t s of Y, X r e s p e c t i v e l y ; YVAR, XVAR are i n the co r r e s p o n d i n g v a r i a n c e u n i t s . T A B L E ' 2 : • ' C A S E 1 - F R E Q U E N C Y D I S T R I B U T I O N F O R 1 0 C L A S S c S X - C E L L L I M I T S F R E Q X T O T 6 8 . 0 - 1 2 7 . 2 8 7 7 7 1 2 7 . 3 - 1 0 6 . 5 1 8 2 8 3 3 1 3 6 . 6 - 2 4 5 . 8 9 5 2 1 1 6 3 2 4 5 . 9 - 3 0 5 . 1 2 2 3 6 1 9 2 7 3 0 5 . 2 - 3 6 4 . 4 2 0 7 6 9 6 8 0 3 6 4 . 5 - 4 2 3 . 7 2 2 2 8 7 9 5 9 4 2 3 . 8 - 4 8 3 . 0 1 7 1 7 6 8 6 3 4 8 3 . 1 - 5 4 2 . 3 4 6 2 3 5 L 1 5 4 2 - 4 - 6 0 1 . 6 9 5 1 8 5 6 0 1 . 7 - 6 6 0 . 9 1 6 6 0 X V A R Y T O T Y V A R R H O 2 9 7 . 6 1 5 1 6 5 . 0 — 0 1 0 9 3 1 7 7 . 2 2 4 3 8 8 . -J "** * 0 5 9 4 2 3 8 . 0 1 8 9 0 1 0 3 . 2 0 . 1 1 7 0 2 8 8 . 7 5 7 7 5 8 7 . 0 0 . 2 7 5 2 3 2 3 . 3 7 2 0 9 1 4 8 . 3 0 . 0 4 1 0 2 9 9 . 1 8 3 7 6 8 6 . 0 0 . 4 1 3 6 2 5 4 . 7 8 1 5 3 5 8 . 0 0 . 1 3 7 2 2 8 5 . 7 2 4 2 3 3 4 . si 0 . 4 5 2 2 2 9 3 . 6 4 8 3 8 6 - 9 ~ • 4 2 3 9 0 . 0 7 1 0 . 0 **** T A B L E 3 : C A S S 2 - F R E Q U E N C Y X - C E L L L I M I T S F R E Q X T O T 4 7 . 0 — 1 0 1 . 4 3 2 3 0 1 0 1 . 5 - 1 5 5 . 9 8 1 1 0 3 1 5 6 . 0 - 2 1 0 . 4 4 1 7 6 8 5 2 1 0 . 5 - 2 6 4 . 9 1 2 2 2 8 9 4 9 2 6 5 . 0 - 3 1 9 . 4 1 9 1 5 6 0 4 9 2 1 9 . 5 - 3 7 3 . 9 2 3 7 8 1 3 6 3 3 7 4 . 0 - 4 2 3 - 4 2 0 0 8 0 0 5 1 4 2 8 . 5 - 4 8 2 . 9 1 3 1 5 9 2 5 1 4 8 3 . 0 - 5 3 7 . 4 5 2 2 6 2 0 5 5 3 7 . 5 5 9 1 . 9 1 5 8 4 6 5 T A B L E 4 : C A S E 3 - F R E Q U E N C X - C E L L L I M I T S F R E Q X T O T 5 4 . 0 — 1 1 0 . 8 7 5 7 1 1 1 0 . 9 - 1 6 7 . 7 2 4 3 5 6 0 1 6 7 . 8 - 2 2 4 . 6 6 7 1 3 5 1 6 2 2 4 . 7 - 2 8 1 . 5 1 6 3 4 1 8 9 0 2 8 1 . 6 - 3 3 8 . 4 2 0 3 6 3 6 7 3 3 3 8 . 5 - 3 9 5 . 3 2 5 3 9 2 6 5 7 3 9 5 - 4 - 4 5 2 . 2 1 7 6 7 4 5 4 6 4 5 2 . 3 - 5 0 9 . 1 7 3 3 4 8 3 6 5 0 9 - 2 - 5 6 6 . 0 2 6 1 3 7 0 1 5 6 6 . 1 - 6 2 2 . 9 8 4 7 7 0 D I S T R I B U T I O N F O R 10 C L A S S c S X V A R Y T O T Y V A R R H G 7 5 0 . 3 9 3 1 2 4 . 0 1 9 6 7 1 4 6 . 7 1 7 3 2 2 6 . 5 0 . 1 7 1 0 2 7 3 . 5 1 1 1 3 1 5 0 . 3 0 . 1 7 1 4 2 6 6 . 0 3 8 4 3 1 3 3 . 3 0 7 6 6 2 7 7 . 3 5 8 8 1 1 6 7 . i 0 . 1 1 3 4 2 3 8 . 5 8 2 1 6 1 6 3 . 5 0 . 2 2 3 4 2 4 0 . «+ 8 1 4 7 1 2 3 . 2 0 . 1 3 3 8 2 5 3 . 8 5 6 2 5 1 3 3 . 8 « 1 0 1 2 2 0 7 . 2 2 5 5 1 1 4 7 . 6 0 . 1 9 9 2 2 8 6 . 0 3 1 8 2 2 3 . 3 0 . 2 2 9 3 D I S T R I B U T I O N F O R 10 C L A S S E S X V A R Y T O T Y V A R R H O 5 4 1 . 0 1 2 4 1 7 3 . 2 0 . 1 9 0 1 2 0 3 . 5 6 0 3 2 8 8 . 5 0 . 1 4 0 7 2 6 6 . 4 1 7 8 7 1 7 9 . 3 0 . 0 0 3 9 2 5 9 . 7 5 4 6 2 1 2 0 . 0 0 . 0 1 9 1 2 6 4 . 3 6 1 7 1 1 7 7 . 7 0 . 1 5 7 3 2 7 5 . 9 3 3 4 2 0 6 . 5 J . 1 5 7 4 2 5 4 . 3 7 1 7 4 3 0 9 . 3 0 . 0 9 7 3 2 4 3 . 7 3 0 6 7 1 7 2 . 2 0 . 1 2 1 4 1 3 8 . 7 1 3 3 5 4 2 6 . _ 0 . 0 2 4 9 3 1 2 . 5 4 6 1 4 8 9 . 1 5 1 8 6 R E M A R K S : S A M E C O M M E N T S A S F O R T A B L E 1 ( F O O T N O T E 2 ) TABLE 5: CASE 4 - FREQUENCY DISTRIBUTION FOR 10 CLASSES X-CE LL LIMITS FREQ XTOT XVAR YTOT YV AR RHO 53. 0 - 114.6 2 156 1250.0 9 4.5 114. 7 - 176.3 22 3405 291.3 526 235.8 - . 0479 176. 4 - 238.0 117 25139 264.4 3760 187.2 0. 2640 238. 1 - 299.7 214 57536 292.3 7237 163.4 0338 299. 8 - 361.4 22 8 75473 297.2 8663 270.U - . 0174 361. 5 - 423. L 23 7 92950 291.2 9934 290.3 0. 0614 423. 2 - 484.3 120 54135 274.4 4594 303.0 « 0686 484. 9 - 546.5 48 24630 313.4 2 502 764.9 0. 1353 546. 6 - 608.2 8 46 30 426. 5 527 1297.0 0. 3536 608. 3 - 669.9 4 25 5 3 467.7 252 280.7 u. 2834 TABLE 6: CASE 5 - FREQUENCY DISTRIBUTION FOR 10 CLASStS X-CE LL LIMITS FREQ XTOT XVAR YTOT YVAK RHO 62. 0 128.7 9 996 487.0 55 7 0.9 -.7749 128. 8 - 195.5 32 5592 320.6 788 275.1 -.1956 195. 6 — 262.3 163 33239 321.4 5579 182.3 0.0282 262. 4 - 329. 1 233 68805 422. 5 8606 245.7 0.2679 329- 2 — 395.9 275 98930 347.1 11372 453.0 0.1726 396. 0 - 462. 7 209 83600 41 0.4 8787 616.9 -.0732 462. 8 - 52 9.5 63 3 0869 282.5 3032 860.3 0.3551 529. 6 — 596.3 12 6637 306.4 741 490.3 0.2 2 04 596. 4 - 663. 1 3 1913 346.3 258 31.0 -.7093 663. 2 729.9 1 729 0.0 97 0.0 TABL q 7 : CASE 6 - FP EQUrNCY DISTRIBUTION . i = 0R 10 CL ASS cS X-CE LL LIMITS FREQ XTOT XVAR YTOT YVAii RHO 65. 0 _ 122.9 6 540 400.0 123 77.9 0.1994 123. 0 — 180.9 18 2784 213.8 443 3 44.8 0.1317 181. 0 - 238. 9 89 19333 231.8 3233 257.0 0.2152 239- 0 - 296. 9 219 58792 241.1 7322 285.6 . 0.0616 297. 0 - 354.9 264 86119 2 50.4 11237 393.5 j.0160 355. 0 - 412. 9 207 79332 282.3 9292 564.3 -.1434 413. 0 - 470.9 108 47279 256. 5 4763 1002.0 0.0092 471. 0 - 52 8.9 54 26751 212.3 3329 1164.4 0.1106 529. 0 - 586.9 25 13746 293. 1 2243 1440.2 -.2056 587. 0 - 644. 9 10 6083 326.6 947 869.3 -.4042 REMARKS: SAME COMMENTS AS FOR TABLE 1 (FOOTNOTE 2) TABLE 8: CASE 7 - FREQUENCY X-CELL LIMITS FREQ XTOT 123.0 - 178. 5 16 2504 178. 6 • - 234.1 94 19972 234. 2 - 289. 7 210 57991 289- 8 - 345.3 275 87995 345. 4 - 400. 9 170 62706 401.0 - 45 6. 5 129 54603 456. 6 - 512. 1 64 30906 512. 2 - 567.7 22 11794 567. 8 - 623.3 10 5965 623.4 - 678.9 2 1321 DISTRIBUTION FOR 10 CLASSES XVAR YTOT YVAR RHO 240.4 406 346.4 0. 3038 226.9 3264 233.j 0. 2031 221.8 7864 333.1 0. 0266 256. 1 12906 456.1 0. 14 32 283.2 9110 884.9 • 1737 301.0 5754 1383.J 0. 1432 224.3 463 3 12 56.9 0. 1109 217.1 2 53 + 3022 .2 0. 1453 315. 8 1179 3060.o • 2759 612.5 113 5000.J TABLE 9: CASE 8 - FREQUENCY DISTRIBUTION FOR 10 CLASSES X-CELL LIMITS FREQ XTOT XVAR YTOT YVAR RHO 110.0 - 168.2 13 1909 345. 5 523 193.5 2586 168.3 - 226.5 42 8587 294.2 2 402 190.2 0. 3744 226. 6 - 284.8 174 44754 290. 7. 11573 137.1 0. 1704 284.9 - 343.1 200 63076 303.7 15779 121.2 0. 4506 343. 2 - 401.4 242 90416 269.3 22926 121.0 0. 3324 401. 5 - 459.7 208 83956 232. 7 22073 93. i 0. 4659 459. 8 - 51C.0 92 44428 273.9 10794 80.1 0. 4631 518. 1 - 576.3 24 12946 172.2 3194 73.9 0. 1535 576.4 - 63 4.6. 4 2349 141.6 592 224. 7 ~ ' * 2131 634.7 - 692.9 1 692 0.0 170 0.0 TABLE 10: CASE 9 - FREQUENCY DISTRIBUTION FOR 10 CLASSES X-CcLL LIMITS FRFQ XTOT XVAR YTOT YVAR RHO 22.0 - 35. 4 2 101 1624. 5 21 84. 5 **** 85.5 - 143. 9 10 1276 197. 4 . 343 193. 7 ' 0. 0092 149. 0 - 212. 4 35 6563 305. 1 1914 206. 0 1615 212. 5 - 275. 9 • 170 42271 334. 1 11439 156. 6 0. 4224 276. 0 - 339. 4 232 71394 331. 3 17621 199. _ 0. 3339 339. 5 - 402. 9 253 94173 356. 6 23711 224. 2 u -3635 403.0 - 466. 4 202 3 7607 305. 6 21611 150. ) 0. 3119 466. 5 - 529. 9 70 34113 298. 4 8328 122. 6 0. 3841 530.0 - 593. 4 21 11572 329. 2 2920 231. i 0. 4442 593. 5 - 656. 9 5 3135 420. 0 809 302. 7 0. 5841 REMARKS: SAME COMMENTS AS FOR TABLE 1 {FOOTNOTE 2) TABLE 11: CASE 10 - FREQUENCY DISTRIBUTION FOR 10 CLASSES X-CE LL LIMITS FREQ XTOT 10. 0 - 7 2.4 1 10 72 . 5 - 134.9 7 799 135. 0 - 197.4 39 6753 197. 5 - 259.9 104 24352 2 60. 0 - 322.4 191 56141 322. 5 - 384.9 303 107004 385. 0 - 447.4 217 90110 447. 5 - 509.9 103 48696 510. 0 - 572.4 30 16021 5 72. 5 - 634.9 5 3035 TABLE 12: CASE 11 - FREQUENCY X-CE LL LIMITS FREQ XTOT 66. 0 - 12 5.7 6 612 125. 8 - 185.5 15 2421 185. .6 - 245.3 82 17963 245. 4 - 305.1 201 55341 305. 2 - 364.9 220 73628 365. G - 424.7 24 2 95326 424. 3 - 484.5 166 74869 484. 6 - 544.3 48 24344 544. 4 - 604.1 18 10153 604. 2 - 663.9 2 1300 TABLE 13: CASE 12 - FREQUENCY X-CELL L IMITS FREQ XTOT 79.0 - 139.7 9 998 139. 8 - 200.5 31 5431 200. 6 - 261.3 120 28539 261. 4 - 322.1 232 68096 322. 2 - 382.9 24 5 35970 383. 0 - 443.7 204 83899 443. 8 - 504.5 115 53887 504. 6 - 565.3 32 16956 565. 4 - 626.1 10 5336 626.2 - 686.9 2 1326 XVAR YTOT YVAR RHO 0.0 22 0.0 £ -£ # a$c 314. 8 23 4 446 .0 0. 5230 377.1 1930 405.3 0. 3723 307. 7 6554 276.6 0. 0356 357.9 14153 302.5 0. 2624 317. 3 26834 197-0 0. 2621 360.2 22418 293.o 0. 3315 263. 5 11576 223.3 0. 12 43 304. 2 3 73 9 392.4 0. 1 846 377. 5 753 274.3 0. 4048 DISTRIBUTION FOR 10 l CLAS S £S XVAR YTOT YVArC RHO 475.6 157 404.6 0. 2841 251.3 686 300.6 • 0063 256.3 4837 353.-+ 0. 4213 289. 5 14410 2 96.9 0. 1927 269, 5 18542 384.0 0. 2078 272.3 23837 356.3 0. 13 74 301.8 17848 407.2 0 3 64 270.1 5636 532.7 0. 19 67 317.4 2520 1340.0 0. 5947 333.0 330 2.0 DISTRIBUTION FOR 10 CLASSES XVAR YTOT YVAR RHO 466. 1 220 307. o 0. 8057 311. 3 1680 300. i 1296 250. 3 7862 3 70. o 0. 0734 277. 0 17129 364. 3 0. 0512 323. 9 21144 462. 5 j . 2054 317. 5 20879 555. t 0. 1574 293. 5 12077 690. 1 0. 1136 279. 1 4239 662. o 0. 0 435 252. 9 1736 3093. 8 0. 7101 058, 0 328 578. 0 REMARKS: SAME COMMENTS AS FOR TABLE 1 I FOOTNOTE 2) TABLE 14: CASE 13 - FREQUENCY x - c c LL LIMITS FREQ XTOT 92. 0 - 145.8 1 1 1294 145. 9 - 199.7 30 5327 199. 8 - 253.6 99 22876 253. 7 - 307.5 184 51778 307. 6 - 361.4 224 74724 361. 5 - 415.3 2 02 78382 415. 4 - 469.2 153 67328 469. 3 - 523.1 69 33801 523. 2 - 577.0 19 10286 577. 1 - 630.9 9 5397 DISTRIBUTION FOR 10 CLASSES XVAR YTOT YVAR RHO 253. 1 323 136. 1 0. 0935 229. 1 1655 333. 5 0. 1523 218. 0 6773 293. 5 0. 0971 230. 7 13921 525. 0 0. 2274 192. 4 18862 800. 5 0. 1531 241. 4 19354 886. 1 0. 0507 230. 9 16117 1283. 5 u. 0020 212. 5 8199 1262. 4 0. 2903 250. 7 2804 1395. i 0. 4922 348. 0 1855 1580. 4 ~ • 0386 TABLE 15: CASE 14 - FREQUENCY X-CELL LIMITS FRFQ XTOT 83.0 — 140.0 11 1276 140. 1 - 197.1 37 6353 197. 2 - 254.2 115 26344 254. 3 - 311.3 182 51318 311.4 - 368.4 220 74161 368. 5 - 425. 5 23 3 92390 425.6 - 482. 6 137 61949 432. 7 - 539.7 55 27751 539. 8 - 596. 8 7 3940 596-9 653.9 3 1889 TABLE 16: CASE 15 - FREQUEN X-CELL L IMITS FREQ XTOT 53. C — 111.0 7 646 111.1 - 16 9. 1 16 2434 169. 2 - 227.2 59 1 2176 227. 3 - 285. 3 16 1 41 796 285.4 - 343.4 216 68126 343. 5 - 401. 5 199 74568 401.6 - 459. 6 226 96509 459. 7 - 517. 7 91 43359 517. 8 - 575. 3 20 10330 575.9 - 633.9 5 3015 DISTRIBUTION FOR 10 CLASS £S XVAR YTOT YVAR RHO 279. 6 280 107.1 0. 1243 334.7 1732 505.9 0. 0532 245. 4 7609 419.9 0. 2398 251.1 12923 743.7 0. 0565 244.0 19625 1079.3 0 . 0539 237. 7 23131 1449.0 0. 0720 277. 3 13 3 73 1841.0 1929 298.2 6798 3295. 7 0. 3418 99. 8 1007 2233.3 — . 0597 417.3 471 9247.0 0. 3431 DISTRIBUTION FOR 10 CLASSES XVAR YTOT YVAR RHO 342. 9 277 371. 0 * 1183 221. 2 1114 297. 8 0. 0607 220. 2 5106 213. 4 0. 5913 273. 2 1714 3 130. o 0. 4752 233. 6 27162 127. 0 0. 4619 283. 7 29964 132. ? 0. 6542 273. 7 38564 121. 2 0. 5393 203. 1 1691 7 97. 3 0 . 6302 313. 4 4324 175. 7 0. 6759 508. 5 1203 161. 3 0. 8655 REMARKS: SAME COMMENTS AS FOR TABLE 1 (FOOTNOTE 2} TABLE 17: CASE 16 - FREQUENCY DISTRIBUTION FOR 10 CLASoES X-CELL L IMITS FREQ XTOT 96. G — 14 9.8 9 1104 149.9 - 203. 7 27 4909 203. 8 - 2 5 7.6 97 2 2739 257. 7 - 311.5 197 56376 311.6 - 365.4 224 75907 365. 5 - 41 9.3 217 35200 419.4 - 473. 2 149 66117 473.3 - 527.1 58 28610 527. 2 - 531. 0 18 9866 5 31. 1 634.9 4 2453 TABLE 13: CASE 17 - FREOUENi X-CELL LIMITS FREQ XTOT 62. 0 — 117.0 6 557 117. 1 - 172. 1 20 3146 172.2 - 227.2 53 10841 227.3 - 282. 3 141 3655 1 282.4 - 337.4 221 68987 337. 5 - 3 92.5 248 90405 392. 6 — 447. 6 170 71254 447. 7 - 502.7 92 43590 502. 8 - 557. 8 33 20026 557.9 612. 9 11 63 5 9 TABLE 19: CASE 18 - FREQUENt X-CELL LIMITS FREQ XTOT 53. 0 — 115.9 5 385 1 16. 0 - 178. 9 21 3227 179.0 - 241.9 99 21562 242. 0 - 3 04.9 216 59607 305.0 - 367. 9 237 79363 368.0 - 430.9 246 97474 431. 0 - 493. 9 123 58393 494.0 - 556.9 40 20830 557. 0 - 619.9 7 4122 620.0 - 682. 9 1 682 XVAR YTOT YVAK RHO 459.3 367 1027.9 0. 1365 244.9 1937 190.4 0. 1918 216.9 9680 171.0 0. 3603 231.0 22601 186.0 0 . 2117 241. 8 30037 238.1 0. 5294 235.4 33973 Z06.Z 0 . 4867 232. 9 25934 143.0 0. 4005 227.2 11326 243.4 0. 6273 157. 8 3932 191.3 0. 2422 481. 7 987 224.3 0. 8 5 65 DISTRIBUTION FOR 10 ( CLASSES XVAR YTOT Y VAR RHO 598.2 296 3 2 3.9 0. 5663 10 8.3 1288 488.0 0. 3506 253.2 4651 344.8 0. 5 50 7 241 = 2 14996 257. L 0. 3485 223.6 27890 239.0 0. 3365 253.9 36065 314. a 0. 3398 232. 5 28617 309.6 0. 4153 263.8 16735 236.5 0. 0840 241.7 7829 425.3 0. 4530 245.3 2473 781.0 0. 6222 DISTRIBUTION FOR 10 CLASSES XVAR YTOT YVAR RHO 706.5 146 415.2 * 1145 257.0 1265 330.6 0. 5128 292.7 9290 363.5 u. 6150 314.7 24727 319.4 0. 3843 318. 3 31878 447.9 0. 3369 296.3 38737 402.3 0 . 3840 306. 7 22431 338.* 0. 2192 339.2 7995 473.3 0. 2304 46.8.8 1684 707.0 0. 49 8 7 0.0 256 0.0 **** REMARKS: SAME COMMENTS AS FOR TABLE 1 (FOOTNOTE 2} TABLE 20: C A S C 19 - FREQUENCY D I S T R I B U T I O N FOR 10 C L A S S E S X-CELL L I M I T S FREQ XTOT 71.0 — 127.0 7 655 127. 1 - 133. 1 22 362 1 183. 2 - 239.2 67 14700 239. 3 - 295.3 167 45281 295. 4 - 3 51.4 217 70261 351 . 5 - 407.5 239 90461 407. 6 - 46 3. 6 185 8C065 463 . 7 - 519. 7 75 36412 519.8 - 575. 8 18 9600 575.9 631. 9 3 1314 TABLE 21: CASE 20 - FREQUSN! X-CELL L I M I T S FREQ XTOT 81 .0 — 140.3 8 911 140. 9 - 200. 7 34 5976 200. 8 - 260.6 126 29390 260.7 - 320.5 225 65533 320. 6 - 330. 4 228 80325 330. 5 - 440.3 203 35153 440. 4 - 500.2 128 59530 500. 3 - 560, 1 34 17972 560.2 - 620.0 6 3449 620. 1 - 679.9 3 1963 XVAR YTOT YVAR RHO 268.0 208 123.9. 0. 7538 234. 7 1628 367.2 0. 43 73 137.4 6137 331.7 u. 2718 258.2 18265 403.5 0. 3237 263.0 27919 643 .4 0. 3114 270.3 37202 634.9 0 . 13 59 244. 9 31072 739.4 0 . 2743 276. 8 13562 887.8 0. 4272 101.2 3 85 6 1603.o 0. 1547 702.3 715 842.3 0. 4743 D I S T R I B U T I O N FOR .10.CLASSES XVAR YTOT YVAR RHO 40 5.0 268 515.1 0. 4243 344.2 2 581 394.7 0. 4074 300.3 12392 535.o 0. 1567 335.9 2 6128 684. a 0. 1324 280. 0 32493 831.o 0. 3156 276.2 33245 1047.6 0. 1146 242.1 22663 1599.7 0. 0937 365.2 6612 213 7.^ 0. 1323 91.4 • 1313 2839.U U . 6479 637.0 860 2921.3 •> 3365 TABL . E 22: CASE 21 - FREQUEN X-CE ;LL . L I M I T S FREQ XTOT 73. 0 - 128.8 9 906 128. 9 - 184.7 26 4165 184. 8 - 240.6 95 20333 240. 7 - 296.5 177 47906 296. 6 - 325.4 224 73220 352. 5 - 40 8.3 199 75400 408. 4 - 464.2 173 75249 464. 3 - 520.1 71 34610 520. 2 - 576.0 18 9764 5 76. 1 - 631.9 8 4844 D I S T R I B U T I O N FOR 10 C L A S S E S XVAR YTOT YVAR RHO 276.3 406 317.6 0. 7471 202. 5 1762 389.3 0. 409 5 232. 4 89 91 569.3 0. 4868 260.2 19714 902.3 0. 1505 255.2 29566 1349.2 0. 1841 269.2 30509 1287.3 0. 0487 251.1 28546 2028.3 0. 0638 224.6 12214 2564.o 0. 3425 147.7 3750 3730.4 0. 1321 213. 7 2033 10393.8 • 1206 REMARKS: SAME COMMENTS AS FOR TABLE 1 (FOOTNOTE 2) 62. from the non-linear approximation of the re l a t i o n s h i p . V(Yj ) = k(XMIDj) P , (4.1.10) The computer programme BMD.X85 (adapted by Jason Halm of the University of B r i t i s h Columbia from the University of C a l i f o r n i a , Los Angeles' BMD documentation centre) involving user s p e c i f i c a t i o n of the derivatives of V(y_.) with respect to k and then p, and i n i t i a l guessed values, was used. One procedure for determining the guessed values that was used, involved the f i t t i n g of the l o g -transformation of equation (4.1.10), and the use of a least squares regression estimation procedure. Programme MREG (A. Kozak, the University of B r i t i s h Columbia) was used. I n i t i a l summary for cases A1-A3 are provided i n table 23, where R" = YBAR/XBAR. INITIAL TABLE SUMMARY 23 FOR CASES A1-A3 Case YBAR XBAR P R" R ' Al 39.34 343.64 1. 8808 Q u i l l 0.10 A2 88.67 352 .74 2. 5000 0.251 0.25 A3 137.83 350.59 2. 5000 0.393 0.40 Total 88.61 348.99 1. 6232 0.254 — Frequency d i s t r i b u t i o n tables for 10 classes, and h i s t o -grams for 5 and 15 classes are presented as Tables 24-26, 63. and Figure 3 resp e c t i v e l y . Except for case A3 (Table 26), the remaining two combined cases gave reasonably small class c o r r e l a t i o n c o e f f i c i e n t s . TABLE 24: CASE A l - FREQUENCY DISTRIBUTION FOR 10 CLASSES X-CE LL LIMITS FREQ XTOT XVAR YTOT YVAR RHO 47. 0 - 115.2 27 2299 420.0 496 137.2 • 0755 115. 3 - 133.5 177 27996 335.8 4119 264.1 0. 1869 183. 6 - 251.8 891 2C0500 311.4 28117 196.2 0. 1257 251. 9 - 320.1 1775 503223 398.0 60938 255.0 0. 1328 320. 2 - 383.4 1931 680745 368. 1 77975 3 5 5 . 6 o. 0582 338. 5 - 456.7 1471 616379 380.0 62574 466.3 0. 0311 456. 8 - 525.0 560 271631 354.8 28559 7 01.7 0. 2062 5 25. 1 - 593.3 136 74939 383. 7 9931 1533.4 0 . 04.0 3 593. 4 - 661.6 29 17909 236.3 2425 1304.9 1938 661. 7 - 729.9 3 2076 1047.0 132 2112.3 0. 5766 TABLE 25: CASE A2 - FREQUENCY DISTRIBUTION FOR 10 CLASSES X-CE LL L IMITS FREQ XTOT XVAR YTOT YVAR RHO 10. 0 - 78.2 3 98 869.3 43 86.3 • 9977 78. 3 - 146.5 71 8614 354.2 2275 276.4 0. 3804 146. 6 - 214.8 321 60589 386.7 17015 349.6 0. 1506 214. 9 - 233.1 1 246 316526 36 3.7 35421 293.2 u. 1367 233. 2 - 351.4 18 34 585287 370.4 146254 436.0 0 . 1803 3 51. 5 - 419.7 1849 712361 360.9 179997 535.5 0. 1343 419. 3 - 488.0 1266 568867 365 .0 135153 626 .o 0. 0014 433. I — 556. 3 33 5 172504 364.6 42360 897.3 0. 1747 556. 4 - 624.6 63 36531 236.0 10074 1824.9 0. 2901 624. 7 - 692.9 12 7781 489.7 2 03 3 322.8 2970 TABLE 26: CASE A3 - FREQUENCY DISTRIBUTION FOR 10 CLASSES X-CE LL LIMITS FREQ XTOT XVAR YTOT YVAR RHO 53 . 0 - 115.9 40 3714 345. 8 1570 425.9 0. 4000 116. 0 - 178.9 145 22735 23 3.0 9523 457.6 0. 43 47 179. 0 - 241.9 604 130972 275.3 56131 363.7 0. 5029 2 42. 0 - 304.9 1401 386231 314. 1 156840 446.2 0. 2636 305. 0 - 367.9 1777 597469 333.0 239739 632.2 0. 3409 368 . 0 - 430.9 1679 668524 221.9 266278 • 550.7 0. 2462 431. 0 - 493.9 1016 464402 301. 4 178250 815.9 0. 1121 494. 0 - 556.9 275 142939 323.7 55307 U l I . i 0. 2178 557. 0 - 619.9 53 30739 344. 1 12314 2228.8 0.2033 620. 0 - 632.9 10 6420 530.4 2670 1170.* 0. 0173 REMARKS: SAME COMMENTS AS FOR TABLE'1 [FOOTNOTE 2) AL. 5 classes A2. 5 classes A3. 5 classes 3000 \ 2000 1000 2 3 4 x-classes 1 2 3 x-classes 2 3 4 x-classes 1500 1000 500 Al . 15 classes s A2. 15 classes A3. 15 classes 7 11 x-classes 15 11 .15 1 11 15 x-classes x-classes Fig.3 Histograms for Al, A2 and A3 - 5 and 15 classes. 3 66. 4.2 IFS Log scal i n g and weight data These data made up of several samples from an i d e n t i c a l area (Vavenby, B r i t i s h Columbia) were put together to constitute a period's weight scaling data. They were a c t u a l l y made up of f i v e 'operator s t r a t a ' , and involved mainly Spruce (Picea spp.), Douglas-fir (Pseudotsuga menziesii (Mirb.) Franco) and "Balsam" (Abies lasiocarpa (Hook.) Nutt.). Some were pure samples but a majority of cases were mixed— Spruce/Balsam and firmwood re j e c t s made up several but unspecified species. No attempt was made to i d e n t i f y the species volume components with regards to the use of part i c u l a r sampling schemes because species proportions are determined operationally from the volume scales. Further analysis might be j u s t i f i e d . Net load weights (x^) were given in pounds (lbs.) and 4 the scale or volume (y^) were given in firmwood cubic f e e t . Thear.e_ationship between the t o t a l load scales and the net weight was of the order 1 to 50, so that a r a t i o of 0.02 or thereabout may be expected. Tables 27A-27C provide segmented summaries for the 1060 paired data used. In Table 2 7A, the data were grouped i n 20's 3 Data were provided by the Ind u s t r i a l Forest Service Ltd. consulting firm (IFS). 4 The firmwood cubic scale i s intended to provide a measure of the cubic foot volume of firmwood in the log. Primary volume deductions permitted i n scal i n g are rot and voids. as determined by the order of a r r i v a l s of the truck loads to the weighing s t a t i o n . Thus set 1 represents the f i r s t 20 a r r i v a l s , set 2 the next 20 a r r i v a l s and so on. Similar interpretations could be given to Tables 27B and 27C, except that the la s t 20 a r r i v a l s were not included. Y c o e f f i c i e n t s of v a r i a t i o n for sets 16 andc-29 (Table 27A) , fo r i sets-.8'and-10 t( Table' 27B) ahd^ f or»set'' 4 ' '(Table- 27C) ' • were p a r t i c u l a r l y higher than others. Except for these, the YCV% were f a i r l y constant, ranging from about 8% to 25%. The XCV% appeared to be more consistent than the YCV's. The range for the XCV% from Table 27C was 6.01%, that i s 9.05% to 15.06%. The groups c o r r e l a t i o n c o e f f i c i e n t s were exception-a l l y low or negative for the sets with high YCV%. These were either due to exceptionally large variances of y or negative covariance. Using the same format as for previously given frequency d i s t r i b u t i o n s , Tables 28, 29 and 30 give such r e s u l t s for the weight scaling population for 5, 10 and 15 classes re s p e c t i v e l y . As can be observed, there were very few loads i n the i n i t i a l classes. Such loads represent h a l f -loads which are quite few in practice. These w i l l , however, r e s t r i c t the f u l l u t i l i z a t i o n of some of the sampling schemes, p a r t i c u l a r l y DXZ, IXZ, Types 2-6. The determination of p (from equation 4.1.10) yielded TABLE 27A 6 8 . POPULAT ION PARAMETERS FOR DATA OBTAINED FROM I F S - G R O U P E D IN 2G«S SET YBAR YVAR YCV5? RHO XBAR XVAR XCVIS 1 1 3 8 8 0 25 3 7 5 5 7 . 9 4 1 3 . 9 6 0 . 9 3 9 2 2 6 6 7 8 5 3 , 0 0 9 9 2 7 9 3 9 6 1 4 . 6 8 2 1495o 60 2 5 7 0 3 . 8 4 1 0 . 7 2 0 . 9 1 4 2 9 4 7 2 1 4 7 . 0 0 3 2 4 8 6 8 3 3 7 . 9 0 3 1479o 20 1 8 5 9 0 . 5 6 9 . 2 2 0 . 7 5 0 5 7 1 7 3 1 7 3 . 0 0 4 3 0 5 7 1 4 6 8 . 9 7 4 1 4 5 7 , 6 5 1 4 7 4 2 . 8 8 8 . 3 3 0 . 6 6 7418 7 3 4 8 1 . 0 0 2 7 8 6 7 1 5 7 7 . 1 8 5 1 3 6 0 * 50 4 1 6 5 7 . 3 5 1 5 . 0 0 0 . 5 7 6 0 9 6 6 6 3 6 8 . 0 0 6 9 4 6 4 0 6 4 1 2 . 5 6 6 1 4 5 6 0 65 3 4 7 4 1 . 9 8 1 2 . 80 0 . 7 7 9 7 6 0 7 3 4 3 9 . 0 0 6 8 3 9 1 7 6 9 1 1 . 2 6 7 1 4 7 8 0 80 1 8 6 6 6 , 5 6 9 . 2 4 0 . 8 1 4 5 7 9 7 2 5 2 3 . 0 0 3 0 1 7 8 9 2 5 7 . 5 7 8 1504o 30 2 5 7 6 1 . 5 1 1 0 . 6 7 0 . 9 3 3 1 4 8 7 4 5 4 4 , 0 0 4 9 4 8 5 5 6 8 9 . 4 4 9 1450o 60 3 2 4 8 2 . 8 4 1 2 . 4 2 0 . 3 2 6 7 6 6 7 1 8 6 6 . 0 0 6 8 2 2 6 0 6 3 1 1 . 4 9 10 1 4 3 9 . 7 0 3 9 3 7 2 . 7 1 1 3 . 7 8 0 . 9 3 3 5 1 5 6 9 1 7 3 . 0 0 7 4 4 4 3 0 1 5 1 2 . 47 11 1 6 1 2 . 2 0 3 2 7 4 3 . 1 1 1 1 . 2 2 0 . 9 0 6 9 9 6 7 8 0 1 6 . 0 0 6 1 7 8 6 5 2 2 1 0 . 0 8 12 1 5 2 9 o 9 0 4 2 6 1 9 , 5 9 1 3 , 4 9 0 . 8 7 9 7 2 0 7 1 5 0 3 . 0 0 6 5 6 9 3 9 0 6 1 1 . 3 4 13 1386o CO 4 7 0 2 2 . 4 0 1 5 . 6 5 0 . 9 3 3 4 4 0 6 5 2 5 0 . 0 0 8 4 9 3 4 7 8 0 1 4 . 1 2 14 1 2 8 1 . 4 0 1 5 3 9 3 . 2 4 9 , 68 0 . 6 3 8 8 0 9 6 3 4 5 9 . 0 0 3 7 8 2 5 7 1 9 9 . 6 9 15 13 6 9 , 2 5 1 4 3 0 2 . 4 4 8 . 7 3 0 . 5 9 2 9 8 4 6 2 1 9 7 . 0 0 3 3 5 2 6 9 7 0 9 . 3 1 16 9 0 1 , 35 4 2 9 9 2 2 . 5 8 7 2 , 7 4 0 . 1 8 4 4 6 3 72 1 0 4 . 0 0 4 2 4 7 7 6 0 6 9 . 0 4 17 1 2 8 2 , 4 5 1 6 0 7 4 . 0 0 9 . 89 0 . 6 5 4 3 1 7 6 4 2 0 3 , 0 0 2 9 3 3 3 0 5 0 8 , 4 4 18 1 1 5 7 , 7 0 1 9 4 1 8 . 7 1 1 2 . 0 4 0 . 9 2 3 9 7 4 6 1 7 5 6 . 0 0 6 1 6 7 7 6 5 4 1 2 . 7 2 19 1161o 75 3 6 6 2 4 . 1 4 1 6 , 4 7 0 . 9 3 4 9 3 9 6 2 3 6 8 , 0 0 9 9 6 9 3 9 7 8 1 6 . 0 1 20 1 1 2 5 , 6 5 1 3 9 1 3 4 , 4 8 3 3 , 14 0 . 2 7 9 9 1 4 6 5 C 9 2 . 0 0 5 4 3 7 4 6 9 1 1 1 . 3 3 21 1 3 1 8 , 9 5 3 4 9 6 3 . 7 0 1 4 . 1 8 0 . 8 1 6 7 7 5 6 9 2 6 2 . 0 0 3 5 5 7 3 6 6 6 8 . 6 1 22 1 2 2 4 . 1 5 2 6 2 3 2 . 7 8 1 3 , 23 0 . 7 1 5 5 1 9 6 4 5 7 9 . 0 0 5 5 6 3 1 8 8 9 1 1 . 5 5 23 1 3 4 8 . 2 0 2 0 2 9 3 . 5 6 1 0 . 57 0 . 7 7 8 9 9 7 6 5 6 4 5 . 0 0 3 7 1 7 6 5 2 2 9 . 2 9 24 1 3 C 4 . 3 0 1 4 0 2 9 . 5 1 9 , 08 0 . 8 1 5 2 9 2 6 4 6 1 0 . 0 0 3 6 6 4 8 7 7 7 9 . 3 7 25 1 3 0 9 . 5 0 1 8 6 8 6 . 5 5 1 0 , 4 4 0 , 6 9 9 2 8 6 6 6 0 2 6 , 0 0 1 9 5 0 9 1 7 4 6 . 6 9 26 1 2 9 8 . 4 0 4 2 8 1 4 , 2 4 1 5 . 9 4 0 . 8 2 9 6 4 0 6 7 4 9 6 . 0 0 4 6 1 0 0 1 0 9 1 0 . 06 27 1 3 5 4 . 60 1 4 5 8 2 . 8 4 8 . 9 1 0 . 7 7 7 3 5 5 6 8 7 5 6 . 0 0 2 6 7 7 0 8 5 1 7 . 5 3 28 13 .19 . 75 2 3 4 2 3 , 1 4 1 1 . 6 0 0 . 9 4 0 1 5 2 6 9 4 8 9 . 0 0 5 1 9 3 4 8 5 3 1 0 . 3 7 29 1 2 1 5 . C 5 1 1 7 5 6 7 . 9 0 2 8 . 2 2 0 . 5 1 4 3 8 5 6 9 8 2 6 . 0 0 2 8 0 7 7 3 0 8 7 . 5 9 30 1 2 2 9 . 7 5 3 6 9 2 8 , 54 1 5 , 63 0 . 7 6 7 5 3 5 6 4 3 0 6 . 0 0 5 9 4 0 6 0 8 9 1 1 . 9 9 31 1 3 0 0 . 05 2 4 2 3 0 . 0 0 1 1 . 9 7 0 , 4 4 5 1 4 2 6 7 5 2 4 . 0 0 1 1 2 4 7 2 3 5 4 . 9 9 32 1 1 3 4 . 5 0 3 3 7 8 2 . 5 5 1 6 . 2 0 0 . 8 3 2 7 4 5 6 0 9 3 3 . 0 0 9 6 2 4 3 0 7 3 1 6 . 10 33 1 1 6 1 . 65 1 4 9 2 7 . 68 1 0 . 5 2 0 . 7 5 6 6 8 7 5 7 8 3 7 . 0 0 2 9 1 4 9 0 2 8 9 . 3 3 34 1 1 9 3 . 8 0 7 0 8 8 8 . 7 6 2 2 . 3 0 0 , o 3 9 5 6 2 6 4 7 5 1 . 0 0 5 5 2 0 4 0 6 3 1 1 . 4 7 35 1 2 1 0 . 75 2 4 0 8 1 . 2 4 1 2 . 82 0 . 2 7 1 5 7 3 6 5 6 7 8 . 0 0 2 6 0 4 3 5 0 3 7 . 7 7 36 1 3 1 0 . 4 5 1 3 1 9 2 , 0 0 8 . 76 0 , 7 3 1 7 7 4 6 6 4 7 9 , 00 3 7 0 7 2 4 7 9 9 . 1 6 3 7 1 4 0 5 . 4 5 2 1 7 7 5 . 1 0 1 0 . 5 0 0 . 3 7 4 6 8 1 7 0 9 5 0 . 0 0 6 8 7 5 9 9 5 1 1 1 . 6 9 38 1 3 0 1 . 9 0 2 8 4 3 0 . 7 9 1 2 . 9 5 0 , 8 5 0 4 7 7 6 0 4 6 9 . 0 0 6 2 7 6 6 3 4 3 1 3 . 1 0 39 1 2 5 5 . 9 5 3 8 9 8 9 . 6 0 1 5 . 7 2 0 . 3 5 4 1 9 6 5 6 9 3 2 . 0 0 2 9 9 3 8 7 2 3 9 . 6 1 40 1 2 1 3 . 50 314 4 3 . 3 5 1 4 . 6 1 0 . 7 1 2 2 1 5 5 5 5 7 9 . 0 0 3 0 9 6 0 4 2 3 1 0 . 0 1 41 1 3 7 7 . C 5 2 3 7 2 6 . 90 1 1 . 1 9 0 . 7 5 0 4 6 1 7 1 6 3 9 , 0 0 6 1 0 0 4 9 6 2 1 0 . 9 0 42 1 4 3 9 . 7 0 2 0 8 4 3 . 1 1 1 0 . 0 3 0 . 8 3 4 0 4 0 7 3 9 9 0 . 0 0 6 7 4 5 9 6 5 C 1 1 . 1 0 43 1 5 3 9 . 70 3 1 8 9 1 . 1 1 1 1 . 6 0 0 . 8 4 2 6 0 4 7 5 8 6 7 . 5 0 7 0 0 8 6 0 6 6 1 1 . 0 3 44 1 5 2 3 . 2 5 4 8 0 1 8 , 2 4 1 4 , 3 9 0 , 5 1 6 3 8 1 7 5 0 9 8 , 5 0 1 9 1 2 6 1 2 3 7 1 8 . 4 2 45 1 2 7 C . 4 0 3 2 1 2 9 . 4 4 1 4 . 11 0 . 9 2 6 9 6 3 6 4 5 0 4 , 0 0 7 7 4 2 4 1 6 6 1 3 . 6 4 46 1 3 9 4 . 9 5 6 1 3 5 2 . I C 1 7 . 76 0 . 8 9 3 6 6 7 6 9 5 7 5 . 0 0 8 5 3 1 9 7 0 8 1 3 . 2 8 47 1 5 8 7 . 9 5 6 1 8 6 5 . 2 0 1 5 , 6 6 0 . 8 8 6 0 4 4 7 8 2 9 6 . 0 0 7 3 9 2 1 3 7 0 1 C . 9 8 48 1 1 7 C . 90 7 4 5 6 5 . L 9 2 3 . 3 2 - 0 . 2 8 7 7 3 3 5 3 1 2 1 . 0 0 6 0 3 5 7 8 5 8 1 3 . 3 7 4 9 1 3 0 5 . 0 0 4 8 6 3 5 . 80 1 6 . 9 0 0 . 8 4 4 9 0 5 6 8 5 5 5 . 0 0 7 5 6 8 2 6 6 9 1 2 . 6 9 50 1 2 5 0 . 3 5 5 2 3 8 8 . 0 8 1 8 . 3 1 0 . 5 1 5 5 8 6 74 94 .1 .00 5 5 3 3 1 9 5 9 9 . 9 3 51 1 1 8 4 . 10 3 5 8 2 5 . 5 9 1 5 . 98 0 . 4 4 1 6 9 3 6 3 1 4 5 . 0 0 4 8 3 4 0 4 8 9 1 1 , 0 1 52 1 2 8 0 , 3 0 1 8 0 7 1 , 9 1 1 0 . 50 0 . 8 3 7 5 0 3 6 3 9 5 6 . 0 0 6 2 5 0 5 4 8 2 1 2 , 3 6 53 1 4 0 3 , 7 0 3 C 5 9 2 . 7 1 1 2 . 4 6 0 . 7 8 3 2 4 0 7 1 1 3 1 . 0 0 7 2 1 5 1 1 0 5 1 1 . 9 4 T A B L E 278 6 9 . P O P U L A T I O N PARAMETERS FOR DATA OBTAINED i FROM I F S -GROUPED IN 4 0 ' S SET YBAR YVAR YCV2 RHO XB AR XVAR XCVS 1 1 4 4 1 . 9 2 3 4 5 1 1 . 0 9 1 2 . 8 8 0 . 9 2 3 1 6 9 7 0 0 0 0 . 0 0 7 0 4 8 2 8 9 3 1 1 . 9 9 2 1 4 6 8 . 42 1 6 7 8 2 . 02 8 . 82 0 . 7 1 0 2 1 6 7 3 3 2 7 . 0 0 3 5 4 7 6 0 3 7 8 . 1 2 3 1 4 0 8 . 5 7 4 0 5 0 9 . 6 7 1 4 . 2 9 0 . 6 9 4 3 0 7 6 9 9 0 3 . 5 0 8 1 4 1 9 4 8 5 1 2 . 9 1 4 1 4 9 1 . 55 2 2 3 7 5 . 4 0 1 0 . 03 0 . 8 3 4 3 0 8 7 3 5 3 3 . 5 0 4 0 8 4 3 5 2 7 8 . 69 5 1 4 4 5 . 1 5 3 5 9 5 6 0 6 8 13o 12 0 . 8 7 6 8 0 5 7 0 5 1 9 . 5 0 7 3 1 3 2 8 5 6 1 2 . 1 3 6 1 5 7 1 . 1 0 3 9 3 7 7 . 5 9 1 2 . 6 3 0 . 8 8 5 6 3 6 7 4 7 5 9 .5C 7 4 2 3 3 5 3 7 1 1 . 53 7 1 3 3 3 . 7 0 3 3 9 4 1 . 9 1 1 3 . 81 0 . 8 4 2 1 1 4 6 4 3 5 4 .50 6 2 1 6 9 0 6 3 1 2 . 2 5 8 1 1 3 5 o 3 0 2 7 6 8 4 4 . 3 1 4 6 . 25 - O o 1 3 4 4 5 2 6 7 1 5 0 . 5 C 6 2 5 2 9 6 2 0 1 1 . 7 8 9 1 2 2 0 . 0 7 2 1 6 3 6 . 1 9 1 2 . 0 6 0 . 7 9 9 9 1 9 62979.50 4 6 9 8 9 1 9 7 1 0 . 88 10 1 1 4 3 . 7C 8 8 2 0 4 , 3 1 2 5 . 97 0 . 4 7 5 2 6 8 6 3 7 3 0 . 0 0 7 8 8 7 9 5 4 8 1 3 . 9 4 11 1 2 7 1 . 5 5 3 2 8 4 4 . 6 0 1 4 . 2 5 0 . 7 7 1 0 6 0 6 6 9 2 0 . 5 0 5 1 0 7 C 6 5 4 1 0 . 6 8 12 1 3 2 6 . 25 1 7 6 4 2 . 5 4 1 0 . 0 2 0 . 7 9 2 7 7 2 6 5 1 2 7 . 5 0 3 7 1 6 8 9 8 7 9 . 3 6 13 1 3 0 3 . 9 5 3 0 7 8 0 . 0 0 1 3 . 4 5 0 . 7 7 9 7 5 8 6 6 7 6 1 . 0 0 3 3 3 3 1 7 5 9 8 . 6 5 14 1 3 3 7 . 1 7 1 9 3 0 6 . 2 2 1 0 . 3 9 0 . 8 6 4 7 5 7 6 9 1 2 2 . 5 0 3 9 4 7 5 7 0 6 9 . 0 9 15 1 2 2 2 . 4 0 7 7 3 0 1 . 0 4 2 2 . 74 0 . 5 0 9 6 8 5 6 7 0 6 6 . 0 0 5 1 3 4 4 5 5 3 1 0 . 6 8 16 1 2 1 7 . 2 7 3 5 8 5 6 0 3 7 1 5 . 56 0 . 7 4 8 9 7 3 6 4 2 2 8 . 5 0 6.4642367 1 2 . 5 2 17 1 1 7 7 . 7 2 4 3 1 6 6 . 6 2 1 7 . 6 4 0 . 6 1 3 5 5 1 6 1 2 9 4 . 0 0 5 4 1 1 4 6 9 7 1 2 . 0 0 18 1 2 6 0 . 6 0 2 1 1 2 0 . 8 4 11. 53 0 . 4 6 8 4 7 9 6 6 0 7 8 . 5 0 3 1 7 0 6 9 2 3 8 . 5 2 19 1 3 5 3 , 6 7 2 7 7 8 2 . 7 9 1 2 . 3 1 0 . 6 6 4 0 4 3 6 5 7 0 9 . 5 0 9 3 2 1 2 8 8 0 1 4 . 6 9 20 1234o 72 3 5 6 6 5 . 7 7 1 5 . 3 0 0 . 7 8 8 0 0 8 5 6 2 5 5 . 5 0 3 0 8 9 3 4 0 1 9 . 8 8 21 1 4 0 8 o 38 23 26 5 . 86 1 0 . 8 3 0 . 8 1 9 4 3 8 7 2 8 1 4 . 5 0 6 5 6 0 7 5 5 3 1 1 . 1 2 22 1 5 3 1 . 4 7 4 0 0 2 1 . 5 2 1 3 . C6 0 . 6 1 8 9 5 4 7 5 4 8 3 . 0 0 1 3 0 8 1 4 9 3 8 1 5 . 15 2 3 1332c 67 5 0 6 1 8 , 5 4 1 6 . 88 0 . 9 0 6 6 1 6 6 7 0 3 9 . 5 0 8 7 7 9 4 1 4 4 1 3 . 9 8 24 1 3 7 9 . 4 2 1 1 1 7 0 7 . 0 7 2 4 . 2 3 0 . 6 3 2 1 3 8 6 8 2 0 8 . 5 0 1 6 8 8 8 2 5 2 5 1 9 . 0 5 25 1 2 7 7 . 6 7 5 1 2 5 8 . 5 9 1 7 . 72 0 . 5 9 0 0 0 6 7 1 7 4 8 . 0 0 7 5 6 9 2 7 3 3 1 2 . 13 26 1 2 3 2 . 2 0 2 9 2 6 1 . 1 6 1 3 . 88 0 . 5 9 2 2 2 0 6 3 5 5 0 . 5 0 5 5 5 7 2 6 7 0 1 1 . 7 3 TABLE 27C P O P U L A T I O N PARAMETERS FOR DATA OBTAINED FROM I F S ' -GROUPED IN 8C» S SET YBAR YVAR YCV% RHO X BAR XVAR XCV£ 1 1 4 5 5 . 17 2 5 6 2 1 . 9 2 11. 04 0 . 8 4 6 8 0 0 7 1 6 6 3 . 5 0 5 5 7 4 2 6 0 1 1 0 . 4 2 2 1 4 5 0 . 0 6 3 3 1 6 3 . 5 5 1 2 . 5 6 0 . 7 7 2 0 4 0 7 1 7 1 8 . 5 0 6 4 4 2 1 6 3 5 1 1 . 19 3 1 5 0 8 . 1 3 4 1 6 3 2 . 5 8 1 3 . 53 0* 3 8 7 7 3 3 7 2 6 3 9 . 5 0 7 8 2 2 4 3 2 0 1 2 . 1 8 4 1 2 3 4 . 5 0 1 6 5 2 3 3 . 5 5 32.93 0.059290 6 5 7 5 2 . 5 0 6 4 2 9 8 0 1 1 1 2 . 20 5 1181.89 5 6 3 7 8 . 3 4 2 0 . 09 0 . 5 3 8 o 7 9 6 3 3 5 4 . 7 5 6 3 0 6 7 8 1 2 1 2 . 5 4 6 1 2 9 8 . 9 0 2 5 9 9 1 . 3 9 1 2 . 41 0 . 7 3 6 4 9 9 6 6 C 2 4 . 0 0 4 4 9 1 7 7 9 8 1 0 . 1 5 7 1 3 2 0 * 5 6 2 5 3 1 8 . 8 8 1 2 . 0 5 0 . 8 0 9 6 3 6 6 7 9 4 1 . 7 5 3 7 7 9 2 9 8 8 9 . 0 5 8 1 2 1 9 . 84 56584.67 1 9 . 5 0 0 . 5 8 6 9 3 9 6 5 6 4 7 . 2 5 5 9 9 9 9 7 5 8 11.80 9 1 2 1 9 . 1 6 3 3 8 6 0 . 4 0 15.09 0 . 5 9 2 0 4 7 6 3 6 8 6 . 2 5 4 8 6 2 5 4 7 8 1 0 . 9 5 10 1 2 9 4 . 2 0 3 5 2 6 0 . 7 6 1 4 . 5 1 0 . 7 1 2 4 6 7 6 C 9 8 2 . 5 0 8 4 3 9 1 9 3 5 1 5 . 0 6 11 1 4 6 9 . 9 2 3 5 4 3 1 . 8 9 12. 81 0.688780 7 4 1 4 8 . 7 5 9 9 9 8 8 1 9 2 1 3 . 4 9 12 1 3 5 6 . 0 5 81709.20 2 1 . 0 3 0 . 7 2 2 3 0 3 6 7 6 2 4 . 0 0 1 2 8 6 7 6 6 9 3 1 6 . 7 7 13 1254 .94 4 0 7 7 6 . 4 7 1 6 . 0 9 0 . 5 7 3 7 5 3 6 7 6 4 9 . 2 5 8 2 4 2 6 7 1 8 13 .42 a value of 1.2727, which indicated that the variance of y w a s almost proportional'to X j , thus p a r t i a l l y j u s t -i f y i n g the use of the ratio-of-means estimate, r^ , which i s used in weight s c a l i n g . The determination was based on the l a s t eight classes of Table 29, but i d e n t i c a l r e s u l t s should be expected from the l a s t 11 classes of Table 30. 71. TABLE 28: FREQUENCY DISTRIBUTION (5 CLASSES) FOR IFS SCALE DATA X-CELL LIMITS FREQ XTOTAL XVAR YTOTAL YVAR ? RHO (lbs.) (lbs.) ( l b s . ) 2 (c.ft.) ( c . f t . ) 23540.0 - 37996. 1 3 79980 14590696 2598 287959 -.1386 37996.2 - 52452. 3 3 3 1618040 9677278 3 2776 11737 0.6919 52452.4 - 66908. 5 454 27660820 163066 81 552124 24761 0.2142 66908.6 - 81364. 7 495 35958150 15738667 695978 54916 0.3953 81364.8 - 95820. 9 75 6412230 14727967 12 5664 48001 0.4004 TABLE 29: FREQUENCY DISTRIBUTION C10 CLASSES) FOR IFS SCALE DATA X-CELL LIMITS FREQ XTOTAL XVAR 0 YTOTAL YVAR 0 RHO (lbs .) (lbs.) ( l b s . ) 2 (c.ft.) ( c . f t . r 23540.0 - 30768.0 2 49060 1960056 1974 48 8072 **** 30768.1 - 37996.1 1 30920 0 624 0 37996.2 - 45224.2 3 125560 9913717 2649 941 2 0.695 5 45224.3 - 52452.3 30 1492480 4131334 30127 1092 0 0.7186 52452.4 - 59680.4 168 9492240 46705 43 199014 18088 0.2424 59680.5 - 66908.4 2 86 18168580 4855893 353110 2 782 9 0.1210 66908.6 ~ 74136.5 3 23 22666880 4261033 438145 45049 0. 3107 74136.6 - 81364.6 172 132912 70 4369270 257833 60482 0.274 8 81364.7 - 88592.6 57 4768390 3171990 93409 50418 0. 2929 88592.8 - 95820.7 18 1643840 6339871 32255 23980 0.343 8 TABLE 30: FREQUENCY DISTRIBUTION (15 CLASSES) FOR IFS SCALE DATA X—CELL LIMITS FREQ XTOTAL XVAR^ YTOTAL YVAR 9 RHO (lbs .) (lbs.) ( l b s . ) 2 (c.ft.) { c . f t . r 23540.0 — 28358.6 2 49060 1960056 1974 48 807 2 **** 28358.7 - 33177.4 1 30920 0 624 0 33177.5 - 37996.1 0 0 0 0 0 *** * 37996.2 - 42814.8 2 81140 9945 784 1768 18818 **** 42814.9 — 47633.6 8 371140 797522 7081 2 60 5 -. 0031 47633.7 - 52452.3 23 11657 60 1469816 23927 7737 0.4726 52452.4 - 57271.0 93 5099100 1744978 107873 15158 0.1776 572 71.1 - 62089.7 166 9931820 1822225 201509 1643 8 0.0407 62089.8 - 66908.5 195 12629900 2009 553 242742 34303 0.0794 66908.6 — 71727.2 233 16108260 1827974 308424 54335 0. 1948 71727.3 - 76545.8 164 12126420 1857385 235066 44492 0.0044 76545.9 - 81364.5 98 7723470 1799837 152488 34694 0.1381 81364.6 — 86183.2 52 4333700 2292740 84847 50639 0.3013 86183.3 — 91001.9 16 1420150 2135695 27741 27912 0.0042 91002.0 - 95820.6 7 658380 2695591 13076 22853 0.1137 5.0 RESULTS FOR CASE STUDIES 1-21 AND A1-A3 5.1 General introduction Tables 31-34 and 66-72 represent sample outputs for the generated cases and the IFS scale data res p e c t i v e l y . 'SS' symbolizes sampling schemes. S t a t i s t i c s under 'RATIO WITH P' or 'USING EST. P=1.2727', that i s , R, YBAR and SYBA.R indicate the r a t i o , the mean of Y estimat ed through r a t i o estimation with the given p value, and the standard error of YBAR, respe c t i v e l y . The s t a t i s t i c s under 'APPROX RATIO' represent those obtained using the ratio-of-means (p=l) estimate. SYBA and YBAR under 'NO RATIO' are the s t a t i s t i c s obtained without the use of r a t i o estimation, implying no use of a u x i l i a r y inform-ation. RHO i s the sample c o r r e l a t i o n c o e f f i c i e n t . The discussion following was not r e s t r i c t e d to the afore-mentioned tables but to a great number of outputs. The tables, however, indicated most of the general r e s u l t s to be reported, and should therefore serve as suitable references. 5.2 Case studies 1-21 and A1-A3 5.2.i Relationship between RHO and standard errors Examination of the r e s u l t s for a l l the d i f f e r e n t sample .73. TABLE 31 CASE 1- EPF ICIENCY OP SAMPLING SCHEMES ( S S ) ;P»- 1.0 SI2E="50 (SAMPLE OUTPUTI SS R A T R 1 0 b I T YBAR H P SYOA.R A P P . R R 0 X R YBAR A T I 0 SYOA.R N O R A T SYOA. I 0 Y BAR RHO CR .1033 36.23 1.0 5 80 .0991 34.7 3 1.0629 20.37 34.77 .003966 RS . 1022 35. 13 1.8760 .0985 34.53 1.0753 18.20 35.70 .696950 M5 .0909 34. t e 1.6075 . 0964 33. 00 1.8874 17.69 33.30 .600660 HIO . 1C29 36.08 1.5551 .1025 35.92 1.5548 15.09 34. 63 .728695 H15 . 1034 36.26 1.5871 .1013 35.52 1.587Q 17.09 36.30 .773135 M2Q .1056 37.03 1.7723 .1071 37.56 1.7722 15.61 4C.93 .616084 2,5 . 1026 35.98 1.5598 .1011 35.44 1.5588 16.71 36.50 .760249 2,10 .0933 32.69 1.S536 .0922 32.34 1.9542 20.74 30.40 .753232 2,15 • C946 33.17 2.0645 .0958 33.5 8 2.C661 20.02 34. 90 .696912 3,5 .0986 34.55 1.7066 . 1022 35. 82 1.7106 21.40 37.97 .839356 3,10 .1.060 37. 16 2.0620 .1073 37.63 2.0629 22.92 39.03 .777666 3,15 . 101C 35.41 1.6305 .1005 35.23 1.63 02 19.32 36.13 .8C6861 4.5 .lOO'i 37.99 1.5605 .1C54 36.95 1.5638 19.41 37.00 .844065 5,5 .0963 33.77 1.6541 .0926 - 32.46 1.6569 18. 16 35. 53 .617851 5,10 .1039 36.41 1.5 2 84 .1060 37.15 1.5301 17.74 42.40 .8078C3 5,15 .1003 35.17 1.6644 .1013 35.5C 1.6658 19.85 39.40 .812470 6,5 . 1001 35. C8 1.4051 .0976 34.21 1.4036 16.04 31.90 .797664 6,10 .0912 31.99 1.6407 .0883 3 0.95 1.6424 17.72 27.43 .772828 6,15 . 0901 34.39 1.8645 .0949 33.27 1.8630 21.88 29.93 .8C614C 1.5 . 1C62 37.22 1.6830 .1029 36.06 1.6855 10.60 37.23 .604715 1,10 .1021 35.88 1.4053 .1005 35. 22 1.4066 17.11 37.80 .829478 1,15 .1101 38.61 1.7695 .1104 38.70 1. 7695 17.47 39.50 .705338 1X5 .0924 32.38 2.0046 . 0910 31.89 2.0845 16.01 32.46 .329664 1X10 .1035 36.28 1.4025 . 1024 35.91 1.4027 15.90 29.00 .787923 1X15 . 0090 31.20 1.4485 . 0897 31.46 1.4479 10.25 32.77 .831656 0X5 . 0862 30.23 2.7440 .0860 30.15 2.74 41 2C. 70 29.07 .223738 0X10 . 1050 37. C9 1.0865 . 1040 36.47 1.8868 16.68 33.13 .623947 0X15 .0953 33.42 1.7711 .0955 33.46 1.7711 18.07 28.90 .753900 REMARKS : SYBA.R.YDAR.SYBA. ARE IM Tut S A M E UNITS (SEE TABLE I) 74. TAOLE 32 CASE 11- EFFICIENCY OF SAMPLING SCHtM !P = 0.5 SI/*-E=l00 ( SAMPLE OUTPUT I R A T 1 0 W I T H P A P P R O X R A T I 0 N O R A T I 0 SS R YEAR SYDA.R R ' YBAR SYBA.R SYCA. YBAR PHC CR .2446 e 7 . i e 1.4037 .2448 87.25 1.4027 54.28 80.32 .763222 RS .25C5 89.30 1. 5209 .2516 89.67 1.5218 53.72 88.34 .714928 M5 .2488 88.70 1.6937 .2500 89. 11 1. 6945 59.22 89.69 .705749 MIO .2549 90.85 1.7106 .2558 91.17 1.7112 55.42 90.09 .643553 H15 .2505 89.29 1. 4526 .2511 89.49 1. 4527 55.92 88.71 .761660 H20 . 2375 84.66 1.5954 .2386 85.05 1.5964 49.50 66.48 .610684 2.5 .2390 £5.19 1.5683 .2393 85.29 1.5684 55.77 86.24 .72779E 4,5 .2535 90. 36 1.5648 .2543 90.65 1.5654 61.29 93.01 .773662 5,5 .2485 £8.57 1.4939 .2495 88.95 1.4949 49.38 99.03 .671462 5,10 .2368 C4.43 1.7254 .2384 84.99 1.7275 62.47 100.01 .726311 6,5 .2609 93.00 1.5409 .2619 93.34 1.5415 56.62 86.67 .736962 6,10 .2538 90.47 1.4644 .2549 90. 84 1.4651 65.80 76.75 .823057 1,5 .2515 89.65 1.5093 .2529 90.14 1.51C8 52.77 90.00 •7138C5 1.1C . 247C 88.04 1.6030 .2486 80.62 1.6049 51.58 88.39 .652748 1,15 .2464 87.83 1.4756 .2474 88.20 1.4766 5C.90 87.21 .701155 1X5 .2717 56.85 1.3783 .2700 96.24 1.3769 41.60 77. 51 .579253 1X10 .2501 e9.16 1.3908 .2504 89.24 1.3909 48.04 79.08 .6902C4 1X15 .2564 91.41 1.6066 .2568 91.54 1.6067 56. 84 89.01 .708243 DX5 .2751 se.cs 1.3082 .2755 98.21 1.3082 35.70 53.30 .4C2910 DX10 .2560 91.26 1. 5392 .2563 91.37 1.5393 45.96 86,49 .550027 DX15 .2439 86.94 1.5336 .2452 87.4C 1.5349 52.25 88.76 .692975 REMARKS : SY3A.R,YBAS,SYDA. ARE IN THE SAME UNITS (SEE TABLE 1) 7 5 . TABLE 33 CASE 15- EFFICIENCY OF SAMPLING SCHEMdS (S S) ;P = - I . O SIZE=90 (SAMPLE OUTPUT) R A T I C H I T H P A P P R 0 X R A T I 0 N 0 R A T I 0 SS R YBAR SYBA.R R YEAR SYBA.R SYBA. YBAR RhO CP .3964 140.11 0.9817 .3979 140.64 0.9019 47.40 142.13 .957986 RS .4017 141.97 1.1622 .4066 143.73 1.1657 47.79 145.64 .949637 M5 .4035 142.64 C.5858 .4041 142.83 0.9059 42.41 143.41 .945800 M10 .3997 141.27 C.9418 .3997 141.29 0.9416 40. 18 146.44 .961832 H15 .4009 141.65 C.9111 . 4028 142.36 0.9112 44.35 144.11 .9 591e 5 K20 .4044 142.95 1.2005 .4080 144.22 1.2014 47.81 139.53 .940220 2.5 .4009 141.70 1.0872 .4019 142.07 1.0074 50.86 136.57 .954597 4.5 .4009 141.70 1.C076 .4022 142.17 1.0077 55.93 141.41 .967996 5,5 .3962 14C.03 1.3001 .3953 139.72 1.3003 53.44 144.87 .940504 5, 10 . 3966 140.18 1.0176 .3978 140.60 1.0179 65.31 156.79 .976161 6,5 .4062 143.56 i - 2 4 8 1 . 4123 145.73 1.2503 46.19 129.19 •9349C9 6.1C .3588 140.96 1.3889 *4103 145.03 1.4013 60.00 121.03 .956796 1.5 „4026 142.30 1.C792 . 4044 142.94 1.0794 49.21 141.46 .952642 1,10 .3970 140.34 1.C590 .3902 140.75 1.C55C 53.31 141. 52 •9608C4 1.15 .4022 142.16 1.0207 .4025 142.27 1.0206 54.18 141.62 .964613 1X5 .4277 151.18 1.6930 .4304 152.13 1.6930 32.00 108.68 .673185 1X10 .4012 141.81 1.1226 .3995 141.20 1.1228 44.06 118.77 .935115 1X15 .3981 140.71 1.2147 . 3979 140.64 1.2147 52.51 136.04 .946129 DX5 .4322 152.77 1.5860 .4355 153.93 1.586C 27.77 102.88 .605759 DX10 .4051 143.17 1.0460 .4047 143.03 1.0459 38.86 123.07 .926208 0X15 .3933 139.01 0.9611 .3961 140.02 0.9618 48.79 141.76 .963701 REMARKS : SYBA. R,YBAR,SYBA. ARE IN THE SAME UNITS (SEE TABLE 1 ) 76. TABLE 34 - CASE A l - FFFT CIENCY OF SAMPLING SCHEMES I S S J . c S T . P=1.B008 SIZE = 140 (SAMPLE OUTPUT) ss R A T R 1 0 W I T YBAR H P SYBA.R A P P R R 0 X R YBAR A T I 0 SYBA.R N O R A T SYBA. I 0 YBAR RHO CR .1260 43.95 2.1986 .1255 43. 06 2.1919 23.38 42.40 .382278 RS .1252 42.96 2.6942 .1226 42.07 2.6898 28.98 42.20 .381052 M5 . 1220 41.06 2.1101 .1194 40.97 2.1025 22.38 42.20 .282209 M10 .1216 41.73 1.9554 .1199 41. 15 1.9526 21.76 41.40 .442465 M15 .1103 37. EE 1.8160 .1070 36.73 1.8018 17.74 36.27 .25432C M20 .1150 39.49 2.0058 .1125 38. 6C 1.9972 20.38 39.67 .304101 2.5 .1320 45.31 3.1280 .1313 45.07 3.1279 35.30 49.71 .463719 2.10 .1070 36.72 2.4432 .1118 38.36 2.4177 33.07 40.36 .700514 2.15 .1209 41.49 2.4432 .1158 39.7 5 2.41E2 28.39 42.47 .543641 3.10 . 1207 41.45 2.9984 .1167 40.06 2.9909 37.06 43.44 .550712 3,15 .1446 49.63 3.3490 .1320 45.30 2.2574 37.56 48.61 .520928 4,5 .1030 35.37 2.3628 . 1062 36.44 2.3464 3C.29 28.94 .662138 5,5 .1185 40.67 2.1246 .1203 41. 3C 3.1212 35.20 49.82 .472564 5,10 .1244 42.70 3.2696 .1211 41.58 3.2711 39.50 53.55 .568535 5,15 .1294 44.41 2.3237 .1265 43.44 3.3221 39.68 57.06 .547761 6.5 .1108 30.03 2.3823 .1078 37.00 2.3756 25.25 36.14 .356286 6.10 . 132C 45.3 2 2.5194 .1293 44.40 2.5148 3C.59 41.04 .5716E3 6,15 .1233 42.31 2.4013 .1132 38.86 2.3167 23.96 36.10 .414090 1.5 .1158 39.74 1.8958 .1124 38.59 1.8843 19.32 37.77 .316169 1.10 . 1242 42.63 2.0971 .1208 41.45 2.0883 23.14 41.11 .444365 1.15 .1093 37.52 1.8832 .1073 36.82 1.87S1 19.92 36.72 .370537 1X5 . 1292 44. 34 1.658'. .1281 43.98 1.6581 17.16 30.30 .262745 1X10 .1219 41.85 2.5686 .1184 40.64 2.55E7 25.25 40.6-9 .151569 1X15 .1182 40.58 2.0145 .1163 39.93 2.0111 22.44 41. CO .451422 DX5 .1454 49.92 J.5489 .1449 49.72 2.54 8 8 26.31 38.15 .214433 0X10 .1214 41.67 1.9394 .1187 40.74 1.9314 19.57 4C.42 .277751 0X15 .1238 42.49 2.0089 .1197 41.09 1.9919 21.43 42.79 .420167 REMARKS : S Y B A . R , Y B A R t S Y B A . ARE IN THE S4 ME UNITS (SEE T A B L E 11 sizes and for a l l cases, indicated that irrespective of the sample c o r r e l a t i o n c o e f f i c i e n t , there was consider-able gain in precision for r a t i o estimation procedures over non-ratio estimation. In some cases, the standard error for r a t i o estimation was about one-tenth (1/10) that of corresponding non-ratio estimate, for RHO greater than 0.70. This amount was found to reduce to about one-f i f t h (1/5) for RHO less than 0.40. However, the range was wide enough to indicate a clear preference for r a t i o estimation. No par t i c u l a r sampling scheme was found to y i e l d p a r t i c u l a r l y low or high c o r r e l a t i o n c o e f f i c i e n t s (RHO). There was also no discernible trend i n RHO for the d i f f e r -ent number of frequency classes or groups taken within each sampling scheme. However, classes or groups that gave higher RHOs usually gave smaller standard err o r s . It also did not follow that a sampling scheme yie l d i n g samples with low RHOs necessarily gave higher standard errors than another with s l i g h t l y higher RHO. A possible r e l a t i o n s h i p between the way the observations or samples were drawn (that i s , the sampling schemes) and the standard errors was therefore expected. 78. 5 . 2 . i i Estimates of Rp versus Rj_ for same samples No advantage of one over the other in terms of the r e s u l t i n g standard errors from the same samples was observed. As summarized below, there were cases where r n was less than or greater than but both were f a i r l y XT close to the true r a t i o used in the data generation. (Designations <= less than, greater than, >j> = far greater than, > = greater than or equal to, and so on.) a) CS 1: rp ^  .10, r-^ .10 but r ^ and rp f a i r l y close. b) CS 2: r < both f a i r l y close to 0.10. c) CS 3: r < r.. , but r p closer to 0.10 than r ^ . P J- XT d) CS 4: r p < r , ^  > 0.10. e) CS 6 and CS 7: r ^ ^ r ^ , r 1 ^ > 0 - 1 0 ' v i c i n i t y of 0.13 + f) CS 8 and CS 9: r p and rj_ close to each other and to 0.25. No s p e c i f i c trend observed. g) CS 10: rp< rj_' Except for estimates from DXS. and/or 1X5, both rp and r-^ were f a i r l y close •to 0.25. h) CS 11: i d e n t i c a l and close to 0.25. i) CS 13: r >r^, both greater than 0.25. j) CS 14: r ">r^. Generally as for CS 13, except that for some small sized samples, schemes Types 1, 5, 6 gave r e l a t i v e l y small estimates averaging 0.23 for both rp and r]_. k) CS 15: F a i r l y close to each other, and to 0.40. Estimates for DX5 and 1X5 higher than 0.40. 1) CS 16, CS 18, CS 20: As for CS 15. m) CS 17 and CS 21: F a i r l y close to each other. Estimates for schemes DX5, DX10, 1X5 and 1X10 were higher than for other schemes. For the cases A1-A3 where the p values were estimated, the following were found to indicate the general r e l a t i o n -ships. n) CS A l : r p> r]_# r-^0.11. For DXZ and IXZ schemes, the r a t i o s were far greater than 0.11. o) CS A2 : r p » r-]_ b u t r2_ close to 0.25. p) CS A3: Both r p and r ^ f a i r l y close to each other, and to 0.40 (0.38 - 0.43). As expected, estimates of YBAR could be biased for both formulae r e l y i n g on r p and rn_ res p e c t i v e l y . However, general consistencies in the rankings of the standard errors were found. That i s , i f the standard errors were to be ranked using either estimates r e l y i n g on r p or r]_, the same r e s u l t s would be expected. It was also concluded that r-^ could be considered f a i r l y robust since i t did not seem to be affected by either sample size or p. Subsequently, the remaining comparisons were made with the estimates under the 1APPROX RATIO' columns. 5 . 2 . i i i Screening for 'best' sampling schemes The general approach used in the sel e c t i o n process revolv-ed around a ranking procedure based i n i t i a l l y on the standard er r o r s . A rank value of 13 was assigned to the scheme, for a s p e c i f i c sample size and twenty r e p l i c a -tions, with the least average standard error; 12 for the next smallest and so on. This implied that only the f i r s t 13 schemes with the smallest standard errors, were assigned rank values above zero. A l l others were given values of 0. The rank values were then t o t a l l e d for a l l the sample sizes and r e p l i c a t i o n s used. The f i r s t s i x (or seven, i n case of a t i e ) schemes with the highest t o t a l rank values were then advanced to the next screening process. Tables for the i n i t i a l stage are 35A-58A ('A' tables), and those of the corresponding second stage as 35B-58B ('B' t a b l e s ) . The l a t t e r stage involved the computation of the mean deviation (YBAR - true YBAR) to check for magnitude and consistencies in over- or under-est imation. The use of absolute standard errors, as opposed to ranks, may be considered by some readers as a more appropriate procedure in the 'A' screening process. However, standard errors depend to some extent on the sample size and unless the d i f f e r e n t errors are weighted to r e f l e c t such dependence, di r e c t absolute comparisons may be questionable. I t was, therefore, f e l t that the use of ranks, even though i t could camouflage the true differences i n standard errors between schemes, was a better or more appropriate approach than the use of absolute values. In discussing the r e s u l t s , the case studies were broken up into four segments: CS 1-7, CS 8-14, CS 15-21 and CS A1-A3. a) CS 1-7: Tables 35A-41A and 35B-41B Schemes based on p r o b a b i l i t y increasing or decreasing with the magnitude of the a u x i l i a r y variable (DXZ, IXZ) appear-ed i n a l l of the 'B' tables except i n 35B. As indicated by t h e i r magnitudes i n the 'A' tables, schemes DX5 and 1X5 consistently gave very high rank value in d i c a t i n g t h e i r usually smaller standard e r r o r s . These r e s u l t s indicated that i f the samples required for r a t i o estimation were selected so as to give preference to the smaller or larger sizes of the a u x i l i a r y variable in a graduating fashion, these should give minimum variances. Ideally, precision which i s a measure of the c l u s t e r -ing of the samples around th e i r mean, i s not the only a t t r i b u t e to look for from an estimate. An unbiased estimator i s also i d e a l , but as shown i n the 'B' tables, there was a tendency for schemes DX5 and 1X5 to over-estimate the true mean of interest consistently. Although these were random deviations, the same trends were obtain-ed for four additional computer runs of the same sample sizes and number of r e p l i c a t i o n s . For the DXZ and IXZ schemes, the more the frequency classes, say as for DX10 or 1X15, the smaller were nthe" 'deviations. Thus '&£f ?the^pc5pula t i o n were to be large enough for greater number of frequency c l a s s i f i c a t i o n , schemes involving 1X10, DX10 and higher Z's could be recommended for the cases. This would not necessarily imply that 1X10, DX10 and higher Z schemes could lead to minimum-variances. Empirically, the huge random deviations for the minimum-variance schemes were not found to depend on the sample s i z e . The most appropriate schemes for CS 1-3 were CR, M15, and the Type 1 schemes. For these cases, p was less than or equal to zero. For positive p values, most appropriate schemes were 1X10, DX10 and the lower classes of the Type 6 scheme. In cases where the MZ schemes were present, they usually gave smaller average deviations than the others. The JMZ schemes would therefore be preferable, unless the r e l a t i v e l y large deviations in the minimum variance schemes DX5 and 1X5 could be corrected. b) CS 8-14: Tables 42A - 48A and 42B - 48B As obtained in the preceeding subsection a), top honours for the schemes applied to CS 8-14 were taken up by those belonging to the IXZ or DXZ schemes for positive p. Other-wise (p^-O.O), schemes of Types 2, 4 and 5 for only 5 classes dominated with the RS and JMZ types. On minimum deviation, and for negative p, the equal p r o b a b i l i t y schemes and RS were found to be the best. For positive p, and on the basis of the random deviations, schemes of Type 6, MZ, DX10 and 1X10 were found to be better than DX5 or 1X5. I t was observed i n both subsections a) and b) that in most cases where DX5 and 1X5 were selected, they usually picked up values of 11, 12 or 13 ^Tables 38A-41A, 47A-48A*], This might indicate that there was probably no preference in using one or the other. Given the constraints of the available population siz e , i t may be necessary to i n v e s t i -gate for some other population, the optimum number of classes for 'reasonably small' deviation and minimum variance DXZ or IXZ schemes. c) CS 15-21: Tables 49A - 55A and 49B - 55B It was apparent that for negative p, neither schemes DXZ nor IXZ were applicable. Schemes found to y i e l d best r e s u l t s included Types 1, 4 and/or 5. Such schemes, including 1X15, also resulted i n f a i r l y small standard errors and absolute random deviations of the estimate of the mean of y from the true value. For positive p, excluding p = 2.0 i n Tables 55A, 5 5B, a l l schemes advanced into the B tables gave similar r e s u l t s . For p = 2.0, Type 6, 5 gave smaller deviations than the others, even though i t was the s i x t h of the s i x schemes in the f i r s t screening. Schemes DX5 and 1X5 behaved as for previous cases. In general, however, the i n d i v i d u a l rank values for several schemes were independent of sample s i z e . d) CS A1-A3. Tables 56A - 58A and 56B - 58B In a l l three cases, the struggle for advancement into the 'B' tables by the MZ, DXZ and IXZ schemes was evident. If the f i n a l t o t a l rank values for the i n d i v i d u a l CS.1-21 were to be re-ranked and then grouped to correspond with CS A1-A3, the following would be obtained: 85. Combined CS 1-7 : 1X5(1); 1X10(2); DX5(4); DX10(3); CR(6); 6,5(5) Combined CS 8-14: 1X5(4); 1X10(3); DX5(2); DX10(1); RS(6); 2,5(5) Combined CS 15-21: 1X5(5); 1X10(2); DX5(1); DX10(3); CR(6); 6,5(4) The numbers in the brackets represent the order of the maximum t o t a l rank values. That i s , for combined CS 1-7, scheme 1X5 had the largest t o t a l rank value (or ov e r a l l smallest standard e r r o r ) , followed by 1X10 and so on. Comparing Tables 56A-58A to the above summary, indicated the consistent behavior of the IXZ and DXZ schemes. It was also r e a l i s e d that for the aggregated cases, unless the magnitude of the deviation for the IXZ and DXZ schemes was not too d i f f e r e n t from those of the MZ schemes, the IXZ or DXZ schemes may have to be dumped i n favour of the MZ schemes. S p e c i f i c a l l y for CS A l , schemes MlO and DX10 gave smaller deviations than the other four selected with them in the 'B' table. For CS A2, schemes 1, 15 and M20 were o v e r a l l 'best', while from Table 58B and for CS A3 the 'best' minimum deviation scheme was M20. F i n a l l y , i t may be mentioned that i f the selections had been based on mean deviations f i r s t , and then the standard errors, d i f f e r e n t r e s u l t s might have been obtain-ed. TABLE 35A CASE l : RANKS BASED ON STANDARD ERRORS FOR ALL SCHEMES APPLIED SIZE CR RS M5 MO M15 M20 2,5 ,10 ,15 3,5 ,10 ,15 4,5 5,5 ,10 ,15 6,5 ,10 ,15 1,5 , 10 ,15 1X5 10 15 0X5 10 15 30 _ 8 5 _ 7 — _ - - 4 6 2 9 1 12 3 - - 11 - - 13 9 - - -37 12 - 11 - 5 - - 3 - - 13 - 8 10 9 - 6 - 4 - 1 - 2 - - 7 — 40 - - - 7 - - 12 4 11 6 - - 5 - 9 - 10 3 1 13 - - 8 - - 2 - 3 57 - - - 4 2 a - 13 - 1 - - - 3 6 11 - 5 - 9 7 12 - - 10 - -60 7 - - C 8 3 - - 11 12 - 10 - - 1 - - - - 6 13 4 2 - - 9 — 7.5 11 - - 4 - 12 - - 5 - 10 - 8 1 . 3 - - 7 - 9 - 13 - 6 - — 2 -80 - 11 - - 10 8 5 - - 7 - - 3 - - 4 12 - - 6 1 - 2 - - - 13 9 90 13 - - - 11 - - - 9 - - 12 - 2 1 - 4 - - - 8 10 3" - 6 7 - 5 98 - 12 - - 7 11 5 - - 13 4 6 - 8 - - 9 - - - - - 2 1 - - 3 10 TOT 43 23 11 28 43 50 28 20 36 39 27 32 30 16 39 25 47 24 1 47 27 49 19 24 15 19 34 27 TABLE 36A CASE 2: RANKS BASED ON STANDARD ERRORS FOR ALL SCHEMES APPLIED SIZE CR RS ' M5 MO ^ 13 M20 2,5 ,10 ,15 3,5 ,10 ,15 4,5 5,5 ,10 ,15 6, 5 , 10 ,15 1,5 , 10 ,15 1X5 10 15 DX5 10 15 30 _ 2 13 _ 4 _ 12 1 - - - - 9 - - 5 10 8 3 - 7 - 9 6 - - 11 -40 - - 7 - - - - 6 - 9 - - 11 12 - 5 - 13 3 10 - 2 - 4 8 - 1 60 3 11 9 - 3 - - - 4 - - 7 6 10 - - 12 - - 5 - - 2 13 - ' - - 1 80 5 - - - 13 10 3 2 - - • - • - 4 - 7 - - - 1 12 - 11 9 8 • - - 6 -90 12 5 - - 3 4 - - - 1 - 2 - 9 11 - - - 8 6 - 13 - 10 - - 7 95 5 .8 - - 10 6 - - 11 - - - 2 - 13 - - - 3 1 9 4 - 12 7 - 5 100 5 - 6 - 1 8 - - - - - - 4 - 1 - 13 - - 9 - - 12 - 11 3 10 -140 12 - - 8 2 6 5 - - - 10 I 7 4 13 - 11 - 9 178 9 8 - 5 11 1 - - - - - - 4 6 - - 12 3 - - - 7 -io 13 - - 2 — TOT 56 34 35 13 47 29 • 15 9 15 9 1 7 32 34 35 21 47 24 20 56 20 31 74 40 45 18 32 23 ** ** ** ** ** ** TABLE 37A CASE 3: RANKS BASED ON STANDARD ERRORS FOR ALL SCHEMES APPLIED SIZE CR RS M5 MO M15 rV20 2,5 ,10 ,15 3 , 5 ,10 ,15 4,5 5,5 ,10 ,15 6,5 , 10 ,15 1.5 , 10 .15 1X5 10 15 DX5 10 15 30 10k _ — 3 _ _ lot - 2 - 4 - - - 8 9 6 - 13 5 - - - 12 7 1 40 12 - - - - 7 1 13 - 3 - 9 - - - 8 - - 11 10 - 6 - 5 - 4 2 60 4 9 1 - - 2 - - - - - - - 5 6 - 12 3 - - 8 7 - 13 - - 11 10 80 10 - I - - - - - - 3 11 - 6 - - - - - - 7 9 12 4 13 5 2 8 -90 - - 8 10 9 - - 6 - - - 1 - - - - 12 - - 11 3 - 7 5 2 - 13 4 100 10 - - 8 - 7 12 - - - 4 - - - - - 1 - - 3 5 11 - 13 9 - 6 2 120 12 IC 8 - 5 3 - - - - - - - 10 - - - - - - 1 7 6 12 4 2 9 -127 12 2 13 - 7 - - 1 - - 11 - - - - - 6 - - - - 4 8 10 - 9 3 5 133 9 8 1 5 - 13 1 - - - - - - 7 - - 4 11 - - 6 - - 10 12 - 3 -TOT 79i 29 22 23 21 35 14 20 10 i 3 31 i 19 22 6 0 51 23 6 32 55 46 31 76 37 25 64 24 ** ** ** ** *# ** TABLE 35B CASE 1 - DEVIATION FOR BEST SCHEMES (P=-1.0) ZE CR M15 M20 6 ,5 I t 5 I t 15 30 - 0 . 0 3 0 . 76 2 . 80 - 0 . 5 5 I. 30 3 . 94 37 0 .48 - 0 . 4 7 1.10 - 1 . 0 6 - 1 . 52 1. 9 7 40 0 .90 3.41 - 1 . 5 8 - 1 . 7 5 1 . *2 - 2 . 98 57 1 .74 - 0 . 4 4 2 .26 - 0 . 9 3 - 0 . 93 3 . 08 60 - 1 . 1 4 - 1 . 2 6 1.58 - 1 . 7 5 1 . 42 - 2 . 98 75 0.6 1 - 0 . 48 - 0 . 3 1 0 . 2 0 0 . 40 0 . 13 80 0 . 3 7 0 .50 - 1 . 3 3 - 1 . 81 0 . 49 - 0 . 04 90 - 0 . 8 9 0 .36 - 0 . 6 3 - 0 . 9 4 0 . 48 0. 23 93 1.3 8 - 0 . 3 6 - 0 . 74 0 .78 - 1 . 19 0 . 54 TABLE 36B CAS E 2 - DE VIATIOM FOR BEST SCH EMES (P =-0.5} • IZE CR Ml 5 6, 5 I t 5 1X5 0X5 30 1 .15 0 .06 2 . 8 7 4 . 14 23 . 94 13 .94 40 - 3 . 0 2 - 0 . 20 4 . 2 9 - 2 . 3 5 19 . 17 2 0 . 3 3 60 - 0 . 2 0 1.10 3 . 4 6 - 1 . 36 21 . 00 14. 34 80 1.45 - 0 . 7 0 1.76 1 . 03 21 .32 15 .23 90 - 1 . 3 2 3 . 86 2 .93 - 0 . 42 20 .3 5 14 .65 95 - 2 . 7 0 - 1 . 36 2 .31 - 0 . 54 15 . 04 13 .36 100 - 2 . 0 9 - 1 . 3 4 2 .98 - 0 . 02 21 . 43 17 .70 140 - 0 . 3 0 - 2 . 2 5 2 .39 - 1 . 33 19 .05 10 .9 5 173 0 .58 - 0 . 54 2.9.5 - 1 . 78 2 1 . 13 20 . 91 TABLE 373 CAS c 3 - ! DEVIATION FOR B EST SCH EMi_S CP = 0 .0 j SIZE CR 6 , 5 1,10 I t 15 I x i o 0X10 30 1.79 - 0 . 8 1 - 0 . 14 - 0 . 02 1 .33 1 .22 40 0 .41 0 .42 0 .48 1. 0 8 7. 71 3. 62 60 - 2 . 1 2 3 . 1 6 - 0 . 1 2 0 . 31 3.33 - 0 . 24 80 1.04 0 . 82 - 0 . 86 - 0 . 03 1 . 74 1.45 90 - 0 . 2 9 2 . 4 7 0 .75 1. 2 2 1 .39 - 0 . 03 100 2 .21 3 . 5 9 0 .46 1 . 40 2 . 6 3 - 0 . 7 3 120 0 . 1 7 0 . 11 - 0 . 2 7 - 1 . 13 1.62 - 0 . 1 6 127 - 0 . 2 2 3 .27 0 . 0 3 - 2 . 35 - 0 . 5 6 4 . 15 183 0 .65 3 .13 - 1 . 2 7 0 . 90 1 . 00 2 . 2 8 REMARKS: DEVIATIONS IN UNITS OF Y (TABLE 1 ) ; TWENTY REPLICATES PER S I Z E . TABLE 33A CASE 4: RANKS BASED ON STANDARD ERRORS FOR ALL SCHEMES APPLIED SIZE CR RS M5 flQ M15 M2C 2,5 ,10 ,15 3,5 ,10 ,15 4,5 5,5 ,10 ,15 6,5 ,10 ,15 1,5 , 10 , 15 1X5 10 15 DX5 3C - 13 5 - 7 2 8 1 - - - 11 - - - - - 10 - - 6 - 10 12 - 9 40 3 - - 2 1 - - 5 - . - - 7 4 - .9 - 6 10 11 12 13 60 9 6 - 3 1 1 2 - - -• - - - - - - 8 2 - 1 1 - 5 1 3 1 0 - 4 8C 2 - 1 1 4 - - 1 3 - 3 _ _ i _ 9 12 5 8 10 90 - - 1 9 2 4 - - - - - - - - - - 10 2 - 6 - - 12 13 11 7 100 - 2 - 6 - 1 - - - - - - - - - - 9 5 3 4 - - 2 1 0 7 1 3 120 - _ 2 f c - 4 - - - - - - - - - - 10 - 3 5 9 1 12 8 - 13 140 2 5 8 4 - - - - - - - - 6 - - - - 1 - 3 - 10 13 10 7 12 159 - 8 - 3 - 9 5 - - - - - - - - - - - 4 - 2 10 13 11 1 12 TOT 16 34 27 37 12 32 26 6 0 0 0 11 6 0 0 0 47 24 10 39 17 41 97 90 46 93 10 15 3 8 7 7 8 11 11 9 7 71 ** 6 5 8 • 7 6 32 TABLE 39A SIZE CR CASS 5: RANKS BASED ON STANDARD ERRORS FOR ALL SCHEMES APPLIED 3C 40 60 80 90 100 120 140 172 TOT 2 6 16 RS M5 MO M15 M20 2,5 ,10 ,15 3,5 ,10 .15 4,5 5,5 ,10 ,15 6,5 ,10 ,15 1.5 , 10 , 15 1X5 10 15 0X5 10 15 10 _ _ _ _ 8 1 9 5 3 — _ _ - _ 6 - 11 12 - 2 13 7 4 5 _ _ _ 6 — _ 7 - 3 - - - 4 13 10 - - - 11 9 8 12 - 2 _ 3 - - - 8 4 10 1 - - • - - 9 - 6 5 - - 12 7 - 13 11 — _ 4 11 _ _ _ 6 1 - - - - 9 10 7 - 2 5 12 - 8 13 • - — 3 9 6 - - _ - 11 10 8 - - - - - 4 - - - - 12 7 1 13 5 — 2 8 6 7 _ - 3 1 - - - - 4 11 9 5 - - 12 10 - 13 - — 8 _ _ _ _ — 5 2 3 - - - 7 6 1 10 - 9 4 13 - - 12 11 — 8 a 1 9 — - 6 - - 5 - - - - 7 4 - - - - 12 10 - 13 11 — 3 8 _ _ _ _ _ - 4 - - - 9 - - 2 7 1 - 11 12 10 5 13 , - — 20 23 42 7 9 6 13 14 34 38 27 8 3 9 7 39 45 49 17 11 31 108 53 24 115 43 6 ** ** 9* ** ** ** TABLE 40A SIZE CR "CASE 6: RANKS BASED ON STANDARD ERRORS FOR ALL SCHEMES APPLIED 3C 40 60 8C 90 100 120 140 160 TOT IC 4 1 9 24 RS 7 10 8 4 2 4 35 6 3 19 7 IC 7 3 7 7 44 ** M15 M20 2,5 ,10 ,15 3,5 .10 ,15 4,5 5,5 ,10 ,15 6,5 ,10 ,15 1,5 , 10 .15 1X5 10 15 DX5 10 15 _ 2 _ 3 4 _ _ _ 11 _ i t - - •- 9 13 8 ' - 12 7 5 _ 2 9 8 _ _ 8 - - 1 - - - - - 5 6 11 10 - 13 12 -_ 4 _ _ 3 _ - - 7 5 - - 2 8 13 12 - 11 9 6 5 11 2 6 4 13 10 8 12 9 3 •5 1 _ 2 _ _ _ _ - 6 - - 5 - 13 9 7 12 11 5 '_ 2 _ _ _ _ — _ _ - 3 - 4 8 1 12 10 5 13 11 6 1 2 _ _ - _ - - - - - 7 - 11 - - - 12 8 6 13 IC 9 4 5 10 1 - 3 - - 13 9 8 12 11 -8 _ _ _ _ _ _ - - 9 6 - 1 5 - 13 11 2 12 10 -13 27 18 4 12 0 0 10 3 0 12 0 34 27 11 10 31 28 113 87 36 110 90 34 ** ** ** CO C0> TABLE 38B CASE 4 - DEVIATION FOR BEST SCHEMES (P=0.5J CASE 4 - BIAS FOR BE ST SCH EMES ( P=0.5) SIZE 6, 5 1X5 1X10 I XI 5 DX5 DX10 30 1.96 19.51 -3.45 -3.37 8.97 4.73 40 3.82 15.56 3.30 -0.90 0.03 3.10 60 0.23 21.29 5.15 -0.61 11 . 13 2.52 80 -4.58 16. 92 0.08 1.63 11.98 5.62 90 1.21 14. 71 3.52 0.20 13 .47 2.72 100 -3.81 3.37 5.44 -2.48 11.37 0.66 120 1. 33 17. 56 1.07 1. 26 13.58 — 0 .46 140 -2.70 16.48 2.83 1.02 12. 40 0.17 159 1.18 13.65 2.40 -2.39 11 .34 4.58 TABLE 39B CASE 5 - DEVIATION FOR BEST SCHEMES (P=1.0J IZE 6,10 6, 15 1X5 1X10 DX 5 DX10 30 0.36 4. 88 2.72 0.35 10.01 -0.82 40 0.72 1. 93 7.54 1.38 14. 73 5.79 60 5.87 -1. 60 4. 05 -4.69 11.48 -0.2 5 80 0.91 0. 19 6.80 1.34 8.62 -2. 28 90 1.22 3. 46 8.13 0.69 8.94 2. 09 100 -0.55 -1. 20 8.61 -3.69 13.33 2.36 120 2.68 3. 23 3. 11 0.86 11 .98 2. 54 140 0.27 3. 55 12.20 -1.48 8.0 7 0.57 172 -1.24 3. 24 8.81 0.3 9 10.18 2.19 TABLE 403 CASE 6 - DEVIATION FOR BEST SCHEMES (P=1.5) SIZE M10 1X5 I X10 1X15 DX5 DX10 30 -0-64 2.05 -2.72 8.32 15. 91 -3.27 40 -0. 19 14. 01 2. 83 -5.60 7. 00 -6.0 2 60 -5.22 3. 58 -2.21 3.43 15. 02 - i . 2 0 80 -4.76 14.68 -0.80 -1.77 12. 91 -3. 13 90 1.55 8.53 1.90 1.23 13. 14 2. 04 100 1.21 13.03 -0.62 0.94 12. 52 -0. 57 120 -0.32 1.3.14 0.94 1.24 17. 39 -0.04 140 -0.63 14.20 1.62 ~3. 78 10. 30 -0. 04 160 -0.50 17.34 3.57 0.98 9. 72 1. 56 REMARKS: DEVIATIONS IN UNITS OF Y {TABLE 1); TWENTY REPLICATES PER SIZE. TABLE 41A CASH 7: RANKS BASED ON STANDARD ERRORS FOR ALL SCHEMES APPLIED s ize CR RS M5 MO M15 M20 2,5 ,10 ,15 3, 5 ,10 ,15 4,5 5, 5 ,10 .15 6,5 ,10 .15 1.5 .10 .15 1X5 10 15 CX5 10 15 30 11 2 3 4 - 6 2 - - _ 3 - _ - - _ _ - _ 8 10 _ 13 _ _ 12 7 S 40 - 4 - 3 6 5 - 9 7 2 - 10 13 8 - 12 11 1 60 11 - 3 2 8 - 9 7 10 12 4 1 13 6 -80 - - - - 4 2 - - 3 - - - - - - - 6 5 1 7 9 - 13 8 - 12 11 IC 90 4 9 8 11 7 3 2 5 1 13 6 10 12 - -100 10 4 9 - - a - - - - - - - - - - - 2 1 3 - 6 13 7 - 12 11 5 120 3 - 1 7 - 8 2 - 10 9 - 13 11 4 12 6 5 14C 7 9 5 - 4 1 - - - - - 3 - - - - - - - - - 8 13 11 3 12 10 2 160 - 8 4 10 3 - - 1 - - - 2 - - - - - . 6 - - 5 - 13 9 7 12 11 5 TOT 46 36 33 37 29 33 2 10 10 0 3 10 0 0 0 0 6 23 2 41 45 35 118 64 25 109 73 37 *4 ** ** ** ** ** TABLE 42A CASE 8: RANKS BASED ON STANDARD ERRORS FOR ALL SCHEMES APPLIED SIZE CR RS r-5 M10 M15 M20 2,5 4,5 5,5 ,10 6,5 , 10 1,5 ,10 ,15 1X5 10 15 DX5 10 15 30 10 13 - 7 1 6 3 5 9 4 12 — - - 8 2 _ 11 - — 10 40 8 2 - - - 1 11 7 13 10 6 - - 12 9 - - 3 - 4 5 55 10 12 6 - 5 1 4 8 - - 2 3 7 13 - - 9 - - 11 60 - 2 - 6 11 5 12 3 a 1 7 4 - 10 - 9 - - 13 65 2 12 1 - 4 3 - 13 6 - 10 5 8 9 - - 7 - 11 -77 4 - 10 - - - 13 6 9 8 11 - 7 2 1 - 5 - - 12 3 80 - 5 — - - 2 13 7 9 10 11 - - 12 6 - 8 4 - 1 3 90 1 12 - 2 - 7 9 5 11 - - - 13 10 6 - 8 4 - 3 -98 4 6 2 3 5 - 12 8 10 - - - - 11 9 - 7 12 - - 1 TOT 29 62 25 24 21 2? 74 58 83 33 57 7 35 63 62 2 37 50 0 31 46 *# ** ** ** *# TABLE 43A CASS 9: RANKS BASED ON STANDARD ERRORS FOR ALL SCHEMES APPLIED SIZE CR RS f5 MIC Ml 5 M20 2,5 4, 5 5,5 ,10 6, 5 , 10 1,5 .10 ,15 1X5 10 15 DX 5 10 15 30 10 4 _ 7 11 - 13 12 5 2 3 8 — 9 6 _ _ _ 1 33 - 1 7 - 10 12 13 2 3 6 - - 11 4 - - 5 - ei -40 - 3 - - 5 6 11 7 10 13 2 - 12 8 1 - - - - 4 9 60 11 2 - 9 13 1 8 7 12 4 - - - 10 - - 3 6 - 5 -30 6 9 7 2 11 3 4 3 - - - 5 12 - - - 10 1 - - 13 90 3 10 4 9 - 3 7 2 12 1 5 - 8 - - - 6 - - 11 -94 - 5 4 12 8 3 10 - - 11 7 9 - 6 2 - 1 13 - — -122 - 12 5 1 9 11 7 10 4 - - - - - 8 - 6 13 - - 2 140 10 7 8 - - 6 5 3 13 11 12 - 2 - 9 - 1 4 - - -TOT 50 6 0-i 29 47 57 43 77 62 58 45 35 22 34 44 20 0 27 42 0 29* 24 ** ** ** ** #* o TABLE 41B CASE 7 - 1 DEVIATION •FOR B EST SCH EMES (P = 2.0) SIZE CR 1,10 1X5 1X10 DX5 DX10 30 -6.21 2.97 2.39 3.73 -6. 25 -0. 4 8 40 6.82 -1. 57 -3.36 3.03 9.29 -1.36 60 4.65 -4. 78 -5.10 -0.12 4. 86 -5. 56 80 6.46 2.50 0.0 5 -3.91 7.12 3.04 90 -0. 08 1.26 0. 20 1.62 5.70 -3.0 6 100 -1.06 5.69 1.12 -0. 53 5.73 -3.3 7 120 -2.64 0.57 7.13 -1.04 0.49 i . 42 L40 - 1 . 34 2.61 4. 31 -3.24 0.44 0.9 9 160 0.43 -2.99 -0.23 -3.07 -0.04 -3. 86 TABLE 42B CASE 8 - DEVIATION FOR BEST SCHEMES (P=-i.O) SIZE RS 2,5 4,5 5,5 i , 1 0 1,15 30 0.85 0.34 0.19 -2.47 0.01 -0.27 40 -0.58 -2.26 -1.50 -0.23 0.95 0.23 55 0.90 -2.80 -0.49 -1.64 -1.57 -0.4-2 60 1.08 -1.47 -1.52 -1.58 0.21 -0.41 65 - i . 9 9 -0.02 -0.51 -1.08 -2.00 i . 3 4 77 0.03 -0.58 -0.58 -2.29 -0.98 -0.56 80 0.79 0.24 0.12 -0.36 -0.31 -0.03 90 0.23 -0.37 0.52 -2.13 -0.32 -0.82 98 -1.13 -2.70 -2.03 -0.04 1.16 -1.08 TABLE 43B CASE 9 - D E V I A T I O N FOR BEST SCHE ME S {P =-0.5) ;IZE CR RS M15 2,5 4,5 5,5 30 1.74 - 5 . 80 5.23 6.52 2.30 -1.7 0 33 0.87 -0.72 -1.19 1.22 2.34 -1. 14 40 -3.15 -2-13 1.87 1.06 -1.10 -2. 35 60 -0.48 -0.94 l o l 5 4.77 1.8 3 -4. 10 80 -1.91 0.82 -0.23 0.11 1 . 54 0.45 90 -0.59 1.02 0.76 2.09 1 . 94 0.22 94 -0.21 1.40 2.64 0.41 0.68 0.14 122 -0.32 -0.22 1.17 -1.97 0.42 -0.14 140 -0.36 -1.03 -1.01 1.25 0. 72 - 1 . 53 REMARKS: DEVIATIONS IN UNITS OF Y (TABLfc 1 ) ; TWENTY REPLICATES PER S I Z E . TABLE 44A CASE 10: RANKS BASED GN STANDARD ERROR FOR ALL SCHEMES APPLIED SIZE CR RS 3C 7 13 40 10 9 43 - -6C 6 -80 - 7 90 6 3 IOC - 12 119 - 10 120 - 5 TOT 29 59 ** TABLE 45A SIZE CR RS 30 3 -40 -60 7 13 70 9 -80 - 4 90 5 2 100 - -120 10 3 14C 10 4 TOT 44 •5 * ' ** TABLE 46A SIZE CR RS 30 2 9 40 4 6 6C IC 6 70 8 -80 5 2 90 5 -100 - 11 120 7 -140 8 7 TOT 49 41 M5 MIO M15 M20 2,5 4,5 5,5 ,10 6,5 ,10 1.5 ,10 ,15 1X5 10 15 DX 5 10 15 10 11 12 2 6 1 8 - - 5 3 9 - - - - 4 -7 ' _ 12 11 3 8 1 - - 4 5 6 — — — — 2 13 — 8 7 _ 5 1 - 10 - 4 6 2 - 9 - 12 11 — 13 3 5 4 11 8 - - 13 - 2 7 10 - 1 - — 12 9 5 8 6 I 9 12 - - - - 11 10 13 - 2 — — 3 4 8 11 _ — 5 - 7 9 - 2 - 12 10 - 4 13 — - 1 11 1 4 9 - - 6 - 2 - 3 - 8 - 10 13 - 7 5 7 11 9 - 1 6 12 2 - - 5 8 4 - 3 — 13 — 1 8 6 4 7 9 13 3 8 - 10 - 2 - 11 — - 12 — 57 62 49 33 39 49 50 22 27 12 43 46 67 0 43 37 2 77 22 ** ** ** ** ** CASS 11: RANKS BASED ON STANDARD ERROR FOR ALL SCHEMES APPLIED M5 13 4 3 8 8 12 2 50 7 2 12 3 11 35 M15 M20 2,5 4, 5 5,5 ,10 6,5 ,10 1,5 ,10 ,15 1X5 10 15 0X5 10 15 _ _ 12 8 6 5 7 - - 9 4 11 1 - 10 - 2 2 13 - 11 8 - - 9 12 - 5* 3 - 1 10 -_ _ 1 - 4 - - 10 - 3 11 6 5 9 12 2 7 10 _ - 5 - 3 1 - 12 6 - 11 13 4 1 3 _ 2 - 10 9 7 - - 5 13 6 — 11 — 10 _ - - 9 4 - 8 - 13 7 1 11 12 10 4 9 _ — 5 - 7 11 6 - 2 3 - 10 13 8 — 7 8 5 - - - - - 1 - 13 - 4 12 6 9 9 6 - 1 3 5 - 7 12 11 - 13 2 3 .27 37 33 13 31 13 43 32 30 41 13 82i 52 27 76 71 40 ** ** ** CASE 12: RANKS BASED ON STANDARD ERROR FOR ALL SCHEMES APPLIED M5 6 MIO 13 12 21 5 30 M15 10 10 13 5 3 50 ** M20 2,5 4,5 5,5 ,10 6,5 ,10 1,5 ,10 ,15 1X5 10 15 0X5 10 15 _ 3 _ _ 1 5 12 10 - - 11 - - 7 8 _ 1 — _ 5 7 11 2 8 - 9 12 13 10 -7 2 5 - - 1 - 3 4 9 8 2 12 13 11 9 3 _ _ i - 6 2 4 12 9 7 11 - 5 _ _ 9 4 - 8 - 13 7 1 11 12 10 _ 9 3 _ 8 - 1 - 11 13 4 6 12 2 7 4 2 _ 3 - 9 - - 10 3 - 7 6 - 5 -3 _ 9 - - - 4 11 - 13 10 2 12 6 1 4 10 _ 9 - 2 - 12 1 - 6 - - 12 5 11 24 29 9 29 0 36 16 49 44 29 63 65 36 83 60 53 ** ** ** ** ** ro TABLE 44B CASE 10 - DEVIATION FOR BEST SCHEME'S (P=0.0) SIZE RS M5 MIO 5,5 1,15 DX10 30 -2.06 -1.89 -4.34 -1.15 4.06 6.47 40 -1.18 3.96 -3.85 -5.40 -1.80 2.44 43 1.65 -0.35 -0.63 -5.37 0.12 -0.16 60 0.64 -1.62 1.49 0.23 -0.19 4.32 80 -0.98 -1.02 2.57 0.43 -1.12 1.33 90 -2.12 -0.87 1.69 0.22 -1.46 0.78 100 2.65 -1.05 0.61 3.26 0.91 0.55 119 -0.79 -0.04 1.43 -0.03 0.18 i.41 120 1.33 0.96 -1.95 -0.64 1.56 3.73 TABLE 458 CASE 11 - DEVIATION FOR BEST SCHEMES (P=0.5) SIZE CR M5 1X5 1X10 DX5 DX10 30 -3.60 -1.38 18.56 2.33 -3.76 -2.39 40 -0.96 1..6S 4.22 -3.37 6.33 -2. 18 60 6.35 2.26 9.50 0.35 4.47 i.37 70 -4.50 -3.62 18.08 0.48 -5.68 -1.96 80 -4.41 1.48 3.89 2.00 8.74 4.17 90 -2.16 -1.23 3.40 3.42 7.42 - i . l l 100 -1.22 0.73 7.52 6.06 11.66 3.15 120 -0.52 -0.93 9.96 2.69 4.45 2.00 140 -1.65 0.21 7.34 0.34 9.31 2.47 TABLE 463 CASE 12 - DEVIATION FOR BEST SCHEMES (P=1.0) SIZE M15 1X5 1X10 DX5 DX10 DX15 30 -4.83 17.16 3.27 14.59 -1.79 -3.63 40 -1.92 13.70 0.21 12.59 -2.00 - i . 3 1 60 1.06 15.92 0«47 12.33 3.95 0.88 70 -3.47 14.68 2.91 12.32 -1.67 1.59 80 -0.55 17.29 2.55 17.27 0.41 0.04 90 -2.93 15.10 0.56 10.55 1.58 -0.10 100 -0.93 13.47 0.03 13.41 -2.53 -2.14 120 -3.35 19.79 2.25 16.93 2.00 -2.41 140 -0.67 19.27 0.37 18.91 3.33 0.13 REMARKS: DEVIATIONS IN UNITS OF Y (TABLE I ) ; TWENTY REPLICATES PER SIZE. TABLE 47A CASE 13: RANKS BASED ON STANDARD ERROR FOR ALL SCHEMES APPLIED SIZE CR RS M5 M10 M15 M20 2, 5 4, 5 5,5 ,10 6,5 , 10 1,5 , 10 ,15 1X5 10 15 DX5 10 15 30 5 _ 10'. 2 7 - 3 - _ 4 - _ 6 - 9 12 _ 11 13 8 1 40 - - 7- 2 - - 10 3 - - 4 - - 3 5 13 I 11 12 9 6 60 8 - 11 5 - 6 4 - 1 7 - 10 3 - - 12 9 - 13 2 -80 11 9 4 2 - 8 7 - - - 3 5 - - 1 12 6 - 13 10 -90 10 3 - - - - - - - - 5 8 1 2 6 12 11 4 13 7 9 100 3 - - - 1 4 8 - - - 6 9 11 7 2 12 - 5 13 10 -109 1 8 2 - - - - - - 5 9 7 4 - - 12 10 -3 13 11 6 120 1 8 4 2 - 3 10 - - - - 11 - - • 9 13 5 7 12 6 -140 9 6 8 - 5 3 - - - - - 12 1 2 - 13 10 4 12 7 -TOT 48 34 46 13 13 24 42 8 1 16 27 62 26 14 32 I'll 52 45 114 70 22 ** ** ** ** ** ** TABLE 1 48A CAS E 14 : RANKS BASED ON STANDARD ERROR FOR ALL SCHEMES APPLIED SIZE CR RS H.5 M10 M20 2,5 4,5 5,5 ,10 6,5 ,10 1,5 , IC ,15 1X5 10 15 DX5 10 15 30 - - - - 3 - 4 7 8 6 - 5 2 1 - 12 13 - 11 10 9 40 - 1 4 - 5 11 - - - - 7 8 - 2 12 9 10 13 3 6 50 5 - - 2 7 3 - - - 4 9 - 6 8 - 13 10 1 12 11 -55 9 - - 5 - 4 6 - - - 8 - 7 - 1 13 11 3 12 10 2 60 - 11 3 8 5 6 - - - - 7 10 - - 2 12 4 1 13 9 -65 - 3 8 - - - 9 - - - 2 - 4 1 - 13 10 6 12 11 6 70 5 - 9 - 7 4 - - - - 9 - - 2 13 10 1 12 11 6 80 - 3 9 2 - 1 5 8 - - - 7 6 4 - 13 11 - 12 10 -90 - - 4 6 10 - 3 2 - 5 7 9 1 5 - 12 11 - 13 8 -TOT 19 18 37 26 30 32 31 17 8 15 40 48 26 19 5 113 89 22 110 83 11 ** ** ** ** ** ** TABLE : 49A CASE 15 : RANKS BASED ON STANDARD ERROR FOR ALL SCHEMES APPLIED SIZE CR RS M5 M15 M20 2,5 4,5 5,5 ,10 6,5 , 10 1,5 , 10 ,15 1X5 10 15 DX5 10 15 30 - 11 5 6 4 - 9 - 8 10 3 _ 2 12 _ _ 1 7 _ 13 — 37 - 2 11 - 10 3 4 5 11 - 1 - 6 8 7 - - 13 - 9 -40 3 4 10 6 13 - - 1 - 11 - - 7 9 8 - 2 5 - - 12 60 - 10 - 1 - - - 8 6 13 4 2 11 12 9 - - 3 - 7 4 69 8 13 - 7 - 10 4 6 12 5 - 2 9 11 - 3 1 - - -30 - 3 - 10 8 6 4 - 11 12 13 - - 2 9 - - : 5 - 7 1 90 10 - 9 12 13 - 2 8 - 7 - - 3 4 6 - 1 - - 5 11 IOC - 5 4 - 10 - 6 - 11 1 - - 12 13 9 - 8 7 - 2 3 104 - - 7 8 12 11 9 13 5 - 1 - 2 3 6 - - 10 - - 4 TOT 21 48 46 50 70 30 38 41 64 59 22 4 43 72 65 0 15 51 0 43 35 ** ** ** ** *# ** TABLE 47B CASE 13 - DEVIATION FOR BEST SCHEMES (P=1 .5) SIZE CR 6 , 10 1X5 I xio DX5 DXiO 30 5 . 89 -1. 75 12.53 -4 . 00 19 .46 5 .26 40 -9.72 5. 52 7.10 3 . 4 8 19 .26 -5.77 60 -5 .36 0. 24 7 .65 3 .99 18.49 2.02 80 -2. 17 5. 09 11 .94 -1 .39 19.31 0 . 0 0 90 - 1 . 9 4 - 0 . 06 12.55 -0. 77 16.50 2 .41 100 -2.20 1 . 84 13 .58 -5 .06 20.08 - 2 . 3 6 109 - 4 . 4 0 4 . 78 18.07 -1.15 1 3 . 9 9 7. 38 120 .1.8 0 4 . 01 13 .46 -0.25 11.97 3 . 6 9 140 1.04 5. '+3 19. 17 1 .69 21. 73 7.81 TABLE CASE 14 - DEVIATION FOR ZE 6>5 6, 10 1X5 30 -3.14 -1. 52 12.82 40 0.19 5. 34 6.40 50 -4.84 0. 59 5.61 55 2.14 -0. 88 8.11 60 1.69 0. 39 5.99 65 1.92 2. 89 8. 16 70 1.06 5. 66 20.23 80 -1.33 2. 57 9. 16 90 1.12 11. 05 5.37 8EST SCHEMES (P=2.01 1X10 DX5 D X i O 1. 14 22. 19 3. 91 5. 67 16 . 37 11. 39 5. 53 12. 63 7. 50 3. 30 4. 73 - i . 91 10. 76 13. 61 — 2. 17 0. 54 7. 79 9. 58 2. 8 8 4. 24 1. 79 2. 10 17. 71 4. 95 3. 18 9. 72 1. 80 TABLE 49B CASE 15 - DEVIATION FOR BEST SCHEMES (P=-1.0) SI ZE M15 5,5 5, 10 1,10 1 , 15 IXi 5 30 1.62 0.64 1. 16 -0.19 -0. 47 1.50 37 -2.00 1.74 3. 70 1. 54 -0. 57 2.69 40 -Oo 1 1 -1.01 0. 10 -0.46 0. 73 2.29 60 1.75 -0.40 -1. 0 2 1.12 0. 42 0.87 69 0.97 -1.12 -0. 91 -0.09 2 . 49 -0.40 80 -0.64 -0.70 -1. 31 1.10 -0. 30 0. 7 7 90 0.59 -2.05 -1. 17 -1.02 0 . 50 -1.13 100 -0.26 0.39 0. 56 -0.38 -1. 14 1.22 104 -0.26 -0.77 -1 . 01 0.33 0. 71 0.34 REMARKS •* DEVIATIONS IN UNITS OF Y (TABLE 1) ; TWENTY REPLICATES PER SIZE, TABLE 50A CASE 16: RANKS BASED CN STANDARD ERROR FOR ALL SCHEMES APPLIED SIZE CR RS M5 M10 M15 M20 2,5 4, 5 5,5 3C 6 _ 3 _ 9 11 12 - 3 40 9 - - 12 3 10 - 11 2 45 13 - - - 8 - 6 12 2 60 13 1 - 6 - - 8 11 12 78 11 5 - 12 4 10 - 3 -80 1 5 - 7 - - 12 2 9 90 - 4 11 10 7 3 - 8 -100 13 10 12 - 7 8 - 6 5 116 12 - 9 - 10 4 13 - 11 TOT 78 25 40 47 48 46 51 53 49 ** ** ** ** TABLE 51A CASE 17: RANKS BASED ON STANDARD SIZE CR RS M5 M10 M15 M20 2,5 4,5 5,5 23 11 4 1 - 8 - 3 9 -30 9 - 3 - 4 - 8 13 1 40 - - 13 8 7 10 2 4 -60 2 - 11 8 - 13 6 12 -80 - - 11 - 2 1 8 6 9 84 8 1 9 10 3 - 2 4 6 90 13 4 2 8 6 - - - 1 100 1 - - 11 5 9 6 - 7 137 13 - - 8 2 - 12 - 1 TOT 57 9 50 53 37 33 47 48 25 ** TABLE 52A CASE 18: RANKS BASED ON STANDARD SIZE CR RS M5 M10 M15 M20 2,5 4, 5 5,5 27 " — 2 _ - - 8 10 3 6 30 13 - - - - 7 3 11 -40 5 4 - 11 7 - - 6 -60 13 1 - 10 3 11 - 9 5 80 - 2 - 4 - 9 5 6 3 90 - 2 1 - 7 12 - - 3 99 - 1 - 5 - 8 - 4 11 140 2 - 8 9 13 7 - - 1 186 8 10 3 - 9 - 6 7 4 TOT 41 22 12 39 39 62 24 46 33 ** ,10 6,5 ,10 1,5 ,10 ,15 1X5 10 15 DX 5 10 15 7 1 - 13 4 2 5 - - 3 - -13 7 - 6 1 4 - 5 - - 8 -9 - - 1 - 10 • - 4 7 11 3 5 5 - 3 10 4 9 - 2 7 - - -- - - 13 1 9 - - 7 8 2 6 - 3 10 13 11 6 * - 4 - - 8 5 - - 13 2~ - - 1 12 - 9 6 1 - 9 - 4 3 - - 11 - 2 -6 1 3 - 7 - - 2 5 - - 8 45 11 25 69 34 43 5 14 53 22 24 33 ERROR FOR ALL SCHEMES APPLIED 10 6,5 , 10 1,5 ,10 ,15 1X5 10 15 DX5 10 15 5 _ 6 7 12 2 10 - - 13 -- - 2 - 6 - - 12 11 10 5 7 6 5 - - 11 12 3 - - 9 1 1 7 - 3. - 4 - 5 10 - - 9 3 13 - - 7 12 - 10 - - 5 4 7 11 - - - - • - 12 13 - - 5 10 9 - - 7 3 - 5 11 - 12 -- - - 4 12 8 2 13 3 - 10 - 4 11 10 3 5 - • - 6 - 7 9 27 54 13 23 53 56 4 70 54 10 61 35 ** ** *# ERROR FOR ALL SCHEMES APPLIED ,10 6,5 ,10 1,5 ,10 • 15 1X5 10 15 DX5 10 15 4 12 _ • _ - 5 13 9 - 11 • - 7 6 8 - 10 1 9 12 - 2 5 4 -2 10 3 - . 12 - 9 1 8 13 - -4 2 - 7 - - 6 12 - 8 - — - 13 11 - 10 1 12 - - 8 7 -- 11 10 13 - 4 - 3 6 9 — 5 _ 3 9 - 7 - 12 - 2 10 6 13 - 3 12 - 6 4 - - 11 10 - 5 - 2 11 1 - - 12 - - 12 5 -16 64 45 31 36 23 76 25 29 87 22 30 ** ** ** ** TABLE SOB CASE 16 - DEVIATION FOR BEST SCHEME'S lP=-0.5) il ZE CR 2,5 4,5 5,5 I t 5 1*15 30 1»13 2.31 -0.38 0.35 -0.71 -2.17 40 0.04 0.96 -1.12 1.76 -2. 57 1.51 45 0.75 -2.48 -1.92 0.54 -0.31 2.91 60 1.71 -3. 90 -0. 53 0.18 -0.45 -3. 38 78 2.44 0.5 0 0.21 1.89 0.58 1.0 4 80 -1.12 -1.61 -0. 12 -0.43 0.71 0.88 90 3.44 0.00 -2.97 -0. 33 0.70 - l . i l 100 -1,78 -0.30 0.19 -0.30 0. 00 -1.70 116 -1.98 0.3 7 0.48 0.18 -1.40 2. 40 TABLE 51B CAS E 17 - DEVIATION FOR BEST SCH SMES (P = 0.0) • IZE CR 6, 5 1,15 I X10 1X15 0X10 2 3 4.29 2. 28 2.35 2.51 3.16 5. 3o 30 -1.54 5. 16 -0.41 13.73 -2.39 0.16 40 -1.74 1.42 -3.94 2.69 1.17 6.56 60 5. 08 0.2 6 -0.28 -0.38 i . 86 2.08 80 -1.93 0. 78 -1.99 -0.46 -0.13 -0.24 84 1.82 1.20 3.38 3-28 0.32 3. 49 90 -1.66 -0. 01 -0.60 2.31 -0.06 1.35 100 -0.20 0. 29 1.00 4.34 -0.32 3. 49 137 0.18 0.33 1.26 0.66 0.14 I. 83 TABLE 52B CASE 18 - DEVIATION FOR BEST SCHEMES (P=0.5) SIZE M20 4, 5 6,5 6, 10 1X5 DX5 27 -0. 53 -1. 92 3. 33 -1.47 -9.63 7.01 30 -0.66 -0. 69 -3.81 6.35 2.25 -6. 62 40 -6.81 -1. 21 4.55 4.23 -5.19 -0. 34 60 -3.26 -2. 67 1.55 -0.03 1.67 -6.66 80 -2*62 1. 35 0.10 1.92 0. 26 -2.09 90 2. 16 -2. 42 5.54 -0.34 0.35 1.98 99 -0.42 2. 77 -0.34 2.32 -2.31 3. 24 140 -3.88 -1. 14 0.16 -2.07 -2.62 4. H-5 186 2. 77 -1. 10 1.83 1.03 3. 93 1. 00 REMARKS: DEVIATIONS IN UNITS OF Y (TABLE 1); TWENTY REPLICATES PER SIZE. TABLE 53A CASE 19: RANKS BASED CN STANDARD ERROR FOR ALL SCHEMES APPLIED SIZE CR RS M5 MIO 30 40 46 6C 80 90 100 120 159 TOT 11 13 6 7 2 6 45 6 6 3 5 3 2 25 22 4 10 9 4 50 7 20 M20 2,5 4,5 5,5 ,10 6,5 ,10 1,5 ,10 ,15 1X5 10 15 0X5 10 15 - 5 7 3 1 10 3 4 - _ 12 2 _ 13 9 _ 9 5 3 - 11 - 1 - 8 12 10 - 13 - 6 3 — 2 9 4 - - - - - 12 10 - 13. 7 11 — ~ 3 — - 8 Z - 1 - 11 7 - 12 10 6 7 9 4 — 11 - 2 - - 13 3 - 12 3 _ ~ . — 2 - 11 10 - 8 6 13 3 1 12 4 — 10 4 9 1 - 5 8 - 6 13 12 - 11 10 7 7 5 1 — — 3 10 4 - 9 13 11 - 12 6 9 — — — — 8 11 - 2 - 12 3 10 13 5 1 28 27 31 35 6 62 41 19 11 29 111 66 11 111 54 31 ** ** *v ** ** TABLE 54A CASE 20: RANKS BASED ON STANDARD ERROR FOR ALL SCHEMES APPLIED SIZE CR RS M5 MIO M15 M20 2,5 30 3 1 7 _ _ 9 4 40 - 10 6 8 1 5 4 60 4 8 11 2 - 10 -80 - 10 9 - 1 7 4 85 8 - 11 - 6 10 -90 6 4 - 2 - 9 -100 2 11 5 4 3 9 — 120 - 1 - - 11 3 _ 14C - . - 1 - - 5 7 TOT 23 44 50 16 22 67 19 ** 9 27 ,10 »,5 ,10 1,5 ,10 ,15 1X5 10 15 DX5 10 15 2 5 - - . - 12 11 _ 13 10 3 - 2 9 - 12 11 - 13 7 7 3 - 9 - 12 5 6 13 1 -3 6 2 - • - - 12 8 11 13 C -. 5 - - 4 - 12 1 2 13 3 9 — - 5 10 7 12 8 7 13 11 1 8 - - 6 7 13 10 - 12 1 — 5 8 6 9 2 13 7 4 12 10 -- 8 - 3 2 12 11 4 13 10 6 33 30 15 50 18 110 72 34 115 58 16 ** ** ** ** ** TABLE 55A 30 40 60 80 90 IOC 120 140 155 TOT CASE 21: RANKS BASED CN STANDARD ERROR FOR ALL SCHEMES APPLIED CR RS M5 MIO . M15 M20 2,5 4,5 5,5 ,10 6,5 ,10 1,5 ,10 ,15 1X5 10 15 0X5 10 15 - 1 3 5 - - 2 - - - 10 6 - 7 _ 12 11 8 13 4 9 7 — 1 - 3 4 - 2 5 6 8 - 13 11 10 12 _ 9 10 8 6 4 — - 7 - - 9 11 5 1 - 13 3 12 _ 2 ~ 7 6 4 - 2 1 - - 10 3 9 - - 12 11 5 13 8 \ • 5 ~ 3 — - - 4 9 2 1 6 13 11 8 12 7 10 VO — 8 2 5 4 7 - 8 - 3 9 - - - 12 10 6 13 11 1 CO 9 7 3 10 8 — 2 6 - - 4 - - - 1 13 5 8 12 11 • ~ ~ — 3 2 8 — 4 - - 10 9 7 - - 13 11 6 12 5 1 4 9 ' — 6 2 3 7 - - 3 10 - 5 - 12 — 1 13 11 23 45 19 30 28 15 21 28 12 0 55 62 . 29 22 7 113 73 52 112 57 32 ** *# ** ** ** ** 99.' TABLE 53B CASE 19 - DEVIATION FOR BEST SCHEMES <P=1.U) SIZE M10 6,5 1X5 1X10 DX5 DX10 30 -3.21 6.32 12.07 -2.32 11 .37 7.23 40 -1.33 2.95 6.91 -1.65 20. 31 -0.3 9 46 -4.36 1.99 14.02 2. 32 16. 57 1. 71 60 -3.37 3.98 12.20 1.45 7.94 0.69 80 2. 66 -0.41 12.33 4.71 14. 33 -0. 33 90 -2.24 0.57 4.10 1.53 7.52 0. 76 100 2.35 -3. 01 9.86 0.60 13.86 -0. 16 120 3.05 0. 36 9.60 5.55 7. 63 -2.3 0 159 -0.88' 4.29 15.07 3.60 12 .64 3.04 TABLE 54B CASE 20 - DEVIATION FOR BEST SCHEMES (P=1.5) SIZE M5 M20 1, 10 1X5 I X10 0X5 DX10 30 2.99 -1,23 2.34 19.10 1.16 10. 03 -7.98 40 -5.56 0.79 1.58 3.67 4. 4 i 10. 12 2.34 60 -1.64 0.31 -4.06 7.77 -0.07 17.47 2.62 80 0.55 -1.77 3.0 5 12.18 4.57 10. 40 -3. 85 85 2.42 -0.27 -1. 18 13.2 7 1 .73 17.2 5 -0. 06 90 5.33 -0. 32 -3. 12 7.91 -0.22 10.39 -0.51 100 -1.86 1.16 -0.21 11.57 5. 66 14. 97 4. 1 8 120 -1.51 1.37 -0.97 12.24 2 .09 12.96 2.46 140 -4.22 0. 93 2.00 9.48 1 . 33 12. 10 2. 40 TABLE 55B CASE 21 - DEVIATION FOR BEST SCHEMES (P = 2.0) SIZE 6,5 6,10 1X5 1X10 DX5 DX10 30 2.07 11.06 9. 30 10.48 11. 86 3.15 40 - 1.49 -0. 15 5.57 10.97 18. 03 2.31 60 3.80 1. 17 12. 87 9.15 10. 37 4. 74 80 0.72 1.10 16.18 10.70 9.17 6.84 90 5. 16 -5.34 16.22 -0.30 9.22 • -5.81 100 0. 70 7.60 11.64 4.08 24.22 4. 0 8 120 2.43 9.61 11.89 9.36 15.99 4. 00 140 - 0.72 2.79 16.64 10.03 13.21 2.83 155 2.28 0.68 12.58 3.16 11 .57 0.05 REMARKS: DEVIATIONS IN UNITS OF Y' (TABLE 1); TWENTY REPLICATES PER SIZE. TABLE 56A # CR RS CASE A l : RANKED VALUES 3ASED ON STANDARD ERRORS 30 -40 3 50 -70 -80 -90 -100 9 11C 9 120 -TOT 21 4 7 8 10 5 1 3 38 M5 MIO M15 M20 2,5 ,10 ,15 3,5 ,10 ,15 4,5 5,5 ,10 ,15 6,5 ,10 ,15 1,5 ,10 ,15 1X5 10 15 DX5 10 15 2 7 6 11 3 10 9 — - - - - - 5 - - 13 - 1 8 12 -6 9 11 13 - - - - - 5 - 1 - - 4 - - - - - 10 - 7 12 8 2 13 9 12 7 1 - 6 3 - - - - 11 5 2 10 — 4 3 6 5 8 _ - - - - - 1 - - - - - - 9 7 2 11 12 4 13 — 10 1 8 11 2 4 - 5 3 - - 13 9 6 12 - 7 _ 6 11 7 8 3 - 9 - 2 13 4 — 12 10 1 6 2 11 _ — _ — - - - - - 3 - 5 - - - 12 7 4 13 8 10 5 8 12 2 — _ - - - - - - - - - 7 - - 10 4 11 6 - 13 1 -7 6 9 11 - - _ - - 1 - •r - - - 10 4 3 6 13 2 - 12 8 -42 48 71 79 16 0 0 0 0 15 11 1 1 0 25 13 20 30 20 14 107 45 24 105 47 34 ** * * ** ** ** ** TABLE 57A CASE A2: RANKED VALUES BASED ON STANDARD ERRORS # CR RS M5 MIO M15 M20 2 » 5 30 — 11 _ 13 - 12 -40 - - - 12 6 11 -50 1 - 2 6 7 10 -70 - - - 11 12 4 -80 - - - 6 - 13 -90 4 - 6 - 12 11 -100 - 5 9 8 - 12 2 110 5 - - 11 2 1 -120 3 . 5 9 io - - 1 TOT 13 21 26 77 39 74 i ** ** 9 17 24 - 9 -- 1 3 5 - - 1 3 2 - - 7 - - 2 0 27 15 10 8 1 5 2 9 - 11 9 2 9 3 - -4 - - -4 - 10 - 11 8 12 31 26 36 28 20 15 1X5 10 15 0X5 10 15 _ 7 4 1 — 3 - -- 9 7 - 10 3 -- 13 4 - 12 11 -- 13 - 2 9 6 10 - 10 7 1 12 4 -1 13 8 - 10 7 5 3 10 10 - 13 - 1 - 13 7 - 12 6 -4 13 7 - 12 6 -8 102 54 4 90 46 16 ** ** ** ** TABLE 58A CASE A3: RANKED VALUES BASED ON STANDARD ERRORS # CR RS M5 MIO M15 M20 30 2 8 7 3 - 13 40 13 - - 7 - 6 50 - - - 6 11 10 70 13 5 9 - - 11 80 1 6 9 5 - 4 90 - 5 - 10 9 -100 5 9 3 8 2 10 110 - 4 - - 5 -120 - 5 - - 6 11 TOT 34 42 28 39 33 67 ** ** 17 5 2 7 6 1 19 12 14 12 10 12 2 10 1 25 10 ,15 1,5 ,10 ,15 1X5 10 15 0X5 10 15 1 _ _ 9 10 — — 11 4 -- - 4 - - 11 3 10 12 - 5 7 .8 4 - 5 12 - 9 13 2 -- - 6 2 8 12 - 4 10 3 -3 - 7 - 8 13 - - 11 12 -7 - - - 4 11 2 3 13 12 -• - ' - 1 - - 13 6 - 11 4 7 6 11 2 1 10 9 7 - 13 6 12 9 8 - 2 10 12 4 7 12 3 -33 27 24 5 54 103 22 25 107 46 24 #* ** ** ** o o TABLE 569 MEAN DEV. OF BEST SIX SCHEMES FROM TABLE 56A SIZE MIO M15 M20 1X5 DX5 DXIO 30 1.69 -1.84 -4.57 8.96 12.75 -1.09 40 -5.37 -1.89 -2.81 7.49 15.62 -3.09 50 -0.01 -3.89 -6.07 12.92 6.93 3.29 70 -0.94 -3.17 4.42 14.19 8.36 1.60 80 -0.18 -5.10 -1.04 8.62 10.77 0.42 90 3.30 1.54 -1.41 8.76 9.23 -0.46 100 -0.93 1.87 -3.00 8.47 5.57 -1.95 110 -0.98 -1.32 -1.48 12.94 10.22 0.26 120 1.21 1.55 -1.65 12.49 9.13 1.36 TABLE 57B MEAN DEV. OF BEST SIX SCHEMES FROM TABLE 57A SIZE RS M20 1,15 1X5 DX5 0X10 30 6.57 8.76 -0.93 16.45 6,97 3.74 40 7.78 5.69 -5.11 12.29 6.20 2.82 50 7.87 3.37 4.96 15.47 12.87 4.46 70 -2.05 0.76 1.76 3.53 8.80 0.13 80 1.33 3. 07 2.89 8. 82 9.84 -2.47 90 3.56 0.16 5.67 10.46 7.23 1.67 100 -2.36 -0.32 0.64 8.71 6.83 5.21 110 1.17 -0.51 -1.74 6.87 5.28 -0.42 120 2.18 1.91 0.69 1.51 6.88 5.87 TABLE 58B MEAN DEV. OF BEST SIX SCHEMES FROM TABLE 53A SIZE MIO M20 1X5 1X10 DX5 DXIO 30 1.73 1.64 15.16 0.62 7.33 -1.83 40 -3.25 -1.41 15.90 -6.21 16.63 4.46 50 -0.36 2.90 1.15 3.93 14.80 5.94 70 -3.01 -1.99 8.72 1.20 7.86 0.26 80 -1.92 -2.59 -4.22 -0.52 9.42 2.03 90 -2.34 -1.18 5.85 -3.15 5.84 -1.99 100 -1.49 -0.06 4.86 0.98 4.37 4.05 110 -1.46 -1.29 3.73 4.64 11.53 2.56 120 -0.52 0.70 4.76 -0.23 11.47 1.65 REMARKS: DEV. IN UNITS OF Y ; 20 REPLICATES/SIZ£ V102 . 5.2.iv. Mean deviation as function of p and sample si z e , n It had been mentioned that a r e l i a b l e bias estimation or correction procedure on the f i n a l YBAR values would unquestionably make the DX5 and 1X5 schemes the best of a l l . An examination of the scattergrams of the schemes indicated the absence of any r e l a t i o n s h i p between the y mean deviation and R p or R^. Perhaps t h i s may be incon-clusive due to the i n s u f f i c i e n t l y d i f f e r e n t R^ or R^ values. However, for a l l the schemes, the relationships between the deviations, p and n were found to belong to four main categories. Single schemes from such categories were int e n s i v e l y studied and then run through regression analyses. For RS, scattergrams indicated the range of ~f the deviations increased l i n e a r l y with increasing p, and decreased with n (Figures 4a and 4b). With scheme M15, the rafege'-of^the^.dev^ia^igniii.c-riased with' increasing ' h'r (Figures 'Ac and • 4d)'.-^For -the probability-"" scheroes<?rn3JX5 indi<2a€edateaioose*-rg.3!a€(i©ntwi%h^>^bQt the range decreased almost l i n e a r l y with increasing n (Figures 4e 103. F i g . 4 Range of d e v i a t i o n s f o r p and n r e s p e c t i v e l y . (Figures not drawn to s c a l e ) 104. and 4 f ) . The relationships for deviations from D X 5 and p were not very clear but appeared to be minimum at p = 0 . 5 0 . The r e l a t i o n s h i p with size was similar to that of 1 X 5 . In general the relationships between the mean deviation and the sample size were consistent with theory. The general pattern of these relationships suggested the choice of a logarithmic or polynomial regression. The presence of negative deviations prompted the use of the polynomial regression. Logarithmic trans-formation could have been used by changing the scale of the deviations. Second degree polynomials of p, R and n, and the i r r e c i p r o c a l s were applied to the four schemes (whose scattergrams had been studied) with the mean deviation as the dependent v a r i a b l e , (number of observations = 1 8 9 ) . The best r e l a t i o n s h i p for the deviation of RS, D [ R S ] involved only the sample siz e , but t h i s was not s i g n i f i c a n t at <* = 5% l e v e l of s i g n i f i c a n c e . For M 1 5 the function, with the independent variables arranged in decreasing variance r a t i o s (or p a r t i a l F's) within the parenthesis, was of the form: ( 5 . 2 . 1 . ) ( 5 . 2 . 2 ) 105 2 Although the regression was s i g n i f i c a n t , p was i n s i g -n i f i c a n t but i t s removal resulted i n a drop from 4.47% 2 to 2.76% for the c o e f f i c i e n t of determination (100R ), and an increase in standard error of estimate, SEg from 2.6758 to 2.6924 (Table 59). For 1X5, the r e l a t i o n s h i p was of the form, D [ l X 5 ] = f(p, p , R , n) = f(l,2,3,4) . . (5.2.3) A l l the variables i n equation (5.2.3) were highly s i g n i f i -cant with i d e n t i c a l p a r t i a l c o e f f i c i e n t s of determination. Elimination of 1/p and 1/p from the o r i g i n a l model caused a reduction of approximately 33% i n the c o e f f i c i e n t of multiple determination, and an i n s i g n i f i c a n t difference i n the standard error of estimate. The best equation for D [ DX5] , contrary to expect-ation, involved only the p transformations; D [DX5] = f(p, 1/p2, 1/p, p 2) = f(l,2,3,4) , (5.2.4) Elimination of any of the variables at a time resulted i n a reduction in the c o e f f i c i e n t of determination by as much as 50%, and sometimes more. TABLE 59. REGRESSION COEFFICIENTS FOR DEV. R, P AND n RELATIONS • c o e f f i c i e n t o f independent v a r i a b l e s S E E Schemes I n t e r c e p t 1 2 3 4 ( c . f t ) RS -0.7773 • 0.00847 2 .2396 10 OR2 F 2 .01 3.835* 4.47 4.350 16.93 9.372 31.34 20.996 * * M l 5 0.9676 -7 .25330 -2 .26870** - - 2.6758 1X5 6.6569 2.87707 -2.11048 -22.72070 0.35203 6.6569 DX5 11.1389 9.29914 0.04316 -4.31462 -4.17138 5.9446 not s i g n i f i c a n t a t 5% l e v e l - the whole r e g r e s s i o n independent v a r i a b l e not s i g n i f i c a n t a t 5% l e v e l Number o f o b s e r v a t i o n s used - 189 t«# E x p l a n a t i o n o f independent v a r i a b l e s : 1 2 3 4 RS n - -M15 R 2 P 2 _ _ |_i 1X5 P P 2 R n g " v c n 1/P 2 1/P P 2 6.0 RESULTS OF APPLICATIONS TO IFS SCALE DATA AND SUMMARY OF RESULTS 6.1 IFS scale data Sample size determinations were done by specifying the precision requirement as percentages of the mean of Y. Since no prior information on the variances was assumed, these variances were determined from the largest of one hundred r e p e t i t i o n s of random samples of size 30 drawn from each of the f i r s t 10, 11-20, 21-30, 31-40, 41-50 sets of the population as given i n Table 27A. For a l l pairs of samples, the difference between the smallest and largest sizes was less than 5. This was an i n d i c a t i o n of constant variances for the various sets. Tab lies 160H and f&l dndicate\.-'sbmel •of" the rfiirfaUe sample sizes determined. Sample sizes used for t h i s study were r e s t r i c t ed to those less than or equal to 20% of the population s i z e . In t h i s connection, additional samples were taken without necessarily specifying the precision required. Not a l l of the schemes used in the generated case studies were applicable to the weight scaling data. In pa r t i c u l a r , DX10, 1X10 and for higher classes could not be applied because of the algorithm's i n a b i l i t y to obtain enough samples from some of the frequency classes. I f reference i s made to Tables 29 and 30, i t could be seen that most of the i n i t i a l and l a t t e r classes d i d not have enough samples. Since the schemes DXZ and IXZ involved sampling i n t e n s i v e l y from either end, i t was not surpris ing the procedure could not work for these data. I f , however, the extremes were to be ignored, the number of actual classes that could be sampled from would reduce to 5 or 6, thus making possible only the use of DX5, 1X5 or DX6, 1X6, r e s p e c t i v e l y . 109. TABLE 60 SAMPLE SIZE DETERMINATION FOR + A% OF YBAR PRECISION REQUIREMENT YBAR ( e f t . ) RATIO P RATIO' 0.025 1448.7 0.050 1447.0 0.100 1445.9 0.500 1449.8 1.000 1453.1 2.500 1452.4 5.000 1443.7 RHO 0.7977 0.8062 0.8094 0.8149 0.8095 0.7997 0.8249 (c . f t . / l b . ) ( e f t . / l b . ) SIZE SAMPLE BLOCK (N/n) 0.020322 0.020271 0.020326 0.020310 0.020316 0.020312 0.020273 0.020314 0.020263 0.020318 0.020302 0.020308 0.020303 0.020267 1055 1041 761 420 166 31 7 1 1 1 2 7 44 189 TABLE 61 SAMPLE SIZE DETERMINATION FOR ±A CUBIC FEET PRECISION REQUIREMENT - A ( e f t . ) YBAR (c.ft.) RHO RATIO P (c . f t . / l b . ) RATIO ( c . f t . / l b . ) SAMPLE SIZE BLOCK ('N/n> 5.00 1456.8 0.8053 0.020360 0.020351 622 1 10.00 1448.4 0.8180 0.020335 0.020328 277 4 15.00 1452 .1 0.7901 0.020346 0.020337 164 8 20.00 1448.9 0.7981 0 .020304 0.020296 97 13 25 .00 1451.6 0.7926 0.020323 0.020313 68 20 30.00 1452 .2 0.8102 0 .020302 0.020294 42 32 35.00 1447.8 0.7959 0.020339 0.020330 36 40 40.00 1451.9 0.7956 0.020348 0.020340 28 49 5 Results based on the largest of f i v e - 100 r e p e t i t i o n s of samples of size 30 6 p = 1.2727 7 Ratio = Rj_ , ratio-of-means estimator; 8 calculated to nearest integer TABLE 62A RANKED VALUES BASED ON STANDARD ERRORS FOR 14 SCHEMES SIZE CR RS M5 MIO Ml 5 M20 M25 M30 M35 2,5 2 ,10 2,15 3,5 3,10 28 7 - - - 9 % - 11 - - 12 - 6 31 - - 13 6 6 11 - - 5 6 3% 2 36 - - - - 10 5 - 2 % - - - - - 9 45 3 3 - 7 - - 8 3 - - - - 5 % 10% 61 - - 2 % 4% 7 6 8 - - 12 - - -78 - - - - - 7 - 5 - 11 8 9 6 4 164 - - - - - 7 - 9 - 4 - 2 - 1 2 166 - - - - - 10% 4 - 10% 5 3 180 2 4 - - - - - - 9 10 - - - 7 TOT 12 7 15% 17% 32% 46% 31 19% 14 55 11% 29% 16% 45% TABLE 62B RANKED VALUES BASED ON STANDARD ERRORS FOR REMAINING SCHEMES SIZE 4,5 4,15 5,5 5,10 5,15 6,5 6,10 6,15 1,5 1,10 1,15 1X5 DX5 28 - 13 - - 8 3% 2 3% 5 - - 9% 31 8 3h - 12 10 - - - 9 36 12 8 - 2h - 4 eh - eh n 13 45 5h - 3 - 9 - - 1 2 - - - 10h 13 61 11 - 9h - - 2h - 4 % - 9*2 - 13 78 13 2% - - 2h - - - - - - 12 10 164 3 5 11 - - 10 6 - - - 8 13 166 - 12 - - 2 6 7 - - - - 8 1 3 180 12 11 - - 4 4 - - - 7 - 7 1 3 TOT 53*5 e2h 14 13 28 25*5 31% 25h 11 12 16 56% 106% T A B L E 63 NUMBER OF +VE £ - V E DEVo FOR BEST S I X SCHEMES M20 2, 5 4, 5 4, 15 1X5 0X5 TOTAL SIZE + - + - + . - + - + - + - + -28 4 6 7 3 4 6 8 2 5 5 8 2 36 24 31 8 2 6 4 7 3 7 3 5 5 8 ? 41 19 36 6 4 3 7 7 3 8 2 9 1 9 1 42 18 45 4 6 6 4 5 5 6 4 7 3 8 2 36 24 61 6 4 6 4 4 6 6 4 9 10 0 41 19 78 5 5 6 4 7 3 9 1 10 0 10 0 47 13 164 6 4 5 5 6 4 6 4 6 4. 10 0 39 21 166 8 2 4 6 4 6 8 2 9 1 9 1 42 18 130 4 6 8 2 9 1 8 2 9 1 10 0 43 12 TOTAL 51 39 51 39 53 37 66 24 69 21 82 8 TA8LE 64 MEAN DEVIATION FOR THE BEST SCHEMES SELECTED SI ZE M20 2,5 4,5 4, 15 I X5 DX5 28 -2.06 17.52 -7. 19 17. 77 4.32 3o.36 31 9.36 9. 56 5. 59 -0. 68 1 . 13 23.91 36 13.63 -7.26 14.67 6. 59 22.31 32. 19 45 -2. 60 7. 72 -10.22 4. 46 16 .45 36.42 61 8.04 3.49 4.30 10. 0 3 21. 01 53. 15 73 6.8 7 7.89 -3.02 12. 83 22.63 43. 75 164 0. 37 2. 07 0.54 3. 51 11.33 43. 97 166 13.19 -4.66 -3.06 9. 13 17.05 36. 86 180 -6.53 6.31 13.97 9. 73 17.33 38.5 3 MEAN 4.53 4. 79 1.79 8. 16 15.02 38. 90 TABLE 65 MEAN OF A BSOLUTE DEVI AT IONS FOR ABOVE SCHEME SIZE M20 2,5 4,5 4, 15 I X5 DX5 28 22. 93 40. 59 28.60 2 0. 7 9 32. 38 43. 61 31 23.23 2 2.30 21.04 18. 27 26.07 35.42 36 31. 61 36. 21 23. 19 31. 93 25. 35 33.04 45 11.55 22. 51 20.69 22. 02 34. 3 5 44. 76 61 22.65 21 .60 22.89 25. 50 28. 8 2 53.15 78 14.63 19.53 22. 72 17. 29 22.68 43. 75 164 7.83 8.93 14. 15 12. 12 18.26 48. 97 166 15.42 12.40 10. 73 11. 58 17.34 42 . 9 0 180 15.2 1 11-27 14.23 15. 73 17.84 33. 53 MEAN 18.34 21.71 i 9 . a o 19. 47 24.90 42.63 REMARKS: DEVIATIONS IN CUBIC FEET ( C . F T ) ; TEN REPLICATES PER SIZE. Appendices I - IX give the rank values of the various schemes obtained for the r e p l i c a t i o n s for each sample s i z e . The f i n a l ranks were then aggregated to form Tables 62A and 62B. The r e s u l t s from the tables i n d i c a t -ed that schemes DX5; 1X5; 4,15; 4,5; 2,5; M20 and to some extent 2,15 gave the o v e r a l l smallest variances. Scheme DX5, in p a r t i c u l a r , had the best o v e r a l l minimum variance as expected. Sampling with p r o b a b i l i t y proportional to the frequency of x (Type 1,Z) and RS were found to be inapplicable to the scale data. Apart from the study of the deviations of the predicted mean of Y from the true mean (as done i n the 'B' tables of Chapter 5, section 2), the extent of the consistencies in over- or under-estimation of the mean was also studied. It was believed that i f such c o n s i s t -encies existed, i t might be easier to correct for the huge deviations that were prevalent with the 1X5 and DX5 types. Table 63 gives the summary of the numbers of over-estimation (positive deviation) and under-estimation (negative deviation) for the ten r e p l i c a t i o n s of each sample size for the respective 'best' schemes. Under the n u l l hypothesis of equal number of positive and negative deviations. On the basis of these r e s u l t s , i t was concluded that a l l the schemes over-estimated the true mean. Further study of the various columns also showed the consistent over-estimation of the mean by DX5 and to some extent 1X5. Although Table 63 provided valuable information, i t did not give any ind i c a t i o n of the magnitude of these deviations. The mean deviations and the absolute deviations for the least variance schemes were therefore compiled and presented as Tables 64 and 65 respectively. These tables indicated the consistently and,relatively high gvgr^stimajtionvby gDX5, and 1X5 . W l [ iIt C )was rpbserygd -that the magnitude of over-estimation was far in excess of the under-estimation cases in absolute units. Averaged over a l l the sample si z e s , the deviation for DX5 exceed-ed those of the others by at least 50%. Perhaps since we stand to lose by either over-or under-estimation, a better c r i t e r i a than that provided by Table 64,would be the use of the absolute deviations (Table 65). Table 65 also showed the consistently large over-estimation by DX5 in pa r t i c u l a r , and i n some cases exceeded those of other schemes by as much as five-times. Using the usual analysis of variance, and the alternative procedure of Box L Scheffe' (1959) pg. 83-87] . the r e s u l t s indicated that the only scheme that gave s i g n i f i c a n t l y d i f f e r e n t r e s u l t s was that of DX5. The choice of the best scheme was therefore between DX5 and any or a l l of M20; 2,5; 4,5; 4,15; 1X5 . tln^re:iativp, terms, M.2 0 . may have -fepicbe .seriously- s qon^sid.ergd-. 4 pe^iatio^s. -.were r e l a t i v e l y Sample outputs for the test of the e f f i c i e n c y of the schemes on the IFS data are given i n Tables 66-72 and the rank values for each of the 10 applications given in Appendices I-IX. The format for these tables i s i d e n t i c a l to those of Tables 31-34. Unlike the previous case studies, p was estimated using the weighted (by number of observations i n class) non-linear least squares procedure mentioned for case studies A1-A3. The value of p = 1.2727 for the IFS scale data d i d not indicate any s i g n i f i c a n t difference between Rp and Rj_. Except for schemes DX5 and 1X5 which gave high c o r r e l a t i o n c o e f f i c i e n t s (greater than 0.80) most of the times, there was not a general consensus on t h i s for the other schemes. For each scheme, increasing the number of classes or groupings did not necessarily r e s u l t in larger or smaller c o r r e l a t i o n c o e f f i c i e n t s . But as were obtained for the case studies, classes within schemes giving the largest c o r r e l a t i o n c o e f f i c i e n t s also gave the smallest error i n a majority of cases. Indications from scattergrams of the r e l a t i o n s h i p between the mean deviation and the sample size for the best schemes and for the combined r e p l i c a t e s are given be low: a) M2 0 and 2,5: Deviations appeared to be c u r v i l i n e a r , but very s l i g h t l y i n the case of 2,5, r e l a t e d with sample s i z e . The spread or variance for each sample size was f a i r l y constant. b) 4,15 and 4,5: No r e l a t i o n s h i p was evident. c) 1X5 and DX5: Similar r e s u l t s as for the deviation and size relationships i n the combined CSl-21 (Chapter 5, section 2, iv.) Regression analyses for mean deviation on several transformations of sample size n did not y i e l d s i g n i f i -cant r e s u l t s for either of the schemes. In fact, for sub-cases a) and b) , the multiple coefficients of determination were less than 1.5% with no par t i c u l a r independent variable predominating. For DX5, n and 1/n appeared to be important, but used j o i n t l y only 1/n was s i g n i f i c a n t . The o v e r a l l regression was, however, not s t a t i s t i c a l l y s i g n i f i c a n t (.s£ = .05). I t had multiple c o e f f i c i e n t of determination of only 6.34%. Similar r e s u l t s for 1X5 were obtained but 1/n was not s i g n i f i c a n t , and the o v e r a l l c o e f f i c i e n t of determination was 4.43%. From these regression r e s u l t s , i t did not seem reasonable to recommend the DXZ or the IXZ schemes e s p e c i a l l y for Z = 5, for weight s c a l i n g . Unless e f f o r t s other than through regression analyses could be made to devise correction procedures for the deviations from DX5 or 1X5, schemes MZ or those of equal p r o b a b i l i t y types could be recommended for weight s c a l i n g . Equal prob-a b i l i t y schemes are however subject to additional constraints of minimum observations within each class, and may there-fore not be suitable for scaling unless there i s intimate knowledge of the population. MZ, on the other hand, i s simple to execute and could be incorporated in present sca l i n g procedures without much modifications and e f f o r t . In the next chapter, further aspects of s t r a t i f i c a -t i o n based on a r r i v a l s or season (in l i n e with MZ) and on the sizes of the a u x i l i a r y variable (in l i n e with the frequency type schemes) are examined. New sets of weight 118. scal i n g data from i n t e r i o r and coastal B r i t i s h Columbia are used. 119. TABIC 66 E F F I C I E N C Y OF SAMPLING SCHC-MCS -POP YUAk = U2J.3fl ; S AMP LF S I z t: = 31 (SAMPLF OUTPUT) S 3 USING E S T . P« R Y B A R 1.2727 SYBA.R USING APPROX ft YD AR P = 1 . 0 0 SYBA.R CR . 0 1 9 7 2 4 1 3 3 4 . 7 1 2 3 . 6 9 6 8 . 0 1 9 7 5 2 1 3 3 6 . 5 8 2 3 . 6 7 5 9 RS . 0 1 9 5 0 0 1 3 1 9 . £ 2 56. 5596 . 0 1 9 4 6 0 1 3 1 6 . e 6 5 6 . 5 4 5 1 K5 . 0 2 0 3 1 7 1 3 7 4 . £ 3 1 8 . 7 7 5 4 . 0 2 0 3 1 6 1 3 7 4 . 7 9 18.7754 m o . 02C4 28 1 3 0 2 . 3 3 1 8 . 5 9 7 4 . 0 2 0 4 0 7 1 3 8 0 . 9 3 1 8 . 5 8 3 1 M15 . 0 1 9 1 4 1 1 2 9 5 . 2 7 4 0 . 6 9 2 2 . 0 1 9 1 1 8 1 2 9 3 . 6 8 4 0 . 6 8 4 8 M2C . 0 2 0 0 9 5 1 3 5 9 . 8 4 3 0 . 2 6 8 9 . 0 2 0 0 8 4 1 3 5 9 . 0 9 3 0 . 2 6 6 9 H25 . 0 2 0 3 1 1 1 3 7 4 . 4 1 2 6 . 8445 . 0 2 0 3 0 3 1373.ee 2 6 . 8 4 3 2 H30 . 0 1 9 5 0 3 1 3 1 9 . 7 7 6 1 . 9 7 3 4 . 0 1 9 4 6 9 1 3 1 7 . 4 6 6 1 . 5 6 2 5 H35 . C 1 9 5 8 7 1 3 2 5 . 4 3 1 7 . 1852 . 0 1 9 5 7 7 1 3 2 4 . 7 4 1 7 . 1 8 1 6 2 , 5 . 0 2 0 0 8 4 1 3 5 9 . 0 5 2 1 . 4 9 5 3 . 0 2 0 0 8 8 1 3 5 9 . 3 1 2 1 . 4 9 4 8 2 , 1 0 . C 1 9 7 7 S 1 3 3 8 . 3 3 2 5 . 0 9 4 4 . 0 1 9 7 5 7 1 3 3 6 . 9 6 2 5 . 0 8 3 9 2 , 1 5 . 0 2 0 2 4 5 1369.. 94 1 9 . 2 6 2 7 . 0 2 0 2 4 6 1 3 7 0 . 0 3 1 9 . 2 8 2 7 3 ,5 •019771 1 3 3 7 . 6 9 4 6 . 9 3 7 1 . 0 1 9 7 3 8 1 3 3 5 . 6 3 4 6 . 9 2 1 6 3 , 1 0 . C 1 9 9 9 6 1 3 5 3 . 1 2 2 3 . 6 2 8 3 . 0 1 9 9 7 2 1 3 5 1 . 5 1 2 3 . 6 1 3 3 4 , 5 . 0 1 9 9 7 5 1 3 5 1 . 7 0 5 0 . 0 0 7 6 . C 1 9 9 1 2 1 3 4 7 . 4 5 4 9 . 9 5 8 3 4 , 1 5 . C 1 9 7 0 1 1 3 3 3 . 16 2 3 . 3 7 8 0 . 0 1 9 6 5 9 1 3 3 0 . 3 1 2 3 . 3 3 2 9 5 , 5 . 0 2 0 0 6 1 1 3 5 7 . 5 3 1 9 . 2 5 8 8 . 0 2 0 0 7 4 1 3 5 8 . 4 1 1 9 . 2 5 2 3 5 , 1 0 . 0 1 7 6 0 6 1 1 9 6 . 7 6 6 9 . 7 6 8 3 . 0 1 7 6 4 0 U53.7C 6 9 . 7 5 0 0 5 , 1 5 . C 1 9 4 1 9 1 3 1 4 . 0 6 4 3 . 7 3 0 8 . 0 1 9 4 2 0 1 3 1 4 . 1 6 4 3 . 7 3 0 8 6 , 5 . 0 2 0 1 8 4 1 3 6 5 . 6 6 2 2 . 8 9 6 1 . 0 2 0 1 6 3 1 3 6 4 . 4 3 2 2 . 8 8 4 8 ft.lC . 0 2 0 3 3 1 1 3 7 5 . 7 8 2 1 . 9 7 8 4 . 0 2 0 3 0 8 1 3 7 4 . 2 4 2 1 . 5 6 5 7 6 , 1 5 . 0 2 0 2 5 5 1 3 7 0 . 6 1 3 5 . 5 1 1 3 . 0 2 0 2 4 0 1 3 7 0 . 1 8 35.51.09 1 ,5 . 0 2 0 0 1 9 1 3 5 4 . 6 3 2 5 . 0 6 3 8 . 0 2 0 0 1 0 1 3 5 4 . 0 5 2 5 . 6 6 2 3 1 ,10 . 0 1 9 7 9 9 1 3 3 9 . 7 5 2 0 . 0 6 6 1 . 0 1 9 8 0 0 1 3 3 9 . 8 6 2 0 . 0 6 6 0 I t 15 . 0 1 9 4 5 2 1 3 1 6 . 2 7 3 9 . 4 5 8 0 . 0 1 9 4 4 5 1 3 1 5 . 6 3 39 .4573 1X5 . C 1 9 6 6 . ' 1 3 3 0 . 4 3 2 0 . 0 4 0 9 . 0 1 9 6 2 2 1 3 2 7 . 8 3 2 0 . 0 0 3 2 0X5 . 0 2 0 2 5 2 1 3 7 0 . 4 2 15 . 5477 . 0 2 0 2 1 4 1 3 6 7 . 8 4 1 5 . 5 0 5 3 NO RATIO S Y B A . Y8AR 3 2 3 . 9 5 4 0 1 . 6 1 2 6 C . I E 2 2 4 . 8 3 3 4 5 . 1 4 2 5 4 . 7 3 250, 35 4 4 1 . 3 6 2 0 6 . 9 7 2 7 8 . 0 9 2 5 2 . 5 3 3 1 6 . 4 0 3 7 7 . 2 2 3 1 0 . 7 1 3 3 3 . 7 2 2 3 3 . 5 0 3 5 2 . 7 1 5 0 C . 2 C 3 8 9 . 4 3 2 7 1 . 5 4 3 1 0 . 9 3 3 4 6 . 0 3 2 6 1 . 0 7 2 4 9 . 0 8 3 4 6 . 6 4 3 0 8 . 7 2 3 6 5 . 1 5 1 3 4 8 . 7 1 1 2 7 2 . 6 1 1 4 0 1 . 4 5 1 4 1 8 . 5 5 1 2 8 7 . 0 6 1 3 5 4 . 1 3 1 4 4 7 . 7 4 1 3 1 1 . 8 4 1 3 0 0 . 5 5 1 3 1 0 . 7 7 1 3 9 4 . 3 5 1 3 7 6 . 6 1 14 0 4 . 4 8 1 3 5 2 . 0 0 1 3 4 2 . 5 2 1 2 7 7 . 42 1 4 5 5 . 5 2 1 2 5 0 . 4 5 1 3 6 3 . 7 4 1342«26 1 3 2 6 . 5 7 1 3 8 8 . 0 6 1 3 5 5 . 1 0 1 3 7 1 . 2 9 1 3 3 0 . 5 8 1 2 4 0 . 5 7 1 1 3 3 . 9 4 R K M A 1 K S : R IS IN C . F T / I C i Y B A R , S Y B A . f t , S Y J A . ArtL IN C . F T . 120. T A B L E 6 7 E F F I C I E N C Y OF S A M P L I N G S C H E M E S - P O " Yi3AK = i 3 2 9 . 3 8 ; S A M P L E S 1 Z E = 36 ( S A M P L E O U T P U T ) S S U S I N G R E S T . P = 1 . 2 7 2 7 Y B A R S Y B A . R U S I N G A F P R C X * ' Y E A R P = l „ 0 0 S Y B A . R NO RAT 10 S Y B A . Y B A R R F C C R . 0 2 0 5 5 2 1 3 5 0 . 7 5 2 4 . 0 9 1 0 . 0 2 0 5 4 7 1 3 5 0 . 4 2 2 4 . 0 9 0 6 2 4 7 . 6 0 1 4 1 0 . 1 9 0 . 7 3 3 1 C 8 R S . 0 1 9 3 3 3 1 3 0 8 . 2 3 4 5 . 8 3 3 5 . 0 1 9 3 1 2 1 3 C 6 . 82 4 5 . 8 2 8 7 3 6 2 . 3 2 1 3 3 8 . 0 3 0 . 4 7 8 3 7 2 M5 . C 1 9 3 73 1 3 1 0 . 9 4 4 2 . 9 3 3 1 . 0 1 9 3 3 9 1 2 C 8 . 6 7 4 2 . 5 1 8 3 3 1 6 . 4 5 1 2 9 1 . 5 7 C . 3 6 5 2 3 3 M I O . 0 1 5 9 3 2 1 3 4 e . 7 7 2 4 . 3 8 8 1 . 0 1 9 9 3 2 1 3 4 0 . 8 1 2 4 . 3 8 8 1 2 5 9 . 5 6 1 2 7 2 . 4 2 C . 7 5 3 5 6 6 H I S . 0 1 9 5 1 7 1 3 2 0 . 7 2 3 8 . 5 4 9 0 . 0 1 9 5 1 5 1 3 2 0 . 5 5 2 8 . 5 4 5 C 3 2 8 . 6 3 1 3 2 6 . 0 3 0 . 5 7 1 5 C C M2C . 0 2 0 4 7 3 1 3 8 5 . 4 0 1 5 . 2 0 3 1 . 0 2 0 4 7 0 1 3 8 5 . 1 7 1 5 . 2 0 2 6 2 6 4 . 2 5 1 3 7 3 . C 8 0 . 9 1 5 7 4 8 K 2 5 c . C 1 9 5 8 3 1 3 2 5 . 1 4 3 8 . 5 5 2 5 . 0 1 9 5 7 3 1 3 2 4 . 4 5 3 8 . 9 9 1 6 3 1 0 . 1 5 1 3 1 6 . 6 7 0 . 4 7 8 3 4 4 H 3 0 . C 1 9 8 9 9 1 3 4 6 . 5 3 2 4 . 8 1 9 6 . 0 1 9 8 9 6 1 3 4 6 . 3 7 2 4 . 8 1 5 5 3 0 3 . 3 0 1 3 1 6 . 0 6 C . 6 1 9 7 3 7 M 3 5 . C 2 0 4 2 2 1 3 8 1 . 9 3 1 7 . C 8 5 1 . 0 2 0 4 0 9 1 3 8 1 . 0 5 1 7 . 0 8 0 1 2 5 4 . 0 0 1 3 8 3 . 6 4 0 . 8 6 6 5 1 5 2 , 5 . 0 2 0 0 8 3 1 3 5 9 . 0 0 1 9 . 8 5 5 7 . 0 2 0 0 7 0 1 3 5 8 . 10 1 5 . £ 5 1 4 2 2 2 . 4 6 1 3 1 3 . 5 6 C . 7 9 0 3 2 9 2 , 1 0 . 0 1 9 7 9 3 1 3 3 5 . 3 5 1 8 . 5 8 5 0 . 0 1 9 7 8 4 1 3 3 8 . 7 4 1 8 . 5 2 2 2 2 2 2 . 8 5 1 3 5 8 . 7 5 0 . 0 1 6 3 5 5 2 , 1 5 . 0 2 0 3 4 5 1 3 7 6 . 71 1 7 . 2 1 3 4 . 0 2 0 3 3 3 1 3 7 5 . 8 5 1 7 . 2 0 5 6 2 4 4 . 4 4 1 3 7 2 . 3 9 0 . 8 7 4 C C 2 3 , 5 . C 1 9 3 1 1 1 3 C 6 . 7 2 3 6 . 6 4 5 9 . 0 1 9 2 9 2 1 2 C 5 . 4 5 3 6 . 6 4 0 6 3 5 0 . 7 9 1 3 5 5 . 1 7 C . 6 8 6 0 0 7 3 , 1 0 • C 1 9 0 8 1 1 2 9 1 . 2 0 3 5 . 6 4 8 2 . C 1 9 0 7 3 1 2 5 0 . 6 6 3 9 . 6 4 7 6 3 8 1 . 3 2 1 2 9 2 . 2 5 0 . 6 6 6 C 6 C 4 , 5 . 0 1 9 6 9 7 1 3 3 2 . 8 5 3 8 . 2 0 4 5 . C 1 9 7 0 0 1 3 3 3 . 0 5 3 8 . 2 C 4 2 3 6 9 . 2 3 1 3 4 7 . 8 1 C . 6 5 C 2 0 3 4 , 1 5 . 0 2 0 0 9 4 1 3 5 9 . 7 4 4 2 . 9 2 9 9 . 0 2 0 0 5 0 1 3 5 6 . 7 9 4 2 . 9 0 6 1 3 8 1 . 5 5 1 3 7 4 . 5 2 0 . 6 2 5 7 2 9 5 , 5 . 0 1 8 7 3 8 1 2 6 7 . 5 5 £ 5 . 5 6 7 8 . 0 1 8 7 2 2 1 2 6 6 . 9 1 5 5 . 5 6 6 3 4 5 8 . 9 8 1 3 4 3 . 1 1 0 . 5 2 1 6 C 4 5 . 1 C . 0 1 9 8 4 6 1 3 4 2 . 9 5 2 2 . 8 7 2 9 . 0 1 9 8 0 8 1 3 4 0 . 2 6 2 2 . 6 3 7 8 2 1 8 . 0 7 1 4 0 4 . 5 4 0 . 7 5 5 5 2 4 5 , 1 5 . 0 1 5 5 6 1 1 2 2 3 . 6 8 3 7 . 5 5 5 4 . 0 1 9 5 5 4 1 3 2 3 . 2 1 3 7 . 5 5 4 7 3 4 8 . 6 3 1 3 5 6 . 2 1 0 . 6 E 7 C 1 C 6 , 5 . 0 2 0 0 7 0 1 3 5 8 . 10 2 3 . e 5 1 7 . 0 2 0 0 5 5 1 3 5 7 . C 9 2 3 . 6 8 7 2 2 8 4 . 4 3 1 4 0 4 . 2 2 0 . 8 1 2 1 4 2 6 , 10 . 0 1 5 9 3 5 1 3 4 8 . 9 5 2 8 . 7 1 9 4 . 0 1 9 9 3 3 1 3 4 8 . 8 3 2 8 . 7 1 9 4 3 0 5 . 0 7 1 3 2 0 . 6 S C . 7 5 2 1 5 6 6 , 1 5 . 0 1 5 7 4 8 1 2 3 6 . 2 4 1 8 . 4 5 2 1 • C 1 9 7 3 4 1 3 3 5 . 4 0 1 8 . 4 4 6 9 2 4 2 . 7 7 1 3 4 0 . 4 2 0 . 8 5 2 C 1 4 1 , 5 . C 2 C 0 0 5 1 3 5 3 . 7 4 1 7 . 2 5 0 6 . 0 1 9 9 9 2 1 3 5 2 . 8 5 1 7 . 2 4 6 3 2 2 1 . 0 8 1 3 4 8 . 2 2 0 . 8 4 3 8 1 7 1 , 1 0 . C 1 9 9 3 3 1 3 4 8 . £ 3 2 0 . 1 3 6 8 . 0 1 9 5 1 2 1 3 4 7 . 4 0 2 0 . 1 2 6 1 1 9 9 . 6 1 1 3 2 9 . 0 6 0 . 7 4 2 6 4 8 1 , 1 5 . C 1 9 3 0 2 1 3 1 1 . 5 8 4 0 . 8 5 2 2 . 0 1 9 3 6 2 1 3 1 0 . 1 8 4 0 . £ 6 7 3 3 2 2 . 7 5 1 2 8 7 . 6 4 0 . 4 7 8 6 1 3 I X5 . 0 2 0 5 6 0 1 3 9 1 . 2 7 1 6 . 8 2 2 4 . 0 2 0 5 4 6 1 3 9 0 . 3 2 1 6 . 8 1 7 5 3 3 5 . 5 3 1 2 1 0 . 3 6 C . 5 2 7 7 C 0 DX 5 . 0 2 0 7 0 2 1 4 0 0 . 8 5 1 5 . 2 4 3 2 . C 2 0 6 6 7 1 3 5 8 . 5 2 1 5 . 2 1 2 0 3 2 3 . 2 3 1 1 3 5 . 0 6 C . 5 5 0 5 0 1 R E M A R K S : R IS IN C . F T / I B ; Y B A R , S Y r t A . R . S Y U A . A R E I N C . F T . 121. T A B L E 6 0 . E F F I C I E N C Y OF S A M P L I N G SCHF.MES - P O P Y l i A * = i 3 2 9 . 3 U ; S A M P L E S I Z E = 3 6 ( S A M P L E O U T P U T ) S S U S I N G R E S T . P = l . Y B A R 2 7 2 7 S Y B A . R U S I N G R ' ; A P P R O X Y B A R P = 1 . 0 0 S Y B A . R NO R A T I O S Y B A . Y B A R Rl -C C R . 0 1 9 4 3 3 1 3 1 4 . 9 9 3 9 . 7 3 7 5 . 0 1 9 4 1 2 1 3 1 3 . 6 2 3 9 . 7 3 2 7 2 8 2 . 8 4 1 2 9 4 . 4 4 0 . 2 7 3 2 3 2 R S . 0 1 9 9 4 5 1 3 4 9 . 6 5 2 1 . 5 7 0 9 . 0 1 9 9 2 7 1 3 4 8 . 4 4 2 1 . 5 6 2 9 2 0 0 . 6 5 1 3 2 6 . 4 7 0 . 6 9 6 5 2 4 M5 . C 2 0 0 6 6 1 3 5 7 . 8 4 2 6 . 2 8 5 6 . 0 2 0 0 3 8 1 3 5 5 . 9 3 2 6 . 2 7 2 1 2 5 4 . 6 6 1 2 8 6 . 5 3 C . 7 1 5 9 7 3 M I O . 0 1 8 9 9 0 1 2 8 5 . 0 6 3 6 . 6 9 9 7 . 0 1 8 9 7 3 1 2 6 3 . 8 8 3 8 . 8 9 6 4 2 9 9 . 7 0 1 2 6 5 . 7 5 0 . 4 3 4 2 2 9 H 1 5 . 0 1 9 7 B 2 1 3 3 8 . 6 6 3 8 . 3 6 4 7 . 0 1 9 7 6 1 1 3 2 7 . 22 2 6 . 3 7 8 9 2 9 2 . 9 2 1 3 5 5 . 5 3 0 . 4 3 3 7 5 3 H 2 0 . C 1 9 5 1 9 1 3 2 C . £ 5 2 7 . 9 5 0 3 . 0 1 9 5 1 1 1 3 2 0 . 2 9 2 7 . 9 4 9 1 2 5 9 . 4 5 1 2 0 9 . 5 0 0 . 6 6 0 2 8 9 H 2 5 . 0 1 9 3 3 2 1 3 0 8 . 2 0 2 9 . 4 6 9 2 . 0 1 9 3 3 6 1 3 C 8 . 4 7 3 9 . 4 6 8 8 3 4 4 . 3 7 1 3 0 7 . 5 3 0 . 5 9 8 9 4 2 M 3 0 . C 1 6 5 8 2 1 2 5 7 . 4 3 4 8 . 0 7 8 6 . 0 1 8 5 8 0 1 2 5 7 . 3 2 4 8 . 0 7 0 6 3 5 2 . 5 9 1 2 4 1 . 7 5 0 . 2 9 9 1 C 6 H 3 5 . 0 2 0 5 2 2 1 2 8 8 . 6 6 2 0 . 3 2 2 C . 0 2 0 5 1 0 1 2 8 7 . 67 2 0 . 3 1 8 6 2 1 2 . 8 6 1 4 0 3 . 1 7 0 . 7 5 4 2 8 8 2.5 . 0 1 8 6 5 8 1 2 6 2 . 5 6 3 9 . 8 3 2 3 . 0 1 8 6 3 9 1 2 6 1 . 2 6 3 9 . 6 2 8 1 3 0 3 . 5 5 1 2 1 8 . 7 2 0 . 4 1 7 2 6 5 2 , 1 0 . 0 1 8 5 9 3 1 2 5 6 . 1 4 5 6 . 5 7 8 1 . 0 1 8 5 6 5 1 2 5 6 . 3 0 5 6 . 5 / 1 6 4 1 2 . 7 9 1 2 8 3 . 6 7 0 . 3 1 4 3 1 6 2 . 1 5 . 0 2 0 2 7 9 1 3 7 2 . 2 9 2 4 . 2 2 2 0 . C 2 0 2 8 1 1 3 7 2 . 2 9 2 4 . 2 2 2 6 3 0 2 . 4 7 1 3 8 6 . 4 4 0 . 8 2 0 1 4 7 3 , 5 . C 1 9 1 3 4 1 2 9 4 . 8 0 3 6 . 4 2 5 3 . 0 1 9 1 1 3 1 2 9 3 . 3 9 3 6 . 4 1 8 9 3 5 5 . 3 9 1 3 8 0 . 3 3 C . 7 C 2 5 9 1 3 , 1 0 . 0 2 0 1 0 8 1 2 6 0 . £ 6 1 9 . 4 6 5 3 . 0 2 0 0 9 6 1 3 5 9 . 65 1 9 . 4 6 1 0 3 0 6 . 0 9 1 4 3 6 . 6 1 0 . 8 9 7 8 8 3 4 , 5 . 0 1 9 9 9 5 1 3 5 3 . 0 3 1 7 . 9 5 0 2 . 0 1 9 9 7 8 1 2 5 1 . , 9 1 1 7 . 5 4 3 0 2 2 6 . 9 1 1 3 5 1 . 8 3 0 . 8 4 3 3 8 3 4 , 1 5 . C 1 S 9 9 0 1 3 5 2 . 7 1 1 5 . 2 6 8 3 . 0 1 9 9 9 8 1 3 5 3 . 2 7 1 5 . 2 8 6 4 3 2 3 . 5 1 1 3 3 5 . 0 0 0 . 9 4 4 2 2 3 5,5 . 0 1 8 8 4 1 1 3 7 4 . 9 6 5 7 . 1 1 4 6 . 0 1 8 8 4 1 1 2 7 4 . 9 3 5 7 . 1 1 4 6 5 1 4 . 4 8 1 3 7 1 . 9 4 0 . 6 2 9 6 9 1 5, I C . C 1 9 9 2 0 1 3 4 7 . 9 9 2 0 . 6 6 7 5 . 0 1 9 9 1 9 1 3 4 7 . 8 7 2 0 . 6 6 7 5 2 8 8 . 7 5 1 4 2 2 . 0 6 C . £ 6 5 4 3 5 5 , 1 5 . 0 2 0 2 8 7 1 3 7 2 . 6 2 2 0 . 0 8 5 6 . 0 2 0 2 8 9 1 3 7 2 . 9 2 2 0 . 0 8 5 7 2 7 9 . 5 7 1 4 1 9 . 8 6 0 . 8 6 4 6 3 4 6,5 . 0 1 9 9 4 5 1 3 4 9 . 6 6 4 9 . 3 9 4 4 . 0 1 9 8 8 6 1 3 4 5 . 6 4 4 9 . 3 5 6 6 3 9 2 . 4 9 1 3 8 2 . 9 4 0 . 5 2 6 7 9 4 6, 10 . 0 1 9 5 7 7 1 3 2 4 . 7 5 2 4 . 9 2 8 3 . 0 1 9 5 6 2 1 3 2 3 . 7 5 2 4 . 9 2 4 3 2 7 1 . . 4 6 1 3 1 4 . 7 8 0 . 7 7 1 1 1 3 6 , 1 5 . 0 1 9 8 9 9 1 3 4 6 . 5 5 3 9 . 7 1 3 8 . C 1 9 8 7 7 1 3 4 5 . 0 6 3 9 . 7 0 7 3 3 5 0 . 5 1 1 3 7 7 . 6 9 0 . 6 1 9 3 0 2 1 , 5 . C 2 C 1 8 7 1 3 6 6 . C O 1 6 . 0 5 7 0 . 0 2 0 1 8 4 1 3 6 5 . 8 3 1 6 . 0 5 6 8 2 3 8 . 6 9 1 3 5 5 . 5 6 • 0 . 0 6 2 3 1 5 1 , 1 0 . 0 1 8 8 3 9 1 2 7 4 . 8 4 5 0 . 8 C 4 6 . 0 1 0 8 1 8 1 2 7 3 . 4 3 5 0 . 8 0 0 8 3 8 8 . 8 1 1 2 7 5 . 1 4 0 . 4 1 5 5 8 2 1 , 1 5 . 0 1 9 4 28 1 3 1 4 . 7 0 4 4 . 2 8 1 7 . 0 1 9 4 3 8 1 3 1 5 . 3 5 4 4 . 2 8 0 4 3 7 5 . 2 2 1 3 2 1 . 4 4 0 . 5 6 7 7 6 8 1 X 5 . 0 1 9 9 7 2 1 2 5 1 . 4 9 1 8 . 8 3 5 0 . 0 1 9 9 4 0 1 3 4 9 . 8 7 1 8 . 8 2 2 1 2 9 8 . 4 9 1 2 7 1 . 2 2 0 . 9 C 2 3 7 9 0 X 5 . 0 2 0 5 8 8 1 3 9 3 . 1 5 1 6 . 2 4 5 0 . 0 2 0 5 7 2 1 3 9 2 . 1 2 1 6 . 2 3 0 7 3 5 0 . 1 5 1 1 1 4 . 5 6 0 . 9 4 7 1 2 6 R E M A R K S : R I S I N C . F T / I B ; Y B A R , S Y B A . R . S Y J A . A K E IN 122. E F F I C I E N C Y OF S A M P L I N G S C H E M E S T A B L E 6 9 - P O P Y U A R = X 3 2 9 . 3 0 ; S A M P L E S U E = « SS CR RS MS MIO M15 M20 M25 M30 M3 5 2,5 2 , 10 2,15 3,5 3,10 1,5 4 , 15 5.5 5,10 5,15 6,5 6 , 1 0 6,15 1,5 1,10 1.15 1 X 5 0X5 U S I N G E S T . P = 1 . 2 7 2 7 U S I N G A P P R O X P = 1 . 0 0 ( S A M P L E O U T P U T ) KG R A T 1 C R . 0 1 9 5 9 4 . 0 2 0 2 3 3 . C 1 5 4 4 8 . 0 1 9 7 9 6 . 0 1 5 0 6 1 . C 1 9 5 1 2 . 0 2 0 2 7 5 . C 1 9 8 2 0 . 0 1 9 4 0 4 . 0 1 9 9 1 9 . 0 1 5 7 1 7 . 0 1 9 5 9 6 . 0 2 0 1 2 1 . 0 2 0 1 9 1 . 0 1 9 8 5 9 . 0 1 8 7 8 7 . 0 1 9 6 0 2 . 0 1 9 2 5 0 . 0 1 9 4 6 9 . 0 1 9 1 3 6 . 0 1 9 3 7 1 . 0 2 0 4 0 2 . 0 1 5 5 7 5 . 0 1 9 5 4 B . C 1 9 9 3 0 . 0 1 9 9 3 3 . 0 2 0 1 4 5 YBAR 1 3 2 5 . £ 5 1 3 6 9 . 1 2 1 3 1 6 . 0 3 1 3 2 5 . 5 7 1 3 4 3 . 9 7 1 3 2 6 . 2 6 1 3 7 1 . 9 8 1 3 4 1 . 2 2 1 3 1 2 . C 6 1 3 4 7 . 6 8 1 3 3 4 . 2 2 1 3 2 6 . 0 7 1 3 6 1 . 5 5 1 2 6 6 . 3 1 1 3 4 3 . 8 2 1 2 7 1 . 2 3 1 3 2 6 . 4 3 1 3 0 2 . 6 4 1 3 1 7 . 4 6 1 2 9 5 . 0 6 1 3 1 0 . C C 1 3 8 0 . 5 6 1 3 2 4 . £ 8 1 2 2 2 . E C 1 3 4 8 . 6 2 1 3 4 8 . 8 4 1 3 6 3 . 2 0 S Y B A . R 1 9 . 3 2 8 1 3 2 . 4 2 1 3 1 7 . 2 8 2 1 1 9 . 2 8 9 4 1 6 . 6 5 3 6 2 4 . 8 1 5 5 1 6 . 1 5 3 8 3 3 . 3 2 6 2 4 C . 6 8 1 8 2 0 . 5 4 6 8 3 7 . 5 1 4 5 3 1 . 7 4 9 0 1 8 . 7 5 4 8 1 7 . 4 5 4 1 1 8 . 8 5 7 7 4 4 . 1 1 4 6 1 8 . 9 4 2 1 2 2 . 2 2 2 3 2 3 . 7 2 1 4 3 1 . 7 5 2 4 4 0 . 3 7 0 8 2 1 . 1 0 2 2 2 7 . 6 4 5 4 2 4 . 4 6 8 6 1 7 . 6 6 5 0 I S . 5 7 0 3 2 8 . 8 5 2 8 R . 0 1 9 5 8 7 . 0 2 0 1 1 5 . 0 1 9 4 4 6 . 0 1 9 8 9 5 • C 1 9 8 6 8 . 0 1 9 5 1 9 . 0 2 0 2 5 4 . 0 1 9 8 1 8 . 0 1 9 3 8 7 . 0 1 9 9 0 2 . 0 1 5 6 8 9 . 0 1 9 5 9 1 . 0 2 0 1 2 6 . 0 2 0 1 9 0 . 0 1 9 8 3 4 . 0 1 8 7 6 3 . 0 1 9 5 9 7 . 0 1 9 2 3 5 . 0 1 9 4 6 1 . 0 1 9 1 4 1 . C 1 9 3 0 0 . 0 2 0 3 7 5 . 0 1 9 5 8 0 . 0 1 9 5 3 9 .019920 . 0 1 9 9 2 1 . 0 2 0 0 7 9 YBAR 1 3 2 5 . 4 0 1 3 6 1 . 1 8 1 3 1 5 . 8 7 1 3 3 9 . 5 4 1 3 4 4 . 4 2 1 3 2 6 . 8 1 1 3 7 0 . 5 8 1 3 4 1 . 0 5 1 3 1 1 . 87 1 3 4 6 . 7 6 1 3 3 2 . 3 5 1 3 2 5 . 6 5 1 3 6 1 . 9 0 1 3 6 6 . 2 2 1 3 4 2 . 1 4 1 2 6 9 . 6 6 1 3 2 6 . 1 0 1 3 0 1 . 6 2 1 3 1 6 . 9 2 1 2 5 3 . 4 4 1 3 C 6 . 0 5 1 3 7 8 . 7 7 1 3 2 5 . 5 0 1 3 2 2 . 2 0 1 3 4 7 . 5 8 1 3 4 8 . 0 2 1 3 5 8 . 7 4 S Y B A . R 1 9 . 3 3 7 0 3 2 . 3 0 5 0 1 7 . 2 8 2 0 1 9 . 2 8 9 5 1 6 . 6 5 2 6 3 4 . 8 1 8 9 1 6 . 1 8 3 5 3 3 . 3 2 6 1 4 0 . 6 7 8 6 2 C . 5 4 2 C 3 7 . 5 0 6 5 2 1 . 7 4 8 8 1 8 . 7 9 4 1 1 7 . 4 9 4 1 1 8 . 6 4 5 0 4 4 . 1 0 9 2 1 8 . 5 4 1 6 2 2 . 2 1 7 4 2 3 . 7 2 0 3 2 1 . 7 4 4 8 4 0 . 3 1 9 9 2 1 . C 8 7 6 2 7 . 6 4 3 8 3 4 . 4 6 7 0 1 7 . 6 6 2 8 1 5 . 9 7 6 0 2 8 . 0 C 8 0 R E M A R K S : R I S IN C . F T / I B i Y B A R , S Y B A . R , S Y B A . A * E IN S Y B A . 2 0 5 . 2 2 2 0 1 . 7 3 2 2 8 . 3 0 2 1 8 . 2 2 2 3 0 . 0 1 2 9 5 . 5 6 1 9 6 . 1 5 2 8 5 . 4 4 3 1 9 . 0 2 2 0 1 . 5 9 2 9 9 . 0 9 3 0 3 . 3 8 3 0 6 . 5 1 3 0 0 . 7 9 2 1 0 . 0 5 3 8 0 . 1 7 2 8 7 . 56 24 ).. 38 2 5 0 . 6 7 2 8 5 . 9 6 2 8 9 . 5 9 2 1 9 . 5 4 2 7 2 . 2 6 2 8 9 . 8 6 2 0 0 . 6 9 3 0 8 . 1 3 3 0 6 . 8 2 C . F T . YBAR 1 2 2 2 . 9 1 1 3 2 0 . 2 8 1 2 7 0 . 2 1 1 3 2 2 . 5 6 1 2 5 9 . 31 1 3 9 3 . 0 4 1 3 7 5 . 4 2 1 3 1 2 . 4 0 1 3 1 7 . 6 4 1 3 C S . 6 5 1 3 8 5 . 7 3 1 3 5 1 . 6 4 1 4 3 5 . 2 4 1 4 4 0 . 6 2 1 3 0 9 . 3 1 1 3 0 5 . 7 6 1 4 1 8 . 9 8 1 2 6 8 . 6 4 1 3 8 3 . 3 6 1 2 8 4 . 2 2 1 2 9 1 . 2 9 1 3 8 6 . 7 1 1 3 3 2 . 5 3 1 3 2 6 . 2 2 1 3 6 1 . 3 6 1 2 4 3 . 8 2 1 1 2 4 . 3 8 RHO 0 . 7 5 5 C 5 6 0 . 3 7 0 6 2 2 0 . 8 4 6 3 7 1 0 . 7 8 5 7 3 2 C . 6 6 3 4 4 6 0 . 5 6 5 5 6 3 0 . 8 3 5 8 8 2 0 . 5 7 6 3 1 4 0 . 4 6 2 4 6 7 C . 7 C 2 3 3 3 0 . 5 C S C 5 " 0 . 6 8 0 8 4 2 C . 5 C 2 6 6 2 • 0 . 5 1 3 3 7 5 C . 8 C C 7 C C 0 . 5 6 8 7 5 C 0 . 6 8 7 6 6 3 0 . 7 737 71 0 . 7 5 0 8 0 3 C . 6 4 3 C 1 2 0 . 4 2 e 4 7 e 0 . 7 7 1 3 5 4 0 . 7 C 9 C C 1 0 . 5 5 6 9 2 4 0 . 7 5 3 8 7 3 0 . 5 2 2 2 C 6 0 . 7 7 6 4 2 1 123. T A B L E 7 0 E F F I C I E N C Y OF S A M P L I N G S C H E M E S - P O P Y B A R = i 3 2 9 . i 8 i S A M P L E SI ZE =61 ( S A M P L E O U T P U T ) S S U S I N G R E S T . P = 1 . 2 7 2 7 Y B A R S Y B A . R U S I N G R i A P P R O X P Y E A R = 1 . 0 0 S Y B A . R NG R A T I C S Y B A . Y B A R R h C CR . 0 1 9 3 6 9 1 3 1 0 . 6 6 1 7 . 2 1 1 6 . 0 1 9 3 5 4 1 3 C 9 . 6 4 1 7 . 2 0 7 6 1 6 6 . 3 4 1 3 4 2 . 4 6 0 . 7 0 7 5 6 3 R S . 0 1 9 0 5 4 1 3 4 3 . 5 2 1 3 . 6 9 5 9 . 0 1 9 8 3 6 1 3 1 2 . 3 0 1 3 . 6 8 9 7 1 7 0 . 2 3 1 3 1 7 . 18 0 . 8 3 8 1 5 4 M5 . 0 2 C 1 9 3 1 3 6 6 . 4 5 1 1 . 4 6 6 8 . 0 2 0 1 8 5 1 3 6 5 . 92 1 1 . 4 6 5 2 1 7 4 . 5 0 1 2 5 4 . 3 4 C . 8 5 C 5 5 2 M I O . 0 1 9 0 7 7 1 3 4 5 . 0 6 2 4 . 6 9 8 2 . 0 1 9 8 6 3 1 3 4 4 . 1 0 2 4 . 6 9 5 7 2 1 7 . 4 2 1 3 5 4 . 5 9 0 . 6 1 6 0 4 2 M 1 5 . 0 1 9 5 0 7 1 3 2 0 . 0 3 3 0 . 3 1 4 9 . 0 1 9 4 9 9 1 3 1 9 . 4 8 3 0 . 3 1 4 3 2 5 7 . 8 2 1 3 1 4 . 6 2 0 . 5 6 9 3 9 8 M 2 0 . C 1 9 B 0 2 1 3 3 9 . 9 e 1 5 . 3 1 9 6 . 0 1 9 7 9 9 1 3 3 9 . 7 8 1 5 . 3 1 9 5 1 8 9 . 5 7 1 3 3 0 . 5 2 0 . 8 2 4 8 C 0 M25 . 0 1 9 8 1 3 1 3 4 0 . 7 5 1 1 . 5 0 1 4 . 0 1 9 8 0 5 1 3 4 0 . 2 2 1 1 . 5 0 0 1 1 5 2 . 1 4 1 3 2 9 . 4 9 0 . 8 5 1 7 6 7 M 3 0 . 0 1 9 1 5 8 1 2 9 6 . 3 8 2 6 . 4 6 5 4 . 0 1 9 1 5 5 1 2 9 6 . 2 1 2 6 . 4 6 5 3 2 3 3 . 6 4 1 2 1 1 . 6 2 0 . 6 C 9 7 3 1 M 3 5 . 0 2 0 3 7 1 1 3 7 8 . 5 0 2 1 . 2 5 7 2 . 0 2 0 2 6 8 1 3 7 2 . 6 5 2 1 . 1 9 0 4 1 8 6 . 1 1 1 3 8 7 . 6 6 0 . 7 C 7 2 9 6 2,5 . 0 1 9 7 8 1 1 3 3 8 . 5 4 1 5 . 4 6 4 4 . 0 1 9 7 7 1 1 3 3 7 . 8 6 1 5 . 4 6 2 7 1 6 2 . 5 4 1 2 8 4 . 8 7 C . 7 5 2 1 7 7 2,10 . 0 1 9 7 9 0 1 3 3 9 . 2 0 1 4 . 8 6 3 2 . C 1 9 7 6 9 1 3 3 7 . 7 6 1 4 . 8 5 3 9 1 6 1 . 4 0 1 3 6 2 . 1 3 0 . 7 9 2 4 6 0 2 , 1 5 . 0 1 9 5 5 5 1 3 2 3 . 2 8 2 5 . 0 6 5 6 . 0 1 9 5 2 6 1 3 2 1 . 3 2 2 5 . C 5 6 0 2 4 3 . 1 5 1 3 4 2 . 7 C C . 7 C 6 3 4 8 3 , 5 . C 1 9 8 C 6 1 3 4 0 . 2 2 1 8 . 2 0 3 1 . 0 1 9 7 7 7 1 3 3 8 . 2 9 1 8 . 1 8 0 9 2 1 1 . 4 8 1 4 0 4 . 8 4 0 . 6 1 7 C 5 2 3 , 1 0 . 0 2 0 5 7 4 1 3 9 2.22 1 8 . 6 4 7 0 . 0 2 0 5 5 6 1 3 9 1 . 0 2 1 8 . 6 4 1 2 2 6 2 . 7 2 1 4 2 6 . 1 1 0 . 8 7 1 7 8 0 4.5 . 0 1 9 5 4 4 1 3 2 2 . 5 1 2 2 . 4 8 7 6 . 0 1 9 5 4 7 1 3 2 2 . 7 6 2 2 . 4 8 7 3 2 6 3 . 5 9 1 2 1 6 . 3 3 C.eC2683 4 , 1 5 . 0 1 9 2 5 2 1 3 0 2 . 7 9 2 7 . 5 1 5 7 . 0 1 9 2 3 0 1 3 C 1 . 2 6 2 7 . 5 1 4 4 2 6 5 . 2 0 1 3 9 7 . 4 3 0 . 6 9 3 7 2 4 5 , 5 . 0 1 9 7 7 4 1 3 3 8 . C 8 1 6 . 9 8 5 0 . 0 1 9 7 5 4 1 3 3 6 . 7 2 1 6 . 9 7 7 1 2 2 5 . 5 6 1 4 2 3 . 5 1 0 . 8 5 7 6 3 6 5,1 0 . 0 1 9 9 9 2 1 3 5 2 . 8 2 1 5 . 0 2 9 0 . 0 1 9 9 9 8 1 3 5 3 . 2 4 1 5 . 0 2 8 1 2 2 4 . 6 0 1 4 5 0 . 4 3 0 . 6 6 4 7 6 2 5 , 1 5 . 0 1 9 1 5 3 1 2 9 6 . C 5 2 6 . 9 3 6 8 . 0 1 9 1 4 3 1 2 9 5 . 3 7 2 6 . 9 3 5 8 2 5 1 . 0 1 1 3 2 6 . 2 8 0 . 6 6 1 9 5 1 6.5 . 0 1 8 9 9 1 1 2 8 5 . I C 3 4 . 5 8 3 0 . 0 1 8 9 7 9 1 2 8 4 . 2 8 3 4 . 5 6 2 1 3 0 4 . 4 4 1 3 0 4 . 9 7 0 . 6 0 7 1 9 4 6 , 1 0 . 0 2 0 0 6 0 1 3 5 8 . C l 1 7 . 5 3 3 5 . C 2 0 0 4 2 1 3 5 6 . 2 1 1 7 . 5 2 2 8 1 9 2 . 0 6 1 3 0 2 . 0 0 0 . 7 7 7 1 0 1 6 , 1 5 . 0 1 9 5 4 7 1 3 2 2 . 7 1 3 2 . 2 4 9 1 . 0 1 9 5 0 3 1 3 1 9 . 7 4 3 2 . 2 3 2 0 2 4 7 . 9 1 1 3 1 9 . 3 1 0 . 4 7 7 4 0 3 1 .5 . 0 1 9 8 3 1 1 3 4 1 . 5 7 1 1 . 7 8 0 3 . 0 1 9 8 3 2 1 3 4 2 . 0 1 1 1 . 7 8 8 3 1 8 5 . 4 6 1 3 5 7 . 4 1 0 . 8 9 5 5 6 2 1 , 1 0 . 0 1 9 5 9 4 1 3 2 5 . 9 1 2 6 . 0 1 4 6 . 0 1 9 5 8 6 1 3 3 7 . 3 7 2 6 . C 1 4 1 2 2 3 . 6 3 1 3 2 8 . 0 8 0 . 5 8 1 8 3 4 1 , 1 5 . 0 2 0 1 2 2 1 3 6 1 . 6 7 1 3 . 9 0 6 8 . 0 2 0 1 1 2 1 3 6 0 . 9 4 1 3 . 9 0 4 4 1 6 7 . 24 1 3 7 0 . 7 5 C . 8 1 9 6 4 7 1 X 5 . 0 2 0 1 9 4 1 3 6 6 . 5 3 1 5 . 7 4 5 7 . 0 2 0 1 5 4 1 3 6 3 . 6 4 1 5 . 7 2 0 6 2 3 9 . 5 7 1 2 4 5 ' . 54 0 . 8 9 9 6 2 4 DX5 . 0 2 0 6 8 6 1 3 9 9 . 7 8 1 4.5925 . 0 2 0 6 4 0 1 3 9 6 . 6 7 1 4 . 5 6 0 6 2 4 5 . 6 9 1 1 5 2 . 6 1 C . 9 2 0 7 8 7 R E M A R K S ! R I S IN C . F T / I O i Y B A R , S Y B A . R , S Y B A . A R E IN C . F T . 124; E F F I C I E N C Y DF S A M P L I N G S C H E M E S TAlJLE 71 - P O P Y8AR = 1 3 2 9 . 3 8 ; S A M P L E S I Z G = 1 6 4 ( S A M P L E O U T P U T ) SS U S I N G R E S T . P - 1 . 2 7 2 7 Y B A R S Y B A . R U S I N G R ' A P P R O X Y B A R P = 1 . 0 C S Y E A . R NO R A T I O S Y B A . Y B A R RHO C R . 0 1 9 7 7 1 1 3 3 7 . 9 1 1 5 . 4 9 C 5 . 0 1 9 7 6 2 1 3 2 7 . 2 6 1 5 . 4 9 0 0 1 3 3 . 2 2 1 3 4 7 . 3 3 0 . 5 8 3 9 5 2 RS . 0 1 9 7 2 7 1 3 3 4 . E 8 1 1 . 4 4 9 2 . 0 1 9 7 2 0 1 3 2 4 . 4 C 1 1 . 4 4 8 8 1 1 5 . 3 4 1 3 5 0 . 6 4 C . 7 2 C 7 7 6 H 5 . 0 1 9 5 2 1 1 3 2 0 . 9 9 1 4 . 7 1 8 7 . C 1 9 5 0 5 1 3 1 9 . 9 1 1 4 . 7 1 7 1 1 2 5 . 5 3 1 2 9 5 . 7 6 0 . 5 8 2 2 4 9 M I O . 0 1 9 4 5 0 1 3 1 6 . 14 1 5 . 5 8 3 5 , . 0 1 9 4 3 3 1 3 1 5 . 0 1 1 5 . 5 6 1 7 1 3 1 . 9 1 1 3 2 8 . 2 0 0 . 5 7 4 1 C 0 M 1 5 . 0 1 9 5 1 5 1 3 2 0 . 5 4 1 4 . 8 6 8 3 . 0 1 9 4 9 8 1 3 1 9 . 3 8 1 4 . 8 6 6 5 1 2 7 . 6 2 1 3 1 2 . 2 1 C . 5 9 0 1 4 1 H 2 0 . 0 1 9 8 7 2 1 3 4 4 . 7 C 1 2 . 7 5 2 0 . C 1 9 6 4 1 1 3 4 2 . 6 2 1 2 . 7 4 7 3 1 1 3 . 6 1 1 3 4 4 . 2 8 0 . 6 2 6 8 9 1 H 2 5 . C 1 9 5 2 8 1 3 2 1 . 4 4 1 5 . 3 2 0 0 . 0 1 9 5 1 2 1 3 2 0 . 2 6 1 5 . 3 1 6 6 1 3 2 . 6 6 1 3 2 6 . 6 6 0 . 5 S 6 3 1 9 M 3 0 . C 1 9 6 7 3 1 3 3 1 . 2 E 1 1 . 5 2 3 4 . C 1 9 6 0 2 1 3 3 0 . 5 1 1 1 . 5 2 2 5 1 2 2 . 3 3 1 3 1 8 . 3 5 C . 7 5 4 5 6 2 H 3 5 . 0 1 9 9 7 1 1 3 5 1 . 4 3 1 1 . 8 7 4 7 . 0 1 9 9 5 7 1 3 5 0 . 4 7 1 1 . 8 7 3 2 1 1 7 . 6 4 1 3 5 3 . 8 0 0 . 7 1 4 2 9 9 2,5 . C 1 9 7 6 1 1 3 3 7 . I E 9 . 6 0 5 3 . 0 1 9 7 5 2 1 3 3 6 . 6 2 9 . 6 0 4 7 1 0 2 . 9 6 1 2 9 0 . 0 7 0 . 7 6 C 4 7 2 2 , 1 0 . 0 1 9 6 4 2 1 2 2 9 . 1 7 1 4 . 7 9 1 7 . 0 1 9 6 2 1 1 3 2 7 . 7 2 1 4 . 7 8 8 6 1 2 8 . 5 1 1 3 7 4 . 6 6 0 . 6 1 C 2 e 5 2 . 1 5 . C 1 9 8 3 6 1 3 4 2 . 3 0 1 5 . 5 6 5 3 . 0 1 9 8 1 3 1 3 4 0 . 6 9 1 5 . 5 6 1 9 1 4 9 . 7 6 1 3 6 5 . 7 2 0 . 6 9 4 C 6 3 3,5 . 0 1 9 5 6 8 1 2 2 4 . 1 5 1 4 . 1 9 0 2 . C 1 9 5 4 4 1 3 2 2 . 5 5 1 4 . 1 8 6 3 1 4 3 . 9 6 1 3 6 5 . 1 4 0 . 7 2 2 4 9 4 3 , 1 0 . 0 2 0 0 1 0 1 3 5 4 . 0 6 1 0 . 0 7 6 6 . 0 1 9 9 8 4 1 3 5 2 . 3 2 1 0 . C 6 9 9 1 4 4 . 2 4 1 4 1 0 . 2 6 0 . 8 7 9 3 2 4 «.5 . 0 1 9 0 6 4 1 2 9 C . C 7 1 8 . 9 2 7 3 . 0 1 9 0 3 4 1 2 8 8 . 0 5 1 8 . 9 2 3 3 1 5 8 . 6 6 1 2 7 4 . 9 1 0 . 5 6 2 7 5 5 4 , 1 5 . 0 1 9 7 8 3 1 3 2 6 . 7 0 1 2 . 3 9 6 5 . 0 1 9 7 7 6 1 3 2 8 . 1 9 1 2 . 3 9 6 1 1 5 1 . 5 2 1 3 8 0 . 0 4 0 . 8 2 0 2 4 3 5,5 . C 1 9 7 4 0 1 3 3 5 . 7 9 1 1 . 0 9 9 3 . 0 1 9 7 2 6 1 3 3 4 . 6 1 1 1 . C 5 7 3 1 3 9 . 7 6 1 4 4 6 . 6 7 C . 8 3 4 3 C 8 5 , 1 0 . 0 2 0 0 2 1 1 3 5 4 . 7 9 1 0 . 7 1 1 8 . 0 2 0 0 2 2 1 3 5 4 . 8 7 1 0 . 7 1 1 7 1 3 0 . 7 8 1 4 4 5 . 2 4 0 . 8 1 9 5 5 7 5 , 1 5 . 0 1 9 7 5 1 1 3 3 6 . 9 3 1 3 . 8 3 2 2 . 0 1 9 7 4 7 1 3 3 6 . 2 6 1 2 . 8 3 1 6 1 3 7 . 9 3 1 3 8 9 . 4 5 0 . 7 1 4 2 7 0 6,5 . 0 1 9 6 4 1 1 3 2 9 . I C 1 4 . 2 6 7 3 . 0 1 9 6 4 3 1 3 2 9 . 2 2 1 4 . 2 6 7 2 1 5 6 . 6 0 1 3 3 0 . 7 3 C . 7 7 C 5 6 5 6 , 1 0 . 0 1 9 8 3 5 1 3 4 2 . 2 2 1 1 . 7 5 4 8 . 0 1 9 8 2 4 1 3 4 1 . 4 6 1 1 . 7 5 3 8 1 2 6 . 9 1 1 2 9 9 . 9 4 0 . 7 6 4 2 9 6 6 , 1 5 . C 2 0 2 C 8 1 3 6 7 . 4 5 1 1 . 6 5 9 6 . 0 2 0 1 9 5 1 3 6 6 . 6 0 1 1 . 6 5 8 5 1 2 8 . 8 3 1 3 6 1 . 9 9 C . 7 7 6 5 4 5 1,5 . 0 1 9 8 2 1 1 3 4 1 . 2 9 1 3 . 2 8 4 1 . 0 1 9 8 1 5 1 3 4 0 . 8 3 1 3 . 2 8 3 8 1 2 5 . 6 8 1 3 3 3 . 8 8 C . 6 7 4 3 9 2 1 , 1 0 . 0 1 9 6 5 4 1 3 2 9 . 9 6 1 4 . 0 4 3 5 . C 1 9 6 4 0 1 3 2 9 . 0 2 1 4 . C 4 2 3 1 1 9 . 7 4 1 3 2 8 . 7 4 0 . 5 8 1 0 3 3 1 . 1 5 . 0 1 9 2 5 8 1 3 0 3 . 1 4 1 6 . 7 8 6 0 . 0 1 9 2 4 7 1 3 0 2 " . 4 5 1 6 . 7 8 5 4 1 3 e . 6 2 1 3 C 4 . 3 0 C . 5 3 5 0 9 2 1 X 5 . 0 1 9 4 9 9 1 3 1 9 . 5 1 1 6 . 4 9 5 5 . 0 1 9 4 5 5 1 2 1 6 . 4 7 1 6 . 4 8 6 2 1 6 7 . 0 6 1 2 2 6 . 2 4 0 . 7 2 2 5 5 7 D X 5 . 0 2 0 4 5 1 1 3 8 3 . 8 7 9 . 7 3 0 3 . 0 2 0 4 1 2 1 2 6 1 . 2 6 9 . 7 1 9 1 1 5 3 . 2 5 1 2 2 1 . 5 7 0 . 9 0 2 1 4 8 R E M A R K S : R ! S I N C . F T / I B ; Y B A R , S Y B A . R , S Y B A . A R E I N C . F T . 125. ' ^  T A B L E 72 E F F I C I E N C Y OF S A M P L I N G S C H E M E S - P O P YB A R = 1 3 2 9 . 3 8 . S A M P L E S I Z E = 1 8 0 ( S A M P L E - O U T P U T ) S S U S I N G R E S T . P=l YBAR . 2 7 2 7 S Y B A . R U S I N G A P P R O X R Y B A R P = 1 . C C S Y E A . R NO 1 S Y B A . R A T I O Y B A R RHO CR . 0 1 9 7 0 9 1 3 3 3 . 7 1 9 . 7 2 9 0 . 0 1 9 6 9 7 1 3 3 2 . 8 9 9 . 7 2 7 8 1 0 8 . 5 7 1 3 3 3 . 4 3 0 . 7 8 3 0 7 4 R S • 0 1 9 6 7 C 1 3 3 1 . C 8 1 4 . 5 0 5 7 . 0 1 9 6 6 2 1 3 3 0 . 4 9 1 4 . 5 0 5 4 1 2 8 . 9 9 1 3 2 5 . 7 2 C . 6 1 8 5 0 5 M5 . 0 1 9 6 4 6 1 3 2 9 . 4 3 1 7 . 3 1 5 7 . 0 1 9 6 1 6 1 3 2 7 . 4 3 1 7 . 3 1 5 4 1 3 0 . 7 2 1 3 1 8 . 7 4 0 . 4 2 3 1 8 9 MIO . 0 2 0 0 2 2 1 3 5 4 . 6 5 8 . 8 5 1 9 . 0 2 0 0 0 2 1 3 E 3 . 4 8 8 . 6 4 8 3 1 0 2 . 3 2 1 3 2 6 . 4 1 C . 8 C 7 7 3 2 M 1 5 . 0 1 5 5 1 7 1 3 2 0 . 6 5 1 3 . 8 1 0 5 . 0 1 9 5 0 8 1 3 2 0 . 0 7 1 3 . 8 1 0 1 1 2 3 . 2 1 1 3 2 2 . 0 5 0 . 6 2 2 3 1 6 M20 . 0 1 9 9 2 6 1 3 4 8 . 3 5 1 3 . 6 5 5 6 . 0 1 9 8 8 6 1 3 4 5 . 6 4 1 3 . 6 5 2 1 1 1 8 . 2 3 1 3 3 8 . 9 6 0 . 6 1 6 7 6 2 M2 5 . 0 1 9 8 3 1 1 3 4 1 . 5 7 1 0 . 9 0 8 7 . 0 1 9 8 1 6 1 3 4 0 . 9 5 1 0 . 5 0 6 9 9 3 . 7 7 1 3 4 7 . 0 7 0 . 6 6 6 5 7 4 M30 . 0 1 9 6 5 0 1 3 2 9 . 7 0 1 3 . 5 5 5 0 . 0 1 9 6 4 2 1 3 2 9 . 1 7 1 3 . 5 5 4 7 1 1 6 . 9 8 1 3 2 2 . 1 8 0 . 5 8 7 5 6 7 M35 . 0 1 9 9 3 3 1 3 4 8 . 8 2 1 0 . 2 2 2 5 . 0 1 9 9 2 1 1 3 4 8 . 0 4 1 0 . 2 2 1 5 9 8 . 9 4 1 3 3 3 . 6 3 0 . 6 5 7 9 4 4 2 , 5 . 0 1 5 8 9 0 1 3 4 5 . 5 6 1 1 . 6 6 8 9 . 0 1 9 8 8 1 ' 1 3 4 5 . 3 4 1 1 . 6 6 8 4 1 1 4 . 2 9 1 2 8 5 . 6 1 0 . 7 C 1 8 8 7 2 , 1 0 . 0 1 9 5 4 0 1 3 2 2 . 2 8 1 4 . 8 8 8 5 . 0 1 9 5 2 1 1 3 2 1 . 0 0 1 4 . 6 8 6 3 1 2 6 . 8 9 1 3 7 7 . 7 8 0 . 5 8 3 8 6 8 2 , 1 5 . 0 1 9 7 4 5 1 3 3 6 . 1 4 1 4 . 7 6 1 8 . 0 1 9 7 2 3 1 3 3 4 . 6 1 1 4 . 7 5 8 8 1 4 1 . 6 8 1 3 5 6 . 1 2 C . 6 5 1 8 7 4 3 , 5 . 0 1 9 7 1 5 1 3 3 4 . 10 1 3 . 3 3 4 0 . 0 1 5 6 9 3 1 3 3 2 . 6 2 1 3 . 3 3 1 0 1 4 0 . 7 9 1 3 8 2 . 5 0 0 . 7 5 4 2 9 2 3 , 1 0 . 0 1 5 6 8 6 1 3 3 2 . 1 5 1 2 . 6 0 9 4 . 0 1 9 6 7 5 1 3 3 1 . 3 7 1 2 . 6 0 8 5 1 5 2 . 9 0 1 3 7 7 . 4 7 0 . 8 1 7 5 8 9 4 , 5 . 0 2 0 0 3 0 1 3 5 5 . 4 C 8 . 7 7 8 0 . 0 2 0 0 2 9 1 3 5 5 . 3 2 8 . 7 7 8 0 1 2 1 . 9 5 1 2 4 5 . 7 4 0 . 6 6 3 7 5 5 4 , 1 5 . 0 2 0 0 2 8 1 3 5 5 . 2 5 8 . 1 7 5 7 . 0 2 0 0 1 7 1 3 5 4 . 5 6 8 . 1 7 4 5 1 3 2 . 1 7 1 3 8 3 . 7 1 0 . 5 0 2 7 6 8 5 , 5 . 0 1 5 8 6 7 1 3 4 4 . 4 1 1 2 . 9 2 7 3 . 0 1 9 8 5 6 1 3 4 3 . 6 3 1 2 . 5 2 6 3 1 4 8 . 4 3 1 4 4 3 . 5 0 C . 7 5 4 2 3 1 5 , 1 0 . 0 1 9 8 7 4 1 3 4 4 . £ 2 1 2 . 2 6 3 5 . 0 1 9 8 6 5 1 3 4 4 . 2 4 1 2 . 2 6 3 0 1 2 6 . 4 2 1 4 3 3 . 8 8 0 . 7 2 5 7 7 5 5 , 1 5 . 0 1 9 9 1 8 1 3 4 7 . 6 6 8 . 4 5 2 3 . 0 1 9 9 1 1 1 3 4 7 . 3 5 8 . 4 5 1 7 1 1 1 . 8 9 1 4 0 7 . 1 1 0 . 8 5 0 C 0 9 6 , 5 . 0 1 5 8 9 5 1 3 4 6 . 2 5 7 . 3 1 5 7 . 0 1 9 8 9 4 1 3 4 6 . 19 7 . 3 1 5 7 1 2 2 . 2 7 1 3 5 7 . 6 e 0 . S C 8 0 7 3 6 , 1 0 . 0 1 9 7 8 5 1 3 3 8 . 8 1 1 0 . 6 6 0 0 . 0 1 9 7 6 7 1 3 3 7 . 6 0 1 0 . 6 5 7 6 1 1 9 . 7 0 1 3 1 5 . 7 8 0 . 7 8 7 7 0 2 6 , 1 5 . 0 2 0 1 9 9 1 3 6 6 . £ 6 9 . 0 4 1 8 . 0 2 0 1 7 8 1 3 6 5 . 4 5 9 . 0 3 7 9 1 0 6 . 5 5 1 3 6 1 . 2 2 C . 8 1 6 1 4 7 1 , 5 . 0 2 0 1 1 8 1 3 6 1 . 3 7 8 . 3 1 6 6 . 0 2 0 1 1 4 1 3 6 1 . C 7 8 . 3 1 6 4 1 1 1 . 4 6 1 3 5 2 . 7 1 0 . 8 5 3 0 9 5 1 , 1 0 . 0 1 5 8 7 7 1 3 4 5 . 0 4 9 . 0 1 7 8 . 0 1 9 8 7 6 1 3 4 4 . 5 6 9 . C 1 7 8 1 0 3 . 7 5 1 3 4 5 . 8 6 0 . 7 5 3 6 2 2 1 , 1 5 . 0 1 9 3 1 4 1 3 C 6 . 5 6 1 4 . 8 7 4 7 . 0 1 9 3 0 7 1 3 0 6 . 4 6 1 4 . 8 7 4 4 1 2 9 . 5 8 1 3 0 8 . 9 0 0 . 5 5 6 8 6 4 1 X 5 . 0 1 9 8 9 6 1 3 4 6 . 3 6 1 2 . 8 4 4 9 . 0 1 9 8 5 9 1 3 4 3 . 8 5 - 1 2 . 8 3 7 3 1 5 1 . 6 3 1 2 4 0 . 1 7 0 . 8 1 2 3 4 9 D X 5 . 0 2 0 3 4 5 1 3 7 6 . 7 3 6 . 0 6 9 6 . 0 2 0 3 1 0 1 3 7 4 . 3 3 6 . 0 5 6 8 1 3 8 . 7 6 1 1 1 2 . 4 6 C . 5 5 6 6 9 1 R E M A R K S : R I S I N C . F T / I B i Y B A R , S Y B A . R , S Y B A . A R E I N C . F T . 126. 6.2 Summary of r e s u l t s The most important r e s u l t s were as follows: a) was f a i r l y robust for a l l the generated cases and the sample weight scaling data. In pa r t i c u l a r the p value of 1.2727 obtained for the scale data (Vavenby) did not invalidate present use of the r a t i o - o f -means estimator i n weight s c a l i n g . b) Use of r a t i o estimation resulted i n consider-able improvement over non-ratio estimation. For the weight scaling data, the standard error of the estimate of the mean of the population of interest thr>ugh non-r a t i o estimation averaged seven-times that obtained from r a t i o estimation. c) For each scheme, the class or groups with the highest c o r r e l a t i o n c o e f f i c i e n t gave the smallest standard error of estimate but not necessarily the best estimate of the mean of Y. d) A scheme with the highest c o r r e l a t i o n c o e f f i c i e n t for a p a r t i c u l a r sample size did not necessarily give the smallest standard e r r o r . Thus, the r e l a t i v e . e f f i c i e n c y o'f^th'e^Scheme's :was+ no!tIrog©mpiefe'eili-y' -du§ t o ^ i h e i r respective sample co r r e l a t i o n c o e f f i c i e n t s . The p a r t i c u l a r sampling process us.ed alsg contributed to t h i s e f f i c i e n c y . e) Sampling with p r o b a b i l i t y increasing (or decreasing) with the magnitude of the a u x i l i a r y variate r e s u l t e d i n the best minimum-variance sampling scheme, but usually over-estimated the mean of Y excessively. Table 7 3 summarizes the re s u l t s obtained in Tables 35A-58A. It could be seen that for p values of less than or equal to 0.0, schemes DX5, 1X5 were not advanced to the 'B1 tables. This indicated c l e a r l y that the DXZ and IXZ schemes, for Z = 5 would not be applicable to populations with negative p values. For such negative p values, the choice of the best sampling scheme was not very clear or at least s p e c i f i c . However, for populations with positive p values, the DXZ and IXZ (in p a r t i c u l a r for Z = 5) were unquestionably the best i n terms of least standard err o r s . For the rows A l , A2 and A3 i n Table 7 3 with estimated p values greater than 0.0, the sup e r i o r i t y of the DX5 and 1X5 schemes to others was confirmed. However, reasonably good r e s u l t s were obtained from the MZ schemes. f) For the generated case studies, the greater the number of frequency classes for DXZ and IXZ, the smaller were the mean deviations. Schemes MZ, Types 2, 4 and 5 gave far smaller mean and absolute deviations than DX5.'i in the case of the IFS weight scaling data. Therefore unless e f f i c i e n t correction procedures could be found for DX5, the others would be preferred. DX5 gave the least standard errors (minimum-variance) for most of the r e p l i c a t i o n s and the d i f f e r e n t sample sizes used (Appendix I-IX) . For the other schemes obtained for the weight scaling data, besides DX5, could be considered as the r a t i o estimation analogue of the regre: sion sampling scheme recommended by Demaerschalk and Kozak (1973). They concluded that one of the e f f i c i e n t ways of sampling for simple regression would be the use of uniform d i s t r i b u t i o n (equal p r o b a b i l i t i e s for fi x e d or known independent v a r i a b l e ) , to cover as wide a range as possible. g) No appropriate mean deviation relationships with sample size were obtained from the scale data. There was a s l i g h t indication of the deviation from the schemes DX5 and 1X5 being r e l a t e d with the sample size and i t s r e c i p r o c a l . For the generated case studies scattergrams indicated a reduction i n the spread of the deviation (for the r e p l i c a t e s of the same size) with increasing sample s i z e . These v i s u a l interpretations were, however, not confirmed through regression analyses TABLE 7 3 SUMMARY OF MINIMUM-VARIANCE SCHEMES USED IN TABLES 35B-58B Case P R Schemes in maximum order t o t a l of decreasing rank values From Table 1 -1.0 0.10 M20 1,15 Ml 5 1,5 6,5 CR 35A 2 -0.5 0.10 1X5 1,5 CR Ml 5 6,5 1X15 36A 3 • -.-IO.O 0.10 . CR 1X10 DXIO 1,10 6,5 1,15 37A 4 0.5 0.10 1X5 DX5 1X10 DXIO 6,5 1X15 3 8A 5 1.0 0.10 DX5 1X5 1X10 6,15 6,10 DXIO 39A 6 1.5 0.10 1X5 DX5 DXIO 1X10 . MIO 1X15 40A 7 2.0 0.10 1X5 DX5 DXIO 1X10 CR 1,10 41A Al 1.9 0.10 1X5 DX5 M2 0 M15 M5 DXIO 56A 8 -1.0 0.25 5,5 2,5 1,10 1,15 RS 4,5 42A 9 -0.5 0.25 2,5 4,5 RS 5,5 Ml 5 CR 43A 10 0.0 0.25 DXIO 1,15 MIO RS M5 5,5 44A 11 0.5 0.25 1X5 DX5 DXIO 1X10 M5 CR 45A 12 1.0 0.25 DX5 1X10 1X5 DXIO DX15 Ml 5 46A 13 1.5 0.25 DX5 1X5 DXIO 6,10 1X10 ' ,CR 47A 14 2.0 0.25 1X5 DX5 1X10 DXIO 6,10 6 ,5 48A A2 2.5 0.25 1X5 DX5 MIO M20 1X10 DXIO 57A 15 -1.0 0.40 1,10 Ml 5 1,15 5,5 5,10 1X15 49A 16 -0.5 0.40 CR 1,5 1X15 4,5 2,5 5,5 5 OA 17 0.0 0.40 1X10 DXIO CR 1,15 6,5 1X15 51A 18 ••0.5 0.40 DX5 1X5 6,5 M20 4,5 6,10 52A 19 1.0 0.40 DX5 1X5 1X10 6,5 DXIO MIO 53A 20 1.5 0.40 DX5 1X5 1X10 M20 DXIO M5;1,10 54A 21 2.0 0.40 1X5 DX5 1X10 6,10 DXIO 6,5 55A A3 2.5 0.40 DX5 1X5 M20 1,15 DXIO CR 58A h) A p r a c t i c a l way of handling equal p r o b a b i l i t y or uniform sampling schemes (as for Types 2, 3, 4) would require intimate knowledge of the d i s t r i b u t i o n of the a u x i l i a r y population. Scheme MZ, however, was based on the order of a r r i v a l of truck loads and was accordingly recommended for use in weight scaling. 7.0 STRATIFICATION, DOUBLE-SAMPLING AND OTHER CONSIDERATIONS Further aspects of the study being examined i n t h i s chapter include the following: a) the use of t h e o r e t i c a l biases as correction factors in the estimators; b) the relationships between pre-, p o s t - s t r a t i f i c a -t i o n and n o n - s t r a t i f i c a t i o n estimators; and c) the use of double-sampling procedures for the frequency d i s t r i b u t i o n of X, and for use with some pr o b a b i l i t y sampling schemes. 7.1 The data and additional notations Data for studying the above objectives were made a v a i l -able by the Research Divi s i o n of the B r i t i s h Columbia Forest Service, and comprised a period's sample data for t hirteen d i f f e r e n t companies code named A to M, operating in nine d i f f e r e n t biogeographical d i s t r i c t s of B r i t i s h Columbia. A summary i s provided in Table 74. Appendix X provides the location, operator strata and prevalent species (coded) in the data for each company. Appendices XI-XXIII represent the 5-15 frequency classes for the respective companies. These also provide indications of the extent of the v a r i a b i l i t y i n the r a t i o estimates for the various classes. As had been done for previous data sets, the p values .weue determined for each company. From Table 74 these estimates could be seen to be f a i r l y close to 1.0 except for that of company A. The numbertof^resp:ectiveasamp'iL'e-~load-svwere .fa i r l y small, and so the uses of the DXZ and/or the IXZ scheme were ignored for most of the companies. For the double sampling procedures, the i n t e r i o r companies A,D,F, J /K /L,M were combined to y i e l d a t o t a l population size of 17 34 loads. Additional notations introduced include MIN (minimum), MAX (maximum), TBIAS, P-SYB.R and S-SYB.R. TBIAS i s the th e o r e t i c a l bias of the estimate of the mean of Y as given for the 'simplified weight scaling' by equation (3.1.12) times the estimated mean of Y, that i s , TBIAS = Bias ( r p ) times YBAR (7.1.1) P-SYB.R i s the p o s t - s t r a t i f i c a t i o n standard error of YBAR, and S-SYB.R i s the ordinary s t r a t i f i c a t i o n (or p r e - s t r a t i f i c a t i o n ) standard error of YBAR using r a t i o estimation in both cases. The formulae used are given in section 3. TABLE 74 SUMMARY FOR COMPANIES A-M COMPANY A B C D E F G H I J K L M KAMLOOPS NELSON VANCOUVER (coast) CARIBOO NELSON KAMLOOPS PRINCE RUPERT (interior) VANCOUVER (coast) PRINCE RUPERT (coast) KAMLOOPS KAMLOOPS CARIBOO PRINCE GEORGE LOADS (#) 258 279 264 225 328 261 611 348 398 312 191 249 238 MIN X MAX X (lbs.) (lbs.) 43620 100040 33100 16800 30350 37840 5980 89940 53500 233050 43820 83100 92200 17300 119950 80500 39700 145500 21200 205600 39660 116450 77520 89900 21500 176450 XBAR (lbs.) 68453 63038 143091 66657 59985 65419 58242 95298 96942 76416 63680 66015 92269 YBAR (c.ft.) 1308 1269 2695 1312 1192 1390 1071 1744 1861 1585 1271 12 81 1853 ESTIMATE OF P 1.3272 1.0781 0.92 88 0.9715 0.8856 0.8322 1.0155 0.9667 0.8524 0.8962 0.9047 1.0187 1.1938 ( A l l companies operating in i n t e r i o r , except where indicated) CO O J r 7.2 Numerical i l l u s t r a t i o n of t h e o r e t i c a l bias This i l l u s t r a t i o n was necessitated by the lack of my a b i l i t y to evolve a correction procedure for the exces-s i v e l y large but consistent random deviations of YBAR through the DXZ and IXZ sampling procedures. It was f e l t that i f these biases could be numerically estimat-ed, they could be used to correct the estimators. I f we consider only the case of p = 1.00, that i s n n the use of the ordinary estimate r ^ = /E y^/ x^, then equation (3.1.12) reduces to equation (3.1.14), that i s 2 Bias ( r x ) = — ? ( R l S x " S x y } (3.1.14) nX The indication of the numerical estimates for a l l sample sizes was that the bias of r ^ (especially p = 1) was n e g l i g i b l e , and therefore using i t could not have resul t e d in any improvement in the estimation of the mean of Y. Referring to the TBIAS columns for 'estimated p' and 'approximated p' i n Tables 75-77, the bias of rj_ or rp could be obtained by dividing the corresponding TBIAS by YBAR. Judging from these r e s u l t s and those of the l a s t two chapters, i t may not be worthwhile to consider the DXZ or IXZ schemes as excellent candidates for weight scaling sampling procedures. However, i t shou be noted that these were judged to be the best minimum variance schemes for a l l the data sets studied. TABLE 75 R-SULTS FOR COMP.G : SAMPLE SIZ^=70 I SAMPLE) SS RATIO YBAR SY6AR.R DEV. TBIAS RATIO YBAR SYBAR.R DEV. TBIAS P-SYB.R S-SYB.R SHG CR 0.018340 1068.15 12.1357 -2.45 0. 14 0.018339 1068.12 12.1357 -2.48 0.03 0.539458 RS 0.018439 1073.95 11.8439 3.35 0. 13 0.018439 1073.93 11.8439 3.33 0.02 0.546165 M 5 0. 018357 1069.18 14.4254 -1.42 0.20 0.018356 1069.11 14.4251 -1. 49 0. 06 47.642 8 49.0726 0.405408 MIO 0.018876 1099.41 13.9260 28.81 0.19 0.018876 1099.37 13.9259 28.77 0.C3 30.5456 30.3339 0. 566437 Ml 5 0.010407 1072.07 13.4726 1.47 0.11 0.018407 1072.07 13.4726 1.47 0.01 22.5360 21.7747 0.562199 M20 0.018205 1060. 31 11.1992 -10.29 0. 12 0.018205 1060.30 11.1992 -10.30 0.01 16.6020 15. 5949 0.623933 M25 0.018252 1063.01 13.0626 -7.59 0. 20 0.018251 1062.95 13.0623 -7.65 0.05 18.7843 17.1055 0.516312 M3 0 0.018094 1053.84 11.6160 -16.76 0.17 0.018094 1053.32 11.6160 -16. 78 0. 02 14.6575 13.3476 0.667339 M3 5 0.018234 1064.88 12.6020 -5. 72 U.13 0.018283 1064.86 12.6020 -5.74 0.02 11.9023 10.1645 0.565791 1.5 0.018657 1086.60 12.5100 16.00 0.18 0.018656 1086.56 12.5098 15.96 0.04 61.3012 64.1738 0.537328 1, 10 0.018717 1090. 11 13.5500 19.51 0.19 0.018716 1090.07 13.5499 19.47 0.04 53.5411 54.6201 0.514590 1. 15 0.018364' 1069.53 13.8367 -1.07 0. 13 0.018363 1069.48 13.8365 -1.12 0.06 41.0979 41.4872 0.393157 2, 10 0.018488 1076.77 14.6290 6. 17 0.29 0.018487 1076.70 14.6287 6. 10 0.07 47.2811 48.6571 0.576842 4, 15 0.018004 1048.56 13.3404 -22.04 0.22 0.018003 1048.53 13.3403 -22.07 0.04 33.4766 36.3759 0.62C821 1X5 0.018551 1080.44 13.6947 9.84 0.28 0.013550 1080.40 13.6946 9.80 0.04 60.6978 71.0098 0.690392 1X10 0.018319 1066.92 13.3706 -3.68 0.26 0.018318 1066.88 13.3705 -3. 72 0.04 44.2945 45.7351 0.659472 DXIO 0.018728 1090.75 12.7976 20. 15 0.24 0.018727 1090.73 12.7975 20.13 0.02 43.2547 44.3457 0.705960 REMARKS 1. DEV. = YBAR MINUS TRUE YEAR 2. TBIAS = THEORETICAL BIAS OF R TIMES XBAR: ASSUME EXPECTED BLOCK OR STRATA YBARS ARE THE SAME 3. P-SYB.P. = POST STRATIFICATION STANDARD ERROR OF YBAR 4.S-SYB.R = PRE STRATIFICATION STANDARD ERROR OF YBAR (PRE STRAT. ASSUMED) 5.BIAS,ABIAS,TBIAS,S-SYB.R,P-SYB.R.SYBAR.R £ DEV. ARE IN CUBIC FEET.RATIO IN CUBIC FEET PER POUND £ TA3LS 76 R E S U L T S FOR C O H P . K : S A M P L E S I Z e = 1 5 Q ( S A M P L E ) < USING EST IMAT ED P =0. 904700 > <- USING APPROXIMATED P = 1.00 > SS RATIO YBAR SY8AR.R DEV. TBIAS RATIO YBAR SY BAR.R DEV. TBIAS P-SYB.R S-SYB.R RHO CR 0.019793 1260.40 3.9683 -10. 60 -0.49 0.019794 1260.47 3.9684 -10.53 0.01 0.812575 RS 0.020038 1279.18 9.8042 3.18 -0.91 0.020090 1279.35 9.8044 8. 35 0.01 0.601329 M 5 0.019770 1258. 92 4.8922 -12.08 - 0 . 81 0.019771 1259.03 4.8924 -11.97 - 0.01 6.0931 12.4032 0.830889 MIO 0.019783 1259.79 5.3410 -11.21 -0.90 0.019785 1259.91 5.3412 -11.09 0.01 4.7912 9.1422 0.821261 M15 0.019940 1269.81 9.7391 -1 . 19 -0.81 0.019942 1269.92 9.7392 -1.08 0.01 7.7106 13.8936 0.596034 M20 0.019993 1273.18 4.8702 2.18 - 0 . 74 0.019995 1273.30 4.8703 2.30 0.01 2.9981 5.1915 0.817810 M2 5 0.019790 1260.23 4.7975 -10.77 -0.70 0.019791 1260.32 4.7976 -10.68 0.01 3.1194 5.0962 0.818941 M30 0.020030 1275.50 9.8561 4.50 -0.86 0.020033 1275.69 9.8563 4.69 0.01 5.7204 8.9557 C.57429C M35 0.020023 1275.08 9.7361 4.03 -8.22 0.020024 1275.10 9.7361 4. 10 0.00 6.6555 9.3207 0.936389 1,5 0.019957 . 1270.83 9.7106 -0.17 -0.66 0.019959 1270.98 9.7108 - 0 . 0 2 0.01 20.2743 42.4353 0.522382 1,10 0.019968 1271.54 9.4976 0.54 - 0 . 75 0.019970 1271.66 9.4977 0.66 0.01 14.8191 30.4272 0.584950 REMARKS 1. DEV. = YBAR MINUS TRUE YEAR 2. TBI AS = THEORETICAL BIAS OF R TIMES XBAR: ASSUME 3. P-SYB.R = POST STRATIFICATION STANDARD ERROR OF 4.S-SYB.R = PRE- STRATIFICATION STANDARD ERROR OF 5.BIAS,ABIAS,TBIAS,S-SYB.R,P-SYB.R,SYdAR.R £ DEV. EXPECTED BLOCK OR STRATA YBARS ARE THE SAME YBAR YBAR (PRE STRAT. ASSUMED) ARc IN CUBIC FEET.RATIO IN CUBIC FEET PER POUND LO cn TABLE 77 RESULTS FOR COM?.A : SAMPLE SIZ==3G 1 SAMPLE) <- — USING ESTIMATED P =1. 327200 > < US I NG APPROXIMATED P = 1. 00 > SS RATIO Y3AR SYBAR.R DEV. TBIAS RATIO YBAR SYBAR.R DEV. TBIAS P-SYB.R S-SYB.R RHO CR 0 . 019051 1204. 12 25.4027 -3.48 6,50 0.019028 1302.55 25.3930 -5.05 0.16 0. 666648 RS 0. 019336 1323.63 24.6322 16.03 5.67 0. 019335 1323.54 24.6322 15.94 0.00 0 . 741979 M 5 0„ 019093 1306. 97 17.9135 -0.63 5 . 43 0.019037 1306.54 17.9124 -1.06 0.05 32.6723 32.3972 0. 807043 MIO 0„ 019321 1322.55 23.9256 14.95 8.64 0. 019321 1322.61 23.9256 15.01 -0.01 28.6554 26.3380 0 . 821195 H15 0. 019356 1325.00 24.0758 17.40 11.98 0.019336 1323.63 24.0653 16. 03 0. 19 15.3795 13.2347 0.821544 M20 0. 019514 1335.£2 19.8286 28.22 7.67 0. 019485 1333.33 19.8094 26,23 0.19 15.2292 13.1053 0. 795116 M25 0 . 019368 1325.80 22.5247 18.20 68. 10 0. 0i9366 1325.65 22.5246 18.05 0.00 0.0 0.0 c . 973091 M30 0 . 019143 1310.42 22.4972 2.32 4.38 0.019136 1309.94 22.4967 2.34 0.03 0 . 0 0.0 0 . 707798 M35 0. 019514 1335.61 19.4963 28.21 8.54 0.019512 1335.69 19.4967 28.09 0.02 0.0 0.0 0 . 861621 1,5 0 . 019125 1309.20 22.7000 1. 60 6.26 0.019133 1309.71 22.6989 2. 11 -0.05 0.0 0.0 0 . 809673 1, 10 o . o i s a 7 5 1292.03 24.4740 -15.57 6.13 0.018886 1292.82 24.4713 -14.73 - 0 . 0 8 0 . 0 0 . 0 0 . 800035 1,15 0. 0188iO 1287.60 20.3979 -20.00 4.62 0. 0.18806 1287.30 20.3976 -20 . 3 0 0 .02 0 .0 0.0 0 . 756628 REMARKS 1 . DEV. = YBAR MINUS TRUE YSAR 2 . TBIAS = THEORETICAL BIAS OF R TIMES XBAR: ASSUME EXPECTED BLOCK OR STRATA YBARS ARE THE SAME 3 . P-SYB.R = POST STRATIFICATION STANDARD ERROR OF YBAR 4 .S-SY3.R = PRE STRATIFICATION STANDARD ERROR OF YBAR (PRE STRAT. ASSUMED) 5.BIAS,ABIAS .T3TAS,S-SYB.R,P-SYB.R,SYBAR.R £ DEV. ARE IN CUBIC FEET.RATIO IN CUBIC FEET PER POUND 7 . 3 S t r a t i f i c a t i o n for greater e f f i c i e n c y S t r a t i f i c a t i o n was based on two separate c r i t e r i a , a) Based on season or order of a r r i v a l s of loads, and b) Based on the size of the net truck load weight. S t r a t i f i c a t i o n c r i t e r i a (a) were applied to those samples obtained through the MZ schemes. I t may be r e c a l l e d that under these schemes, the population was divided by order of a r r i v a l s into Z groups of approx-imately equal size and then equal samples taken randomly from each group. The order of a r r i v a l s could i n fact r e l a t e to the period or time in which the part i c u l a r sample loads were scaled. Each group of samples was then regarded as belonging to a part i c u l a r season or a r r i v a l s t r a t a . S t r a t i f i c a t i o n c r i t e r i a (b) represented the groupings by size of X and was therefore appropriate for the other schemes besides MZ, RS and CR. It should be clear that group or stratum samples obtained by MZ were not necessari the same as those of Type 1,Z for the same Z even i f the same sets of data were drawn by both schemes. Chances of t h e i r being the same were very rare. P o s t - s t r a t i f i c a t i o n referred to the s t r a t i f i c a t i o n performed after the samples had been obtained. Cochran (1963) discussed a formula obtained by Stephan (1941, 1945). Wensel (1974) used the formula to show that i f s t r a t i f i c a t i o n for weight scaling was done by "gale" value, a considerable amount of improvement in precision could be expected. The formula was of the form, P-SYB.R2=Szn_Z^i-S.2 + V Z fl - V ) S . 2 . - (7.3 th ? where i s the size of I stratum, i s the approp-r i a t e variance formula which i n the case of r a t i o estima-2 2 o t i o n i s ( R s x + s y ~ 2RS Xy) times the f i n i t e popula-t i o n correction factor, and a l l the other symbols as previously defined. The formula was based on the assumption that the S t - XT mean of Y was estimated by > _JL YBAR. and that sampling N x was done with proportional a l l o c a t i o n of samples to s t r a t a . Cochran (1953) indicated that t h i s method could be as precise as proportional s t r a t i f i e d sampling (pre-s t r a t i f i c a t i o n ) provided the stratum size was greater than 20 and the proportions were f a i r l y free from e r r o r s . st I t should therefore be expected that for n = n^<.30 and for i > 2, comparisons between post- and p r e - s t r a t -i f i c a t i o n may not be j u s t i f i a b l e . The proportional s t r a t i f i e d sampling or pre-s t r a t -i f i c a t i o n formula for the standard error, S-SYB.R, could be written as: the f i r s t term of equation (7.3.1) without the correc-t i o n factor i s obtained. The second term of equation (7.3.1) i s the extra term brought about as a r e s u l t of n^ not d i s t r i b u t i n g i t s e l f in the same manner as Nj_. If t h i s second term in equation (7.3.1) was very small, then P-SYB.R could be smaller than S-SYB.R because of the inclusion of the o v e r a l l correction factor. Usually, the sampling would be from a r e l a t i v e l y large N, so that (N-n)/N would approach one. In t h i s case, P-SYB.R could be expected to be greater than or at best equal to S-SYB.R. Discussion of the re s u l t s obtained under t h i s section f e l l under two subsections dependent on the manner in which the available company data were treated. 7.3.i Sampling from respective populations Sample of sizes ranging from 30 to 170, increment of 20, were drawn according to the CR; RS; MZ; 1,Z; and in a few cases DXZ or IXZ sampling schemes. Tables 75-77 provide sample outputs selected to r e f l e c t the whole (7.3.2) I f one of i s replaced by , i n equation (7.3.2), N n spectrum of the r e s u l t s . Comparisons between P-SYB.R and S-SYB.R should be avoided at t h i s moment for the following reasons: a) the actual sample loads (which was being referred to as the population s i z e , N) were not large, so that the use of the correction term (N-n)/N i n equation (7.3.1) might have greatly underestimated the value for P-SYB.R i n comparison with that of S-SYB.R e s p e c i a l l y when n was close to N; and b) the minimum stratum size requirement of at least 20 suggested by Cochran (19(D3) was not met. In performing the computations, the algorithm used ignored any stratum with only 1 observation and th i s was deducted from the given n. I f , however, the f i n a l n obtained after a l l the deductions was less than 30, P-SYB.R and S-SYB.R were not computed and in place a value 0.0 of no mathematical significance was written. The only important r e s u l t s were the indications that within each s t r a t i f i c a t i o n c r i t e r i a , the variances decreased with increasing number of classes or groups. That i s , post- or p r e - s t r a t i f i c a t i o n variance obtained for M35 was less than for M5. This was true for a l l the twenty i t e r a t i o n s done for each sample size. 142. 7 . 3 . i i Assuming sample loads from large population sizes To meet the requirements of the stratum size being at least 20, and the t o t a l sample size greater than 30, the respective data were treated as samples of sizes n from very large but unknown N. Thus, n/N could be 9 regarded as n e g l i g i b l e . Also, i t was assumed that N-[/N could be reasonably estimated by n.j_/n. Results for the s t r a t i f i c a t i o n by size of X were compiled and presented as tables 78 and 79, while those for s t r a t i f i c a t i o n by a r r i v a l s were given i n tables 80 and 81. Tables 78 (or 79) and 80 (or 81) could be compared with each other to study the effectiveness of each type of s t r a t i f i c a t i o n . In Table 78, company T referre d to a l l the seven indicated companies combined. The symbol, #, referr e d to the number of sample loads as given i n Table 7 3; p# was the actual number of sample loads used for the computation of P-SYB.R and S-SYB.R; and NC referr e d to the number of frequency c l a s s i f i c a t i o n s based on the size of X. The same notations were used for the other tables. Referring to Tables 78 and 79, comparisons could be made of the four d i f f e r e n t estimators of the standard error—namely SYBAR (no s t r a t i f i c a t i o n , no r a t i o estima-^ Since the sampling f r a c t i o n was deleted from a l l the four d i f f e r e n t standard error formulae, the r e l a t i v e results should remain unaffected. TABLE 78 STRATIFICATION BY SIZE OF X FOR COMPANIES A»D»F»J?K»L»M & T # P# NC SYBAR.R ( c . f t . ) A 258 257 5 117.7336 A 258 257 10 117.7336 A 258 257 15 117.7336 A 258 256 20 117. 7336 D 225 225 5 83.3450 D 225 225 10 83.3450 D 225 224 15 83.3450 D 225 221 20 83.3450 F 261 261 5 201.4938 F 261 261 10 201.4938 F 261 260 15 201.4938 F 261 260 20 201. 493 8 J 312 312 5 109.0183 J 312 312 10 109.0183 J 312 312 15 109.0183 J 312 310 20 109.0183 K 191 190 5 113.4579 K 191 190 10 113.4579 K 191 188 15 113.4579 K 191 186 20 113.4579 « i. 249 249 5 76.9451 L 249 249 10 76.9451 I 249 248 15 76.9451 L 249 247 20 76.9451 M 238 .238 5 94.1663 M 238 237 10 94.1663 M 238 236 15 94.1663 M 238 23 6 20 94. 1663 T 1734 1734 5 130.3680 T 1734 1734 10 130. 3680 T 173 4 1734 15 130,3680 T 1734 1734 20 130.3680 T 1734 1734 25 130.3680 T 1734 1734 30 130.3680 2 3 4 P-SYB.R SYBAR S-SYB.R ( c . f t . ) ( c . f t . ) ( c . f t . ) 67.9975 188.1063 67.7408 51.0381 188.1063 50.6256 41.6133 188.1063 41.0829 36. 5263 188.1063 35.9025 42.0236 132.2769 41.8079 30.6256 132.2769 30.2706 25. 9413 132.2769 25. 5044 23.2131 132.2769 22.7036 110.5425 362.0577 110.1336 82.5884 362. 0577 81.9922 69.4594 362.0577 68.6636 59.3729 362.0577 58.4737 53.9552 379.6192 53.7202 39. 3327 379.6192 38.9564 33.0415 379.6192 32. 5800 29.6168 379.6192 29.0899 35.3U7 138.6077 85.1467 63.8572 138. 6077 63. 5244 56.9331 138.6077 56.4996 49.2039 138.6077 48.6720 56.3923 220.1492 56.2976 42. 3758 220.1492 42.1866 36.263 3 220.1492 36. 0250 31.1331 220.1492 30. 8756 47.73o8 76 5.9244 47.4490 34.8900 765. 9344 34.4394 31.5009 765.9344 30.9997 27. 6783 765. 9344 27.1020 94. 55o 7 426.4659 94.5 32 5 76.0951 426.4659 76.0568 61.4627 426. 4659 61.4059 53.47o6 426.4659 53.4083 49.7149 426.4659 49.6395 44.7J3 2 426. 4659 44.6213 1/2 1/3 2/3 2/4 RHO 1.73 0.63 0.36 1.00 0.7834 2.31 0.63 0.27 1.01 0.7834 2. 83 0. 63 0.22 1.01 0.7834 3.22 0.63 0.19 1.02 0.7834 1. 98 0. 63 0.32 1.01 0.8057 2.72 0.63 0.23 1.01 0.8057 3.21 0. 63 0.20 1.02 0.8057 3.59 0.63 0. 18 1.02 0.8057 1.82 0. 56 0.31 1.00 0.8493 2.44 0. 56 0.23 1.01 0.8493 2.90 0. 56 0. 19 1. 01 0.8493 3.39 0.56 0.16 1.02 0.8493 2. 02 0. 29 0.14 1.00 0.9611 2.77 0.29 0. 10 1.01 0.9611 3.30 0.29 0.09 1.01 0.9611 3.66 0. 29 0.08 1.02 0.9611 1.33 0. 82 0.62 1.00 0. 5862 1. 78 0. 82 0.46 1.01 0.5862 1. 99 0. 82 0.41 1. 01 0.5862 2.31 0.82 0.36 1.01 0.5362 1.36 0. 5 5 0.26 1.00 0.9373 1.82 0.35 0.19 1.00 0.9373 2.12 0.35 0.16 1.01 0.9373 2.47 0.35 0.14 1.01 0.9 373 1.97 0,12 0.06 1.01 0.9925 2.70 0.12 0.05 1.01 0.9925 2.99 0. 12 0.04 1. 02 0.9925 3.40 0.12 0.04 1.02 0.9925 1. 38 0. 31 0.22 1.00 0.9522 1.71 0. 31 0. i 8 1.00 0.9522 2. 12 0.31 0.14 1.00 0.9522 2.44 0.31 0.13 1. 00 0.9522 2.62 0.31 0.12 1.00 0.9522 £ 2.92 0.31 0.10 1.00 0.9522 co TABLE 79 STRATIFICATION BY SIZE OF X FOR 1 2 3 # P# NC SYBAR.R P-SYB. R. S YBAR ,(c.ft.) 104.9644 (C.ft.) ( c . f t . ) B 279 279 5 74.8554 168.5146 8 279 277 10 104.964-4 57.0078 168.5146 B 279 275 15 104.9644 46. 9768 168.5146 B 279 277 20 104.9644 44.0822 168.5146 r 264 263 5 290.8813 138.5416 568.4294 C 264 262 10 290.8813 93.6370 568.4294 C 264 262 15 290.8313 83.8220 568.4294 C 264 262 20 290.8813 70.4715 568.42 94 E 328 328 5 133.9598 80.5023 201.0255 E . 328 328 10 133.9598 61.2073 201.0255 E 328 324 15 133.9598 51.9411 201.0255 E 328 323 20 133.9598 45.1380 201.0255 G 611 611 5 109.4921 70. 80t>5 130. 4288 G 611 609 10 109.4921 54.00b7 130.4288 G 611 607 15 109.4921 44.9472 130.4238 G 611 606 20 109.4921 39.9156 130.4288 H 348 348 5 150.5865 93.5353 296.6869 H 348 348 10 150. 5865 69.6759 296.6869 H 348 347 15 150.5865 56.6699 296.6869 H 348 346 20 150. 5865 50.1175 296.6869 I . 398 3 98 5 253.2246 124.5619 700.6174 I 398 3 98 10 253.2246 87.19+8 700.6174 I 398 397 15 253.2246 72.7418 700.6174 I 398 397 20 253.2246 63.20+9 700.6174 ES B , C , E , G , H , I 4 S-SYB.R 1/2 ( c . f t . ) 74.7289 1.40 56.7652 1.84 46.6418 2.23 43.7234 2.38 137.9246 2.10 92.5441 3.11 82.5487 3.47 68.9402 4.13 80.3277 1.66 60.9128 2.19 51.5658 2.58 44.6742 2.97 70.7321 1.55 53. 8810 2. 03 44.7796 2.44 39.7175 2. 7+ 93.3333 1.61 69.3286 2.16 56.2070 2.66 49.5778 3.00 124.1474 2. 03 86.5078 2.90 71.8952 3.48 62. 1941 4. 01 1/3 2/3 0.62 0.44 0.62 0.34 0.62 0.28 0. 62 0. 26 0.51 0.24 0.51 0.16 0.51 0.15 0.51 0.12 0.67 0.40 0.67 0.30 0.67 0.26 0.67 0.22 0.34 0.54 0.84 0.41 0.84 0.34 0.84 0.31 0.51 0.32 0.51 0.23 0.51 0.19 0.51 0.17 0.36 0. 18 0.36 0. 12 0.36 0.10 0.3 6 0. 09 2/4 RHO 1.00 0.7829 1.00 0.7829 1.01 0.7829 1.01 0.7829 1.00 0.8659 1.01 0.8659 1.02 0. 8659 1.02 0.8659 1.00 0.7722 1.00 0.7722 1.01 0.7722 1.01 0. 7722 1.00 0.5729 1.00 0.5729 1.00 0.5729 1.00 0.5729 1.00 0. 8674 1.01 0.8674 1.01 0.8674 1.01 0.8674 1.00 0.9356 1.01 0. 9356 1.01 0.9356 1.02 0.9356 T A B L E 8 0 1 4 5 . S T R A T I F I C A T I O N BY A R R I V A L S FOR C O M P A N I E S A , D t F t J , K , L , M A » 2 5 8 NG 5 S Y B A R ( c . f t . ) I 8 8 0 I O 6 S Y B A R . R (c.ft.) 1 1 7 . 7 3 5 P - S Y B . R (c.ft.) 3 5 . 2 3 3 S - S Y B . R ( c . f t . ) 3 4 . 9 6 3 R H O 0 . 7 8 3 4 A 2 5 8 1 0 1 8 8 . 1 0 b 1 1 7 . 7 3 5 2 6 . 0 2 9 2 5 . 5 8 8 0 . 7 3 3 4 A 2 5 8 1 5 I 8 8 0 I O 6 1 1 7 . 7 3 5 2 0 . 6 9 0 2 0 . 1 4 4 Oo 7 3 3 4 A 2 5 8 2 0 1 8 8 . 1 0 6 1 1 7 . 7 3 5 1 6 . 5 9 7 1 6 . 0 2 J 0 . 7 8 3 4 D 2 2 5 5 1 3 2 . 2 7 7 8 3 . 3 4 6 2 9 . 8 6 9 2 9 . 6 0 7 0 . 8 0 5 7 D 2 2 5 1 0 1 3 2 o 2 7 7 8 3 . 3 4 6 2 2 . 3 8 7 2 1 . 9 5 3 0 . 8 0 5 7 D 2 2 5 1 5 1 3 2 * 2 7 7 3 3 . 3 4 6 1 7 . 9 2 7 1 7 . 3 9 4 0 . 8 0 5 7 0 2 2 5 2 0 1 3 2 . 2 7 7 8 3 . 3 4 6 1 5 . 3 8 7 1 4 . 7 6 2 0 . 6 0 5 7 F 2 6 1 . 5 3 6 2 * 0 5 8 2 0 1 . 4 9 4 6 9 . 1 8 3 6 8 . 6 5 7 0 . 8 4 9 3 F 2 6 1 1 0 3 6 2 . 0 5 8 2 0 1 . 4 9 4 4 0 . 7 2 0 4 0 . 0 3 2 : 0 . 8 4 9 3 F 2 6 1 1 5 3 6 2 , 0 5 8 2 0 1 . 4 9 4 3 2 . 1 2 8 3 1 . 2 3 0 0 . 8 4 9 3 F 2 6 1 2 0 3 6 2 . 0 5 3 2 0 1 . 4 9 4 2 8 . 8 1 0 2 7 . 8 1 1 0 . 8 4 9 3 J 3 1 2 5 3 7 9 . 6 1 9 1 0 9 . 0 2 0 3 9 . 7 8 1 3 9 . 5 2 7 0 . 9 6 1 1 J 3 1 2 1 0 3 7 9 o 6 1 9 1 0 9 . 0 2 0 2 9 . 4 2 2 2 9 . 0 0 4 0 . 9 6 1 1 J 3 1 2 1 5 3 7 9 . 6 1 9 1 0 9 . 0 2 0 2 4 . 6 7 0 2 4 . 1 3 3 0 . 9 6 1 1 J 3 1 2 2 0 3 7 9 . 6 1 9 1 0 9 . 0 2 0 2 1 . 6 1 8 2 0 . 9 9 7 0 . 9 6 1 1 K 1 9 1 5 1 3 8 . 6 0 8 1 1 3 . 4 5 3 4 4 . 5 8 2 4 4 . 1 2 0 . 0 . 5 8 6 2 K 1 9 1 1 0 1 3 8 , 6 0 8 1 1 3 . 4 5 8 3 3 . 2 4 2 3 2 . 4 8 1 0 . 5 3 6 2 K 1 9 3 . . 1 5 1 3 8 . 6 0 8 1 1 3 . 4 5 8 2 7 . 6 7 0 2 6 . 7 2 6 0 . 5 3 6 2 L 2 4 9 5 2 2 0 . 1 4 9 7 6 . 9 4 7 3 0 . 1 8 9 2 9 . 9 5 0 0 . 9 3 7 3 L 2 4 9 10 2 2 0 . 1 4 9 7 6 . 9 4 7 2 1 . 6 9 7 2 1 . 3 i 5 0 . 9 3 7 3 L 2 4 9 1 5 2 2 0 . 1 4 9 7 6 . 9 4 7 1 6 . 8 8 2 1 6 . 4 2 6 0 . 9 3 7 3 L 2 4 9 2 0 2 2 0 . 1 4 9 7 6 . 9 4 7 1 4 . 0 3 2 1 3 . 5 0 6 0 . 9 3 7 3 M 2 3 3 5 7 6 5 . 9 3 4 9 4 . 1 6 1 3 6 . 4 7 7 3 6 . 1 7 6 0 . 9 9 2 5 M 2 3 8 1 0 7 6 5 . 9 3 4 9 4 . 1 6 1 2 7 . 4 7 7 2 6 . 9 7 4 0 . 9 9 2 5 M 2 3 0 1 5 7 6 5 . 9 3 4 9 4 „ 1 6 1 2 2 . 1 9 8 2 1 . 5 7 0 0 . 9 9 2 5 M 2 3 8 2 0 7 6 5 . 9 3 4 9 4 . 1 6 1 1 9 . 4 2 6 1 8 . 7 0 0 0 . 9 9 2 5 T A B L E B l 1 4 6 . S T R A T I F I C A T I O N B Y A R R I V A L S F O R C O M P A N I E S B , C , S , G , H , I # N G S Y B A R S Y B A R o R P - S Y B . R S - S Y B . R R H O ( c . f t . ) (c.ft.) (c.ft.) ( c . f t . ) B 2 7 9 5 1 6 8 . 5 1 5 1 0 4 o 9 6 5 4 0 o 6 3 9 4 0 . 3 5 1 0 . 7 3 2 9 B 2 7 9 1 0 1 6 8 . 5 1 5 1 0 4 o 9 6 5 3 0 o 5 1 4 3 0 , 0 3 4 0 . 7 8 2 9 B 2 7 9 1 5 1 6 8 . 5 1 5 1 0 4 . 9 6 5 2 4 . 8 4 3 2 4 . 2 5 1 0 . 7 8 2 9 B 2 7 9 2 0 1 6 8 . 5 1 5 1 0 4 . 9 6 5 2 1 . 2 1 3 2 0 . 5 2 7 0 . 7 8 2 9 C 2 6 4 5 5 6 3 . 4 2 9 2 9 0 . 8 7 6 1 1 3 . 2 2 2 1 1 2 . 3 7 0 0 . 8 t > 5 9 C 2 6 4 1 0 5 6 8 . 4 2 9 2 9 0 . 3 7 6 8 2 . 2 1 1 8 0 . 3 2 2 0 . 8 6 5 9 C 2 6 4 1 5 5 6 0 . 4 2 9 2 9 0 . 8 7 6 6 7 . 3 5 6 6 5 . 4 9 5 0 . 8 6 5 9 C 2 6 4 2 0 5 6 8 . 4 2 9 2 9 0 . 8 7 6 5 9 . 4 6 7 5 7 . 4 0 5 0 . 8 5 5 9 £ 3 2 8 5 2 0 1 . 0 2 5 1 3 3 . 9 6 0 5 0 . 4 2 6 5 0 . 1 2 1 0 . 7 7 2 2 £ 3 2 8 1 0 2 0 1 . 0 2 5 1 3 3 . 9 6 0 3 4 . 5 0 3 3 4 . 0 4 0 0 , 7 7 2 2 E 3 2 8 1 5 2 0 1 . 0 2 5 1 3 3 . 9 6 0 2 4 . 0 1 9 2 3 . 5 2 4 0 . 7 7 2 2 E 3 2 8 2 0 2 0 1 . 0 2 5 1 3 3 . 9 6 0 2 1 . 6 3 4 2 1 . 0 1 9 0 . 7 7 2 2 G 6 1 1 5 1 3 0 . 4 2 9 1 0 9 . 4 9 2 4 1 . 1 1 0 4 0 . 9 7 6 0 . 5 7 2 9 G 6 1 1 1 0 1 3 0 . 4 2 9 1 0 9 . 4 9 2 2 9 . 8 1 3 2 9 . 5 9 6 0 . 5 7 2 9 G 6 1 1 1 5 1 3 0 . 4 2 9 1 0 9 . 4 9 2 2 4 . 3 1 6 2 4 . 0 4 3 0 . 5 7 2 9 G 6 1 1 2 0 1 3 0 . 4 2 9 1 0 9 . 4 9 2 2 1 . 4 9 1 2 1 . 1 6 8 0 . 5 7 2 9 H 3 4 8 5 2 9 6 . 6 8 7 1 5 0 . 5 8 7 4 7 . 9 3 9 4 7 . 6 6 7 0 . 3 6 7 4 H 3 4 8 1 0 2 9 6 . 6 8 7 1 5 0 . 5 8 7 3 4 . 3 1 3 3 3 . 8 7 9 0 . 3 6 7 4 H 3 4 8 1 5 2 9 6 . 6 8 7 1 5 0 . 5 3 7 2 7 . 4 9 7 2 6 . 9 5 5 0 . 8 6 7 4 H 3 4 8 2 0 2 9 6 . 6 8 7 1 5 0 . 5 8 7 2 2 . 8 0 9 2 2 . 1 9 6 0 . 8 6 7 4 I 3 9 8 5 7 0 0 . 6 1 7 2 5 3 . 2 2 3 9 0 . 5 3 2 9 0 . 0 8 0 0 . 9 3 5 6 I 3 9 8 1 0 7 0 0 . 6 1 7 2 5 3 . 2 2 3 6 3 . 9 6 8 6 3 . 2 5 9 0 . 9 3 5 6 I 3 9 8 1 5 7 0 0 . 6 1 7 2 5 3 . 2 2 3 5 2 . 1 3 2 5 1 . 2 4 9 0 . 9 3 5 6 I 3 9 8 2 0 7 0 0 . 6 1 7 2 5 3 . 2 2 3 4 4 . 2 8 7 4 3 . 2 6 9 0 . 9 3 5 6 t i o n or a u x i l i a r y information), SYBAR.R (no s t r a f i c a -t i o n , r a t i o estimation), P-SYB.R ( p o s t - s t r a t i f i c a t i o n , r a t i o estimation and S-SYB.R ( p r e - s t r a t i f i c a t i o n , r a t i o estimation). The four columns following S-SYB.R made such comparisons easy. The r e s u l t s were f a i r l y similar for a l l the companies and number of s t r a t a . These includ ed the following: a) the p r e - s t r a t i f i c a t i o n standard error estimate was s l i g h t l y smaller than the p o s t - s t r a t i f i c a t i o n estimate, but as could be seen from column "2/4", t h i s did not seem tofddif'fer. from 1.00. b) For either .typetofistratifica Jt'ioh^ , th^-g-reate'r' the number of classes, the smaller the standard e r r o r s . c) Considerable improvements over SYBAR.R were obtained by using the s t r a t i f i c a t i o n formulae as evidenc ed in column "1/2". SYBAR.R ranged from 1.33 for company K, NC=5 to 4.01 for company I, NC=20 times that of P-SYB.R (or S-SYB.R). d) SYBAR was by far i n f e r i o r to the other three standard error values. For tables 80 and 81, i t may be assumed that the observations had been obtained through the representative sampling schemes for each stratum,. since t h i s was the o r i g i n a l scheme used for the 'n' samples. The r e s u l t s obtained for t h i s s t r a t i f i c a t i o n c r i t e r i a — s e a s o n a l or a r r i v a l — w e r e i d e n t i c a l to those of the previous Tables 79 and 78. For instance, the greater the number of groupings NG, as i n MZ, the smaller the standard errors S-SYB.R and P-SYB.R became. I t may be noted that i n the case of Tables 80 and 81, the l a s t group in each NG may or may not have the same number of observations as the others, and those with less than two observations were ignored. This was as a r e s u l t of rounding up errors in using the r e l a t i o n -ship, NS = n * l./NG + 0.5, where NS i s an integer. As a r e s u l t of t h i s , there was no output for NG = 20, company K i n Table 80. Comparing Tables 7 8 versus 80, and 79 versus 81 for the same companies, indicated c l e a r l y that s t r a t i f i c a -t i o n by season or order of a r r i v a l s was superior to that by s i z e . Smaller standard errors were consistently obtained for the seasonal s t r a t i f i c a t i o n c r i t e r i a compar-ed to those of the- size of X. l t 9 V 7.4 Double-sampling procedures for companies A,D, F,J,K,L,M Double sampling procedures were used to determine the d i s t r i b u t i o n of X (that i s , the proportions) for some of the p r o b a b i l i t y type sampling schemes, and to estimate the mean of x , X in separate exercises. The sample sizes required for a s p e c i f i e d precision — 5 to 20% of X—were determined using the large sample theory normal approximation formula for confidence in t e r v a l s without use of r a t i o estimation. Random samples were then drawn from the combined populations A,D,F, J,K,L,M, that i s , T (Refer Table 73 and Appendix XXIV). At the same time, the proportion for frequency classes 5 to 2 0 (increment by 5) were determined from the sample data. These estimated proportions were then s t a t i s t i c a l l y compared with those given i n Appendix XXIV but there were no disagreements as to th e i r s i m i l a r -i t i e s . In another exercise, these estimated proportions were concealed and then used to sample from the popula-t i o n as had been done for the previous cases. The algorithm used i n t h i s case could be likened to what had been commonly referred to as 'quota sampling' (Cochran, 1953). The sample size for each stratum wwas 0Qbtained by multiplying the desired p r o b a b i l i t y (determined by the sampling scheme) by the t o t a l number of samples expected. Sampling for both x and y then proceeded u n t i l the required number of observations for each weight class or stratum had been obtained. In the process, however, i t was necessary at c e r t a i n stages to terminate and r e -evaluate the proportions so as to adjust the stratum sample sizes where necessary. This resulted i n more computer time than usual. The r e s u l t s were i d e n t i c a l to those enumerated in chapters 5 and 6, but judging from the comparatively greater computer time than those of chapters 5 and 6, t h i s process may even be more d i f f i c u l t to implement in pract i c e . Moreover, the schemes for which these proportions were required never improved on th e i r deviations and standard errors worsened, thus making such double-sampling procedures unnecessary. Similar r e s u l t s were obtained for the procedure in which X was estimated from the f i r s t phase large samples. In computing the s t a t i s t i c s , extensive use was made of the formulae given in chapter 2, sections 3.i and 3 . i i , and the r a t i o variance for p = 1.0. On the whole, there were no improvements on the r e s u l t s while using double sampling procedures for the weight scaling data. In fact, i n most cases, worse r e s u l t s were obtained and judging from the extra comput-ing time required, i t may not be necessary to consider t h i s for use i n weight scaling practices. It i s conceiv-able that the procedure may be s u f f i c i e n t for some other populations, but c e r t a i n l y not for weight s c a l i n g . 152 . 7.5 Summary of res u l t s P o s t - s t r a t i f i c a t i o n and p r e - s t r a t i f i c a t i o n standard error estimates were superior to n o n - s t r a t i f i c a t i o n estimates. I t was also r e a l i s e d that s t r a t i f i c a t i o n by season or order of a r r i v a l s was by far superior to the s t r a t i f i c a t i o n by the size of the a u x i l i a r y variable (weight). Double-sampling procedures for weight scaling were found to be i n f e r i o r , but i d e n t i c a l r e s u l t s for the sampling schemes as for chapters 5 and 6 were obtained. The t h e o r e t i c a l bias of the estimate of the mean of the population of interest was found to be n e g l i g i b l e , e s p e c i a l l y for p = 1.0 and thus served no useful purpose in correcting for some of the deviations from the DX or IX schemes. It also indicated that i t would not be useful to compare the various r a t i o estimates, i n pa r t i c u l a r the r a t i o of averages and the r a t i o of means. 153. 8.0 CONCLUSIONS AND RECOMMENDATIONS The best minimum-variance sampling scheme for r a t i o estimation, and in par t i c u l a r for weight scaling, was the scheme i n which sampling was done with probability-proportional to the r e c i p r o c a l of the magnitude of the a u x i l i a r y variable (DX5). This i s more s p e c i f i c than the generalization reported by Hansen and Hurwitz (1943), Horvitz and Thomson (1952), Raj (1956), and Sukhatme and Sukhatme (1970). This sampling scheme, for fi v e classes, resulted in huge deviations of the estimate of the population parameter of interest from the true value. With increasing number of classes, t h i s disadvantage was reduced but the property of minimum variance was not necessarily retained. Attempts to correct for these deviations from DX5 were not successful. The e f f i c i e n c y of the DX5 and 1X5 schemes for the generated cases 1-21 was dependent on the value of p in the re l a t i o n s h i p , p V(y^) = k(XMIDj) given i n equation (4.1.10) on page 62 For values of p greater than 0.0, these schemes consistently gave the least standard errors (Tables 35A-58A and 73). However, for negative values of p, except for DX5 and 1X5 that performed poorly, there was no clear decision as to which of the schemes gave the most consistent least standard errors or maximum rank values. For the aggregat-ed cases A1-A3, where p was estimated to be greater than 0.0, the e f f i c i e n c y of DX5 and 1X5 over the others was confirmed. The modified random sampling schemes, MZ, also gave f a i r l y small standard e r r o r s . This scheme, unlike the DXZ or IXZ, was independent of the magnitude of the a u x i l i a r y v a r i a b l e . The most appropriate sampling schemes for weight , s c a l i n g — o n the basis of least mean deviations and reason-able precision—were the modified random sampling (MZ) procedures. These procedures r e l i e d on the groupings by order of a r r i v a l s of the truck loads (by season or l o c a l i t y ) and could e a s i l y be incorporated i n present weight scaling procedures without much modification. The modified random sampling procedures, i d e n t i c a l to uniform sampling within a r r i v a l s t r a t a , were superior to the representative sampling scheme used for weight s c a l i n g . There were no agreements regarding the optimum number of group or strata required. However, the data studied indicated that the number of groups should be selected to make the within group or st r a t a variances small and possibly, homogeneous. Numerically, the bias of the ratio-estimator was n e g l i g i b l e . The r a t i o of means estimator, r ] ( was f a i r l y robust. The fact that the p values of the various weight scaling data were not d i f f e r e n t from 1 . 0 j u s t i f i e s the use of t h i s estimator in weight s c a l i n g . It was also observed that the sample c o r r e l a t i o n c o e f f i c i e n t s were not very much re l a t e d to the e f f i c i e n c y of the various sampling schemes. However, for a l l schemes, the frequency class or group with the highest sample c o r r e l a t i o n c o e f f i c i e n t usually gave the best precision. Double-sampling procedures did not r e s u l t i n any improvement, and in some cases gave worse r e s u l t s . They therefore could not be recommended for use in weight scali n g , at least in the form or manner used i n t h i s study. The use of p o s t - s t r a t i f i c a t i o n procedures resulted i n substantial improvement in the p r e c i s i o n . There was some indication that the greater the number of strata (each stratum of more than twenty observations), the better the precision obtained. S t r a t i f i c a t i o n on the basis of season or a r r i v a l of truck loads was far superior to that by the size of the a u x i l i a r y v a r i a b l e . This implied that the best approach to sampling for weight scaling would be to use modified random sampling, and then the use of post-s t r a t i f i c a t i o n procedures. The following suggestions are made for weight scaling sampling: a) Random samples of equal sizes (or approximately equal) should be taken within each group. These groups should be determined by the order of a r r i v a l s of the truck loads. The optimum number of groups required w i l l depend on the v a r i a b i l i t y in the weight of the loads, season of sampling, and the precision expected. b) Further attempts should be made to s t r a t i f y the data after they have been c o l l e c t e d . P o s t - s t r a t i f i c a t i o n by season or order of a r r i v a l s may be used. The greater the number of s t r a t a , the better. These strata need not correspond with the groups that might have been used i n a) above. Other s t r a t i f i c a t i o n c r i t e r i a may be sale value, o r i g i n of loads, timber marks and so on. It i s conceivable that i f p r e - s t r a t i f i c a t i o n ( s t r a t -i f i e d random sampling) procedures are used and the procedure suggested above applied to each pre-strata, better e f f i c i e n c may be obtained. The cost of such extensions may have to be studied for j u s t i f i c a t i o n purposes. c) The use of the ordinary r a t i o of means estimator is appropriate. However, the use of the estimator without any p r e - s t r a t i f i c a t i o n by species, l o c a l i t y and others, may lead to misleading r e s u l t s i f one attempts to set up confidence l i m i t s for the aforementioned c r i t e r i a even i f p o s t - s t r a t i f i c a t i o n i s done. S o l i d wood content varies 157. i n r e l a t i o n t o t h e n a t u r e a n d e x t e n t o f d e c a y a n d v o i d s p r e s e n t . T h e r e a r e s i z e , p o s i t i o n o f l o g s i n t r e e , a g e , g e n e t i c a n d o t h e r e f f e c t s o n a l l t h e s e . D i f f e r e n c e s i n p e r c e n t a g e m o i s t u r e c o n t e n t w i l l c h a n g e w i t h l o c a l a n d s e a s o n a l w e a t h e r , l o c a l i t y , a m o u n t o f s t o r a g e , m e t h o d o f l o g g i n g a n d s o o n . S o m e o f t h e s a m p l i n g s c h e m e s w e r e g e a r e d t o w a r d s e v a l u a t i n g t h e a b o v e f a c t o r s . P r e - s t r a t i f i c a -t i o n , h o w e v e r , m a y l e a d t o a b e t t e r s o l u t i o n i f o n e w i s h e s t o d r a w s p e c i f i c i n f e r e n c e s a b o u t l o c a l p o p u l a t i o n s . E v e n t h o u g h c o s t s a n a l y s e s w e r e n o t p e r f o r m e d i n t h i s s t u d y , t h e r e l a t i v e " c o s t o f e f f e c t i v e n e s s " o f t h e s a m p l i n g s c h e m e s c o n -s i d e r e d c o u l d b e r e l a t e d t o t h e p r e c i s i o n o b t a i n e d f o r a p a r t i c -u l a r s a m p l e s i z e . F e w e r s a m p l e s , f o r e x a m p l e , m a y h a v e t o b e s e l e c t e d f r o m s o m e s c h e m e s ( m i n i m u m - v a r i a n c e t y p e s ) a n d s t i l l o b -t a i n a t l e a s t t h e s a m e p r e c i s i o n . I n t h e p a r t i c u l a r c a s e o f w e i g h t s c a l i n g , t h e c o s t o f m e a s u r i n g a s a m p l i n g u n i t i s u n k n o w n . s i n c e t h i s i s a v a i l a b l e o n h o u r l y b a s i s ( A p p e n d i x x x v i i ) . T h u s i t c o u l d b e s a f e l y a s s u m e d t h a t t h e c o s t o f m e a s u r i n g a u n i t f o r e a c h s c h e m e i s t h e s a m e . E x t r a t i m e , t h e r e f o r e C o s t , m a y o n l y a r i s e i n t h e p r e p a r a t o r y s t a g e s o f d r a w i n g t h e s a m p l e s , b u t t h i s c o u l d b e o f f s e t t h r o u g h e x p e r i e n c e a n d / o r k n o w l e d g e o f t h e a r e a a n d t h e p r e -v i o u s y e a r ' s p r o d u c t i o n . F o l l o w i n g a p p r o p r i a t e f i e l d t r i a l s , t h e i n s i g h t s g a i n e d f r o m t h e s e a n a l y s e s o f b o t h s i m u l a t e d a n d a c t u a l d a t a c o u l d l e a d t o p r a c t i c a l i m p r o v e m e n t s i n o p e r a t i o n a l m e t h o d s o f w e i g h t s c a l i n g . T h e m o s t i m p o r t a n t c a n d i d a t e m e t h o d s f o r f i e l d t r i a l s w o u l d b e t h e " m o d i f i e d " ( o r u n i f o r m ) r a n d o m s a m p l i n g s c h e m e s . 158. LITERATURE CITED Ajgaonkor, S. G. P. 1967. 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Estimators for use in weight scaling of sawlogs. C a l i f o r n i a Agric. Expt. Sta. B u l l e t i n 866. 23p. Williams, D. H. 1972. Bias in least squares regression. Unpublished M.Sc. (Forestry) t h e s i s , Univ. of B r i t i s h Columbia. 44p. Williams, W. H. 1961. Generating unbiased r a t i o and regres-sion estimators. Biometrics 17:267-274. Yates, F. 1960. Sampling methods for censuses and surveys. G r i f f i n , London, 3rd e d i t i o n . Yates, F. and P. M. Grundy, 1953. Selection without replacement from within strata with p r o b a b i l i t i e s proportionate to s i z e . Jo.ur. Roy. Stat. S o c , B15 (1) :253-261. APPENDIX I IFS LOG SCALc OATA ! RANKED S . E . , VALUES FOR SIZE = 28 • REP CR RS M5 VI0 Ml 5 M20 M25 M30 M35 2, 5 ,10 ,15 3,5 • 10 4 ,5 ,15 5, 5 ,10 ,15 6,5 , 10 ,15 1,5 ,10 .15 1X5 0X5 1 - 2 12 - 4 - - - 13 - - 6 - - 7 - - 11 - 3 - 10 8 1 9 5 2 - 8 - 1 7 - - - - 3 - 2 - - - 12 IC 5 - 6 11 9 - 13 4 - -3 12 - - 6 1 - 11 - - 13 - 4 10 - - - 3 7 - 2 ' - - 5 - 8 - 9 4 13 - - - 12 6 11 1 7 - - 8 10 5 4 2 3- 9 5 ' - IC 5 - - 7 - 13 11 8 1 - 2 - 12 4 3 6 9 6 - 10 - - 7 12 - - 6 - 1 3 2 4 9 - 5 13 - - 8 11 - - - -7 13 • - 3 1 - - - 5 4 - - - 8 12 7 IC - - - 6 - 11 - - - 9 2 8 2 3 - - - 12 - 8 13 11 10 4 - - 7 - 6 - 5 - - - 9 - 1' -9 2 v. -. - 9 11 10 - 8 - 12 - 7 - - 4 - 5 - 3 1 - 6 - 13 IC - •- 5 '- 11 - 6 - 2 - - 8 - 1 - 9 - - 12 - 10 - 4 3 - 7 13 TOT 42 23 30 13 51 18 58 15 33 61 19 38 31 24 11 67 28 26 47 36 34 36 31 37 21 29 51 * * * * * * * * * APPENDIX II IPS LOG SCALE DATA : RANKEC S . E . VALUES FOR SIZE' =31 REP CR RS M5 MIO M15 M20 M25 M30 M35 2, 5 ,10 ,15 3,5 ,10 4 ,5 ,15 5,5 ,10 ,15 6 ,5 ,10 ,15 1,5 .10 .15 1X5 CX5 1 6 - 13 10 - 2 - 1 - - - 8 - 4 7 - - 5 9 12 11 - - - - 3 -2 2 - 9 - - 4 - 7 5 - - - - 10 11 1 - - - 13 3 - -- 8 6 12 -3 - 10 12 9 11 - - - 1 - 3 7 - 2 - 13 - - - 6 5 - - - 4 8 - -- - 7 - 13 9 8 1 - 11 10 - - 12 - - 3 - - - 4 5 6 2 - -5 - . 2 - 6 - 8 - 9 - 13 - 12 1 - 5 - - 7 - 10 11 - 4 ~ 3 6 1 - - - 6 12 - - 13 10 3 - - 9 8 - - .- 7 11 5 - - - 2 4 7 - - 2 •- 5 13 - - - 8 1 - 6 - - • 7 • - 11 - - 9 IC - 4 - 12 8 - - IC 11 - - - - 12 5 - 8 - 1 - 2 9 - - 3 4 - - 6 - 7 13 9 - - 1 5 - 13 6 10 - 11 2 - 9 - - - _ • • 7 8 4 - - 12 3 - - -10 - - 8 - : . 6 - 10 13 - 7 4 - - - 5 3 1 11 - 9 2 - - - - - 12 TOT 9 12 62 41 41 53 32 32 40 41 37 33 27 27 43 31 10 37 24 57 46 29 27 27 16 32 44 ik * * * * * * * * * APPENDIX III IFS LOG SCALG DATA : RANKED S.E. VALUES FOR SIZE=36 REP CR RS M5 MIO M15 *20 M25 M30 M35 2,5 ,10 ,15 3,5 ,10 4,5 ,15 5,5 ,10 ,15 6,5 ,10 ,15 1,5 .10 ,15 1X5 DX5 1 8 2 4 - 1 1 9 - - - - - - 3 - - 7 - 1 6 - - - 1 0 5 1 3 - 1 2 2 4 - - 6 13 - 10 8 6 - 9 5 - 3 12 - 7 - - - - 1 - - - 2 11 3 - 4 1 - - - - - 6 - - 3 - 8 10 1 3 - 5 7 - 2 - 12 - - 9 11 4 - - 2 - - - 7 - - 1 - 5 8 6 4 13 - - 12 - - 9 - 10 - 3 11 5 2 - 7 - 11 - - 13 - - 3 1 - 5 10 4 - - - - - 8 - - 9 6 12 6 1 - - - - 13 - - 10 5 6 9 - - - - - 3 - 2 - 7 8 4 - 11 12 7 3 - 10 4 11 - 12 - - 2 - - 7 9 6 - 5 - - - - 1 - - 8 13 -8 11 9 - 7 8 13 - - - 12 - - - 2 6 - - 3 1 5 IC 4 -9 - 5 - 4 - 3 1 13 - 12 - - 9 11 - 7 - 6 8 - - - 10 2 - - -10 2 9 - 8 - - - - - - _ 1 2 1 - 5 10 - 1 3 - - 4 3 11 - - - 7 6 TOT 31 29 24 29 54 38 30 34 22 32 30 24 27 49 58 44 25 18 34 6 5 37 40 26 40 55 86 * * * * * * * * * * * * APPENDIX IV IFS LOG SCALE CATA : RANKEC S .E. VALUES FOR SIZE=45 REP CR RS M5 MIO M15 M20 M25 M30 M35 2, 5 , 10 , 15 3,5 ,10 4,5 ,15 5,5 ,10 ,15 6,5 ,10 ,15 1, 5 ,10 , 15 1X5 DX5 1 - 1 0 - - - - 5 - - - - 9 12 6 - 1 - 3 13 2 - 7 4 - - 8 11 2 2 - - 1 - - - - - - 7 - 6 8 1 2 - - - 5 10 11 9 - 3 4 - 13 3 - 13 - 11 - - 12 - - - - 10 2 7 - - - - 6 8 4 5 - 3 - 1 9 4 - - - - 7 2 4 13 - - 3 - - - - - 9 - 6 8 5 11 1 - - 10 12 • 5 - 3 2 6 - - - - - - 4 1 7 8 5 - - - - - 13 12 - 10 - 11 9 6 2 4 1 - - 5 - 12 - 7 - - - - - 11 10 a - - - - 9 6 - 13 - 3 7 3 - 10 4 11 - 12 - - 2 - - 7 9 6 - 5 - - - - 1 - - 8 13 -8 6 4 i 3 2 - - 6 8 9 7 - - - 13 12 - 11 10 9 11 - - 6 8 3 10 9 5 2 - - 1 4 - - - 5 - - - 7 - - - - 12 10 10 - - 11 6 8 1 - 13 13 - - - 12 3 7 4 - 9 - - - - - 2 TOT 34 34 13 39 32 18 44 34 18 24 15 23 37 54 37 24 34 17 46 23 33 61 24 28 27 54 79 * * * * * * * * * * * * * cn CT> APPENDIX V IFS LOG SCALE DATA : RANKED S.E. VALUES FOR SIZE=61 REP CR RS M5 Ml0 M15 M20 M25 M30 M35 2 r 5 . 1 0 , 1 5 3 . 5 . 1 0 4, 5 , 15 5 , 5 , 1 0 , 1 5 6 , 5 , 10 , 15 1, 5 , 1C , 1 5 1X5 CX5 1 - - - 12 8 - 11 12 - - - - 7 - - 5 - 3 - 1 - - 2 10 6 9 4 2 10 - - 9 - - - 1 2 4 1 1 7 - - - 6 2 - 8 5 1 - - - 3 - - 1 3 3 12 - - 6 1 - 1 1 - - 1 3 - 4 10 - - - 3 7 - 2 - - 5 - 8 - 9 4 - - - - - 12 - - 5 9 2 - 3 10 4 - - 6 - - 11 1 - - 7 3 13 5 2 10 13 - - - 12 - - 5 7 - - - - - 3 6 - - ~1 - 1 1 9 4 8 6 - - 3 - 1 C - - 7 13 11 - - - - - 4 1 - - 9 8 5 12 - - 2 6 7 1 - 11 8 9 - - - 7 - - 8 2 - 12 - - 4 10 - - 3 - - 5 - 13 8 - - 3 - 11 9 - - - 6 4 - 2 8 7 12 - 10 - - - - 5 1 - - 13 9 1 13 - - - 8 9 - 4 3 - 12 - 6 - 11 10 - 2 - 7 - - - - 5 -10 - 1 4 - - 8 - - - - - - 5 - - 1 3 6 - 3 11 7 IC - - 9 2 12 TOT 26 24 34 35 39 37 43 32 33 58 20 24 29 24 29 47 23 44 20 24 34 19 35 14 44 30 91 * * * * * * * * * * * * APPENDIX VI IFS LOG SCALE DATA : RANKED S . E . VALUES FOR SIZE=78 REP CR RS M5 MIO M15 M2C M25 M30 M35 2 , 5 , 1 0 , 1 5 3 , 5 , 1 0 4 , 5 , 1 5 5 , 5 , 1 0 ,15 6 ,5 , 1 0 , 1 5 1,5 ,10 ,15 1X5 DX5 1 1 0 8 - - 11 5 - 3 9 - 2 - 7 12 - 6 . - - 1 - - 4 - 13 2 - - - - - _ 9 10 12 6 11 5 13 - - - 7 1 - 3 8 4 - 2 -3 7 - - - 10 - - 6 4 3 - 12 1 - 11 2 - - 13 - - - 9 8 - 5 4 - - - - - - 5 9 - 13 6 - - 11 - 12 10 - 1 7 - 4 - - 3 8 2 5 - 2 - - - 5 9 - - 7 8 - 6 1 0 - 4 - - - 3 - 1 - - 11 13 12 6 - 1 4 - 2 3 5 - - - - - 8 7 12 - 11 - 13 - - 10 - - 6 9 -7 - 1 - - - 6 3 7 - 9 4 5 8 - 12 2 - 11 - - - - - - - 10 13 8 - 6 - 11 - 13 - 12 - 9 5 10 8 - - - 1 - - - - - 4 7 2 - 3 9 - 1 7 - - 3 - 3 1 3 1 0 1 1 9 4 - - 1 2 - 2 - - ~ - - - - 6 5 10 - - 1 - 2 7 - 3 - - - 10 - - 9 6 - ' - 4 1 2 5 11 - - - 1 3 8 TOT 17 19 12 11 25 47 22 43 35 61 48 52 46 40 69 38 22 19 38 23 5 30 21 19 26 66 56 APPENDIX VII IFS LOG SCALE DATA: RANKED S .E . VALUES FOR SIZE=164 REP CR RS »5 MIO M15 M20 M25 M30 M35 2,5 ,10 ,15 3,5 ,10 4,5 ,15 5,5 ,10 ,15 6,5 ,10 ,15 1,5 .10 ,15 1X5 0X5 1 9 - 1 - - - 8 - - 2 6 - - 7 13 - - 11 3 5 - - 10 4 12 2 3 - - 5 - 4 10 - 6 8 7 1 - - - 9 - - 12 2 1 1 - - - - - 13 3 1 - - - 2 1 3 - 5 - 6 - - -' 3 ' 12 - 1 0 - ' - 9 7 11 - - - 8 4 4 - 8 - - - 2 - 7 4 13 - - - 11 - 3 9 10 - - 5 6 1 - - - 12 5 - - 8 - - - - - - - 6 1C 2 9 11 3 - - - - 12 7 - 1 4 5 13 6 - - - 7 - 1 0 - 12 6 T - 3 - 8 - 4 - 1 . 9 5 - 2 - - - 13 11 7 2 - - - - - - 5 - 7 - 4 3 13 2 9 12 11 - - - 6 - 1 10 - 8 8 _ 5 _ 9 8 - - - - 4 - 7 10 11 1 - 6 - - - - - - 3 2 12 13 9 7 9 - - - 11 - 12 - - - 8 - 10 3 2 1 - - - 8 4 - - - 5 13 10 11 - 9 - 12 2 4 5 6 - - - - - 7 3 s - - - 10 - - - 1 - 1 3 TOT 33 22 18 21 22 42 14 54 22 38 13 35 21 65 36 40 59 22 21 27 56 41 1 5 27 47 100 * * * * * * * * * * APPENDIX VIII IFS LOG SCALE DATA : RANKED S.E. VALUES FOR SIZE=166 REP CR RS M5 Ml 0 M15 M20 M25 M30 M35 2, 5 , 10 , 15 3,5 , 10 4,5 , 15 5, 5 , 10 ,15 6, 5 , 10 , 15 1,5 ,10 ,15 1X5 DX5 1 - - 1 C - - 7 - - 6 4 - 8 12 - - 5 - 3 - 11 1 2 9 - - - 13 2 12 - 8 - - 5 - - 9 - 7 - 3 2 1 3 1 1 - - 1 * •- - 1 0 - - 6 3 13 - - - 9 - - - 3 2 1 0 - 1 1 - 8 1 - 4 - 5 - 7 - 1 2 6 4 - - - 9 - - 3 4 - 7 - - 11 13 5 - - - 8 2 10 6 - - 1 - 12 5 - 6 - 1 - 1 0 - - - 3 - - 1 1 - - - - • 5 12 9 7 2 - 8 4 \13 6 3 - - 5 - 10 - 2 7 - - 8 6 - 11 - 9 - 1 - - - - 4 - 12 13 7 - - - - 7 - 2 - 11 - - 10 - - 4 5 - - 13 12 3 8 1 - - 6 9 8 - - 4 - 8 - 10 - 1 2 1 3 - - 6 1 2 5 9 - - 7 - 11 3 9 1 3 - 11 6 7 9 - - - 13 - - 2 - 12 - - 4 - 6 - - - 8 5 10 10 - - 11 4 5 12 9 7 - - - - 3 8 - 12 - - - 6 - - - 2 - 1 10 TOT 29 9 32 30 26 56 38 13 25 26 17 56 43 37 28 67 26 3 35 44 47 23 19 23 17 51 95 APPENDIX IX IFS LOG SCALE DATA : RANKED S.E. VALUES FOR SIZE = 180 REP CR RS M5 MIO Ml 5 M20 M25 M30 M35 2 ,5 , 10 ,15 3,5 , 10 4, 5 ,15 5, 5 ,10 ,15 6, 5 ,10 ,15 1,5 ,10 ,15 1X5 DX5 1 - 10 6 - - - 1 - 13 8 5 - - 12 - • - 2 - 3 - 4 11 - - - 9 7 2 8 2 6 4 3 10 - 1 - - - - 7 - 5 9 11 12 12 3 - 11 S - - - - - 9 7 3 - 1 - 12 13 4 5 - - - - - 6 2 - 10 4 12 - - - 6 - 3 - 11 10 13 - 5 - 1 2 8 - 9 - - 7 - - - - - 4 5 - 8 7 - - - 1 - - 9 - - 11 6 10 12 - - 13 - - - 3 4 2 5 -6 - - - 8 - - 10 - - - 6 1 - 5 2 13 - 4 9 7 3 - - - 12 11 7 - - ' 10 - - 8 4 - 3 - - 9 - 6 12 5 - - 7 2 - 1 11 • - - - 13 a — 5 - 4 - - - - 7 - - - 12 11 9 10 - - 1 - 6 - 13 3 2 8 9 4 - - 7 - - 1 - 3 - - - - - 8 11 - - 9 12 2 5 10 6 - - 13 10 8 - - - 7 - 1 10 11. - 4 - 6 3 13 - - - - 12 - - - 2 5 - 9 TOT 32 36 31 17 15 i l 8 31 49 50 16 23 28 40 74 60 29 14 36 36 27 26 29 40 23 40 88 170. APPENDIX X SPECIES FROM SAMPLE LOADS FOR COMPANIES A-M Company Operator Species Codes Stratum A 01 FI LO SP BA YE CE WB HE WH 02 BA LO SP WB FI WH F/R 03 FI LO YE BA SP AS F/R 04 LO SP . AS FI BA F/R 05 YE FI LO F/R 06 FI YE AS BA LO SP F/R B A l l BA CE FI HE LO SP WB WH F/R C 01 BA CE CY FI HE SP F/R D A l l FI LO SP BA F/R E A l l BA CE FI HE LA LO SP WB WH F 01 FI HE LO YE CE WH SP BA F/R 02 FI HE SP WH BA CE F/R 03 CE FI HE SP WB F/R 04 BA SP LO FI WH F/R 05 SP WH FI LO CE HE BA YE F/R 06 BA CE SP FI HE WH F/R 08 BA SPA FI LO F/R G A l l BA CE HE LO SP F/R H 01 AL BA CE CY FI HE CO MA WH I A l l HE CE CY SP F/R J A l l SP FI LO BA HE CE R/R K 01 SP FI LO BA HE CE AS F/R 02 SP FI LO BA HE CE CO BI AS L 01 LO SP BA FI F/R 03 " LO SP FI AS BA F/R 04 LO BA FI SP F/R • 06 FI LO SP AS BA F/R M A l l SP FI LO BA HE CE F/R YE F/R 171. APPENDIX X (Continued) EXPLANATION OF SPECIES CODES AL - ALDER AR - ARBUTUS AS - ASPEN BA - BALSAM BI - BIRCH CE - CEDAR CO - COTTONWOOD CY - CYPRESS FI - FIR HE - HEMLOCK LA - LARCH LO - LODGEPOLE PINE MA - MAPLE OA - OAK SP - SPRUCE WB - WHITEBARK PINE WH - WHITE PINE YE - YELLOW PINE F/R - FIRMWOOD REJECT ( A l l species) 172. APPENDIX XI LOG SCALING OATA ANALYSIS FOR COMPANY A, KAMLOOPS DISTRICT X3AR = 63453o2 POUNDS (LBS), YBAR = 1307*6 CUBIC F££T.(C.FT1 NUMBER OF CLASSES = 5 CLASS WIDTH = 11284,20 LBS •CELL LIMITS ( LBS) 43620,0 54904. 2 66188.4 77472.6 88756.3 54904.1 66188.3 77472.4 88756.6 100040.8 FREQ (LOADS) 16 79 125 37 1 XVAR I ANCS (L3S**2) 12102710o8 6449812. 5 9818163.1 4497961.3 0. 0 YV ARiANCE (C.FT**2) 18663.2 9048.0 17904.0 26278.9 0.0 RATIO (C.FT/LB) 0.019516 0.019327 0.019015 0.018969 0.016513 LOG SCALING DATA ANALYSIS FOR COMPANY A, MHRRIT {INTERIOR) XBAR =• 68453.2 POUNOS(LBS), YBAR = 1307,6 CUBIC FEET (C.FT ) NUMBER OF CLASS ES =10 CLASS WIDTH = 5642.10 LBS X-CELL LIMITS FREQ XVARIANCS YVARIANCE RATIO (LBS) (LOADS) a a s * * 2 > (C»FT**2) (C.FT/LB) 43620.0 - 49262. 0 8 3393503.9 8775.9 0.019083 49262.1 - 54904.1 8 2097818.3 17198.7 0.019900 54904,2 - 60546.2 20 1287291.8 926 2.0 0.019862 60546.3 - 66188.3 59 2493230.7 3507.9 0.019159 66188.4 - 7183 0.3 75 2 734125. 0 12789.4 0,018917 71830.4 - 77472.4 50 2518323.6 17100.6 0.019151 77472.5 - 83114.4 32 2083195. 6 24149.3 0.018893 83114.6 - 83756.5 5 1535744.0 30443.3 0.019431 88756. 6 - 94398.6 0 0.0 0.0 0.0 94398.7 - 100040.6 1 0. 0 0.0 0.016513 LOG SCALING DATA ANALYSIS FOR COMPANY A, M5RRIT (INTERIOR) XBAR . = 6 8453.2 POUNOSILBS), YBAR = 1307.6 CUBIC FEET(C.FT) NUMBER OF CLASS 5S =15 CLASS WIDTH = 3761.40 LBS X-C£LL LIMITS FREQ XVARIANCE YVARIANCc RATIO (LBS) (LOADS) <L'BS**2) (C.FT**2) (C.FT/L8) 43620.0 - 47381.3 6 2047001.1 12177.1 0.019394 47381.4 - • 51142.7 4 2123661.0 17866.0 0.019590 51142, 8 - 54904.1 6 1724857.6 19626.3 0.019575 54904.2 - 58665. 5 7 307541.0 9119.1 0.019789 58665.6 - 62426.9 30 1448608.6 10684.3 0.019852 62427. 0 - 66188.3 42 1280332.7 7419.2 0.018904 66188,4 - 69949. 6 55 1333847. 9 14668.6 0,019007 69949, 8 - 73711.0 44 1222936.0 8434.6 0.018669 73711.1 - 77472.4 26 1032379.4 17855.9 0.019589 77472,5 - 81233.8 23 1069241.4 20964.6 0.018849 81233,9 - 84995.1 12 1018914.4 37129.8 0.019202 84995.3 - 88756.5 2 163320.0 1250.0 0,018906 88756.6 - 92517.9 0 0,0 0,0 0.0 92518. 0 - 96279.3 0 0.0 0.0 0.0 96279.4 - 100040.6 1 0. 0 0.0 0.016513 APPENDIX XII 173. LOG SCALING D A T A ANALYSIS FOR COMPANY Bt. NELSON DISTRICT X B A R = 630: 3 7.9 P O U N D S ( L B S ) , Y B A R = 1 268.7 C U B I C FEET(Co FT) NUMBER Q F 1 CLASS £S = 5 CLASS WIDTH =11368.20 LBS X-CELL LIMITS FREQ XVARIANCE YVARIANCE RATIO (LBS) (LOADS) (LBS**2) (C.FT**2) (C.FT/LB) 331O0„0 - 44468.1 7 14593931.0 13873.3 0.021932 44468.2 - 55836*3 14 7154343.1 11205.9 0.020112 55836.4 - 6 72 04.4 196 6316138.4 11-78 9.7 0.020069 67204.6 - 78572.6 59 8192569. 6 19882.5 0.020181 78572.8 - 39940.8 3 23788981.3 76512.3 0.019957 LOG SCALING DATA ANALYSIS FOR COMPANY B, NELSON DISTRICT XBAR = 63 03 7 .9 PQUND5(LBS)t Y B A R = 1268.7 C U B I C FEET(C» FT) NUMBER D F ( : L A S S ES =10 C L A S S WIDTH = 5684.10 L B S X-CELL L I M I T S F R E Q XVARIANCt YVARIANCE RATIO { LBS) (LOADS) 1LBS**2) (C.FT*#2) (C.FT/LB) 33100.0 - 38784.0 3 1823221.3 4666.3 0.021422 38784.1 - 44468.1 4 612133.0 4834.3 0.022332 44468.2 - 50152.2 1 0. 0 0.0 0.019912 50152.3 - 55836.3 13 2175474.4 9617.6 0.020125 55836.4 - 61520.4 71 2322760.4 7704.6 0.020216 61520.5 - 67204.4 125 1991064. 2 12300.5 0.019992 67204o6 - 72883.5 44 2140145.4 13094.5 0.020134 72883.6 - 78572.6 15 2559934.1 14229.4 0.020310 78572.7 - 84256.6 42803.0 92020.5 0.019473 842 56. 8 - 89940.7 1 0.0 0.0 0.020825 LOG SCALING DATA ANALYSIS F O R COMPANY B, NELSON DISTRICT XBAR = 63037.9 POUNDS(LBS), YBAR = 1268.7 CU6IC FEET C C. FT i NUMBER OF CLASSES =15 CLASS WIDTH =. 3789.40 LBS X-CELL . LIMITS FREQ . XVARIANCE YVARIANCE RATIO (LBS) (LOADS) (LBS**2) (C.FT««*2> (C.FT/LB) 33100.0 - 36889.3 3 1823221.3 4666.3 0.021422 36889.4 - 40678.7 1 0. 0 0.0 0.020371 40678.8 -• 44463.1 3 332264.0 626.3 0.022966 44468. 2 - . 48257. 5 1 0.0 0.0 0.019912 48 257.6 - 52046.9 1 0. 0 0.0 0.019415 52047. 0 - 55836.3 12 1579783.6 9630.2 0.020181 55836.4 - 59625.7 42 1007631.7 8668.4 0.020439 59625.8 - 63415.1 102 366339. 0 931 6. 0 0.019388 63415.2 - 672 04.4 52 100021.1.3 12863.2 0.020140 67204.6 - 70993.8 3 5 1067799.7 15549.6 0.020079 70993.9 - • 74733.2 15 1218137.5 2063 9.5 0.020202 74733.3 - 78572.6 9 1600366.0 11747.3 0.020508 78572.7 - 82361.9 2 42803. 0 92020.5 0.019473 32362.1 - 86151.3 0 0.0 0.0 0.0 86151.4 - 89940. 7 1 0.0 0.0 0.020825 APPENDIX X I I I 174. LOG SCALING DATA ANALYSIS FOR COMPANY C, V A N C O U V E R (COAST) XBAR = 14309L3 POUNDS(LBS), YBAR = 2695.1 CUBIC FEET (C. FT ) NUMBER OF CLASSES = 5 CLASS WIDTH -35910.20 L3S X-CELL LIMITS (LBS) 53500.0 89410.2 125320.4 161230. 6 197140.8 89410.1 125320.3 161230.4 197140. 6 233 050.8 FREQ (LOADS) XVARIANCE (LBS**2) 1 0.0 82 60901469.2 101 110525784.2 69 101503455.0 11 110883313.5 YVARIANCc (C.FT*+ 2) 0.0 91159.6 32750. 8 99830.5 151867.8 RATIO (C.FT/LB) 0.060692 0.019356 0.018561 0.018574 0.018877 LOG SCALING DATA ANALYSIS FOR COMPANY C, PORT MACNEIL (COAST) XBAR = 143091.3 POUNDS(LBS), YBAR = 2695.1 CUBIC FE5T(C S NUMBER OF CLASSES =10 CLASS WIDTH =17955.10 LBS FT) X-CELL LIMITS FREQ XVARIANCE YVARIANCE RATIO (LBS) (LOADS) (LBS**2) (C.F7**2) (C.FT/LB) 53500.0 - 71454.9 1 0. 0 0.0 0.060692 71455.1 - 89410.0 0 0.0 0. 0 0.0 39410.1 - 107365.1 28 15769922.2 188139.5 0.020419 107365.2 - 125320.1 54 17322003.3 . 3 9 54 7. 1 0.018871 125320.3 - 143275.2 54 23197595,8 40438.4 0.018406 143275.3 - 161230.3 47 22336424,8 49412.3 0.018718 161230.4 - 179185.3 47 2 968 8362. 3 51406,4 0.018669 179185.4 - 197140.4 22 18610547.5 149144.2 0.018391 197140.5 - 215095.4 10 32203876.2 131794.6 0.018860 215095.6 - 233050.5 1 0. 0 0.0 0.019026 LOG SCALING DATA' ANALYSIS FOR COMPANY C, PORT MACNEIL (COAST) XBAR = 143091,3 POUNOS(LBS), YBAR = 2695.1 CUBIC F E E K C.FT) NUMBER OF CLASSES =15 CLASS WIDTH =11970.07 LBS X-CELL LIMITS FREQ XVARIANCE YVARIANCE RATIO (LBS) (LOADS) (LBS**2) (C.FT**2) (C.FT/LB) 53500.0 — 65470.0 1 0. 0 0.0 0,060692 65470.1 - 77440.0 0 0,0 0.0 0.0 77440. 1 - 89410. 1 0 0.0 0.0 0.0 39410.2 - 101380.1 12 5635703. 3 8428 9,9 0.019860 101380.3 - 113350.2 36 14098022.8 123672.3 0.019820 113350.3 - 125320.3 34 6541858.6 44165.5 0.018759 125320.4 - 137290.3 40 11687765. 9 26292.5 0.018228 137290.4 - .1492 60.4 29 16342072.6 31328.9 0.018815 149260.5 - 1612 30.4 32 10413904,7 57153.6 0.018701 161230.6 - 173200.5 34 9297630.3 55164.4 0.018738 173200. 6 - 185170.6 21 14091415.5 93137.7 0.018517 185170.7 - 197140.6 14 9076007, 3 126277.0 0.018307 197140. 3 - 209110.7 8 8126836.3 153912.8 0.018871 209110.8 - 221080.8 2 7796782.0 1682.0 0.018817 221080.9 - 233050.8 1 0,0 0.0 0.019026 1 7 5 . APPENDIX XIV LOG SCAL ING DATA ANALYS I S FOR COMPANY D, CARIBOO D I S T R I C T XBAR = 6 6 6 5 7 o 3 P O U N O S ( L B S ) , YBAR = 1 3 1 1 . 6 CUBIC FEET ( C. FT) NUMBER OF C L A S S E S = 5 CLASS WIDTH = 7 3 5 6 . 2 0 LBS X - C E L L L I M I T S FREQ XVARIANCE YVARIANCE RATIO ( LBS) ( L OA DS ) ( L B S * * 2 ) ( C . F T * * 2 ) ( C . F T / L B ) 4 3 8 2 0 . 0 - 5 1 6 7 6 . 1 5 8 5 6 7 2 4 4 . 0 1 6 3 8 2 . 5 0 . 0 2 3 4 9 7 5 1 6 7 6 . 2 - 5 9 5 3 2 . 3 23 3 9 0 7 4 6 7 . 5 9 3 2 3 . 7 0 . 0 2 0 4 1 8 5 9 5 3 2 . 4 - 6 7 3 8 8 . 4 85 4 8 5 1 7 7 2 . 6 7 2 5 1 . 3 0 . 0 1 9 6 4 9 6 7 3 8 8 . 6 - 7 5 2 4 4 . 6 S3 4 6 0 7 8 6 4 . 3 7 4 3 3 . 8 0 . 0 1 9 4 9 4 7 5 2 4 4 . 8 - 8 3 1 0 0 . 8 24 3 2 9 9 7 2 0 . 3 4 5 8 2 . 9 0 . 0 1 9 2 1 8 LOG S C A L I N G DATA ANALYSIS FOR COMPANY D, CARIBOU D I S T R I C T XBAR = 6 6 6 5 7 . 3 P O U N D S t L B S ) , YBAR = 1 3 1 1 . 6 CUBIC F E E T ( C . F T ) NUMBER OF CLASSES =10 CLASS WIDTH = 3 9 2 8 . 1 0 LBS X - C E L L L I M I T S FREQ XVAR I ANCE YVARIANCE RATIO ( LBS ) (LOADS) ( LBS*#2) ( C . F T * *2) ( C . F T / L B ) 4 3 3 2 0 . 0 - 4 7 7 4 8 . 0 • 3 3 5 7 3 1 8 1 . 3 1 1 6 4 1 . 0 0 . 0 2 4 0 4 7 4 7 7 4 8 , 1 - 5 1 6 7 6 . 1 2 4 3 8 0 3 0 0 . 0 4032 8 . 0 0 . 0 2 2 7 4 4 5 1 6 7 6 . 2 - 5 5 6 0 4 . 2 9 1 1 3 9 9 1 6 . 4 6 5 2 2 . 0 0 . 0 2 0 5 4 8 5 5 6 0 4 . 3 - 5 9 5 3 2 . 3 19 6 7 6 6 5 6 . 7 9 6 4 5 . 1 0 . 0 2 0 3 6 0 5 9 5 3 2 . 4 - 6 3 4 6 0 . 4 34 1 0 0 8 1 0 6 . 3 6 2 1 4 . 7 0 . 0 1 9 7 2 3 6 3 4 6 0 . 5 - 6 7 3 8 8 . 4 51 12677 7 3 . 3 6 1 3 0 . 3 0 . 0 1 9 6 0 3 6 7 3 8 8 . 6 - 7 1 3 1 6 . 5 52 1 1 6 0 1 8 2 . 3 5 0 2 5 . 2 0 . 0 1 9 4 5 4 7 1 3 1 6 . 6 - 7 5 2 4 4 . 6 3 1 1 2 8 2 6 6 6 . 7 7 6 1 2 . 3 0 . 0 1 9 5 5 8 7 5 2 4 4 . 7 - 7 9 1 7 2 . 6 19 1 4 7 7 8 3 0 . 8 3 3 7 6 . 4 0 . 0 1 9 1 6 6 7 9 1 7 2 . 8 - 8 3 1 0 0 . 7 5 2 0 2 5 6 0 3 . 0 2 3 7 4 . 7 0 . 0 1 9 4 0 7 LOG SCAL ING DATA ANALYS IS FOR COMPANY D, CARIBOU D I S T R I C T XBAR = 6 6 6 5 7 . 3 P O U N D S ( L B S ) , Y BAR = 1 3 1 1 . 6 CUBIC F E E T ( C . F T ) NUMBER OF CLASSES =15 CLASS WIDTH = 2 6 1 8 . 7 3 LBS X - C E L L L I M I T S FREQ XVARIANCE YVARIANCE RATIO ( LBS ) ( LOADS) ( L B S * * 2 ) ( C . F T * * 2 ) I C . F T / L B ) 4 3 8 2 0 . 0 — 4 6 4 3 8 . 6 2 3 1 9 7 1 2 . 0 2 2 8 9 8 . 0 0 . 0 2 4 8 0 8 4 6 4 3 8 . 7 - 4905 7 . 4 2 2 7 3 7 8 4 . 0 3 6 9 3 . 0 0 . 0 2 1 5 5 3 4 9 0 5 7 . 5 - 5 1 6 7 6 . 1 1 0 . 0 0 . 0 0 . 0 2 4 8 6 3 5 1 6 7 6 . 2 - 5 4 2 9 4 . 8 2 7 1 3 6 . 0 1 5 8 4 2 . 0 0 . 0 2 0 0 8 0 5 4 2 9 4 . 9 - 5 6 9 1 3 . 5 8 3 7 6 3 6 7 . 7 4 6 2 5 . 3 0 . 0 2 0 4 2 0 5 6 9 1 3 . 6 - 5 9 5 3 2 . 3 18 5 2 8 0 1 7 . 5 913 2 . 9 0 . 0 2 0 4 5 1 5 9 5 3 2 . 4 - 6 2 1 5 1 . 0 26 4 5 3 1 4 3 . 2 4 4 1 0 . 9 0 . 0 1 9 7 2 2 6 2 1 5 1 . 1 - 64 7 6 9 . 7 22 3 7 5 7 6 4 . 5 8 3 7 9 . 6 0 . 0 1 9 7 3 0 6 4 7 6 9 . 8 - 6 7 3 8 8 . 4 37 5 7 4 9 6 8 . 1 5 7 0 4 . 0 0 . 0 1 9 5 2 7 6 7 3 8 8 . 6 - 7 0 0 0 7 . 1 41 5 9 6 3 3 8 . 0 5 3 5 8 . 4 0 . 0 1 9 5 0 7 7 0 0 0 7 . 3 - 7 2 6 2 5 . 8 26 5 3 7 1 4 3 . 4 7 3 8 3 . 7 0 . 0 1 9 4 2 9 7 2 6 2 5 . 9 - 75 2 4 4 . 5 16 9 1 3 0 8 3 . 4 5 7 5 2 . 1 0 . 0 1 9 5 6 5 7524<+.6 - 7 7 8 6 3 . 2 11 8 1 6 1 8 3 . 9 4 3 1 9 . 8 0 . 0 1 9 2 5 3 7 7 8 6 3 . 3 8 0 4 8 1 . 9 11 4 1 6 5 5 3 . 9 370 7 . 7 0 . 0 1 9 1 6 2 8 0 4 8 2 . 0 - 8 3 1 0 0 . 6 2 2 9 9 4 2 2 2 . 0 2 7 3 3 . 0 0 . 0 1 9 3 3 4 176. APPENDIX XV LOG SCALING DATA ANALYSIS FOR COMPANY E , NELSON DISTRICT XBAR = 59934.9 POUNDS(LBS), YBAR = 1191,9 CUBIC FEET(C.FT) NUMBER OF CLASSES = 5 CLASS WIDTH =15080,20 LBS X-CELL LIMITS FREQ XVARIANCE YVARIANCE RATIO (LBS) (LOADS) (L3S**2) (C.FT-*2) (C.FT/LB) 16800,0 - 31880. 1 2 14579936.0 9248.0 0.018308 31880,2 - 46960.3 28 18566972.3 21328.6 0.023466 46960,4 - 62040,5 139 16392022,9 21658.2 0.019950 62040, 6 - 77120.6 152 14635400.1 19734.6 0.019445 77120,8 - 92200.8 7 31034133. 3 13755.6 0.01 9206 LOG SCALING DATA ANALYSIS FOR COMPANY E, NELSON DISTRICT XBAR = 59984,9 POUNDS(LBS), YBAR = 1191,9 CUBIC FEET(Co FT) NUMBER OF C LASSES =10 CLASS WIDTH = 7540.10 LBS X-CELL LIMITS FREQ XVARIANCE YVARIANCE RATIO (LBS) (LOADS) (L3S**2) (C.FT**2) (C.FT/LB) 16800,0 - 24340. 0 2 14579936.0 9248.0 0.018308 24340,1 - 31880,1 0 0.0 0,0 0.0 31380.2 - 39420. 2 10 4575072.0 18893.9 0.023434 39420.3 - 46960,3 18 4340886.5 11114,3 0.023480 46960,4 - 54500.4 53 4487377.4 18364,7 0.020357 54500.5 - 62 040. 5 86 3763210.7 19234.9 0.019730 62040.6 - 69580.5 115 3960950. 1 14877, 7 0.0193 75 69580,6 - 77120.6 37 4188021.8 14134.3 0.019642 77120.7 - 84660.6 5 2262163.0 10624.7 0.019413 34660.8 - 92200. 7 2 17698142.0 1003 2.0 0.018739 LOG SCALING DATA ANALYSIS FOR COMPANY Et NELSON DISTRICT X8AR = 59984.9 POUNDS(LBS), YBAR = 1191.9 CUBIC FE£T( Co FT) NUMBER OF CLASSES =15 CLASS WIDTH = 5026.73 LBS X-CELL LIMITS FR EQ XVARIANCE YVARIANCE RATI 0 ( LBS) (LOADS) (LBS**2) IC.FT**2) (C.FT/LB) 16800.0 - 21826.6 1 0. 0 0,0 0.01 72 02 21826.7 - 26853.4 1 0.0 0.0 0.019144 26353.5 - 31880. 1 0 0.0 0.0 0.0 31880.2 - 36906.8 7 2785682.3 11997.0 0.022822 36906, 9 - 41933.6 8 2597078.9 15520.1 0.025047 41933.7 - 46960.3 13 1759039. 9 12145.4 0.022367 46960.4 - 51987.0 29 2178443,3 20409.1 0,021178 51937. 1 - 57013. 7 50 2579755.7 18856.2 0.019510 57013.8 - 62040,5 60 1633262. 9 17416.6 0.019790 62040.6 - 67067,1 93 2027614.9 15089.8 0.019471 67067.3 - 72093.8 33 2183626.2 19466.3 0.019368 72093.9 - 77120.5 21 2489588. 8 8749.3 0.01 9476 77120. 6 - 82147.2 5 2262163.0 1062 4.7 0.019418 32147.3 - 87173.9 1 0. 0 0.0 0.018562 37174.0 - 92200.6 1 0.0 0.0 0.018905 APPENDIX XVI 177 . LOG S C A L I N G DATA A N A L Y S I S FOR COMPANY l-i KAMLOOPS D I S T R I C T XBAR = 6 5 4 1 9 . 3 POUNDSILB • S ) , YBAR = 1 3 9 0 . 2 CUB IC F E E T ( C F T ) NUMBER OF C L A S S E S = 5 C L A S S WIDTH = 2 0 5 3 0 . 2 0 LBS X - C E L L L I M I T S FREQ XVAR.I ANCE YVARIANCE RAT IO ( L 8 S ) ( LOADS) {LBS * * 2 ) ( C . F T * * 2 ) ( C . F T / L B ) 1 7 3 G 0 . 0 - 3 7 8 3 0 . 1 15 4 6 7 2 4 8 5 7 . 4 4 1 5 0 7 . 3 0 . 0 2 2 9 3 0 3 7 8 3 0 . 2 - 5 8 3 6 0 . 3 63 2 9 6 2 9 7 1 3 . 7 56 7 0 5 . 6 0 . 0 2 3 3 9 5 • 5 8 3 6 0 . 4 - 7 8 8 9 0 . 4 132 2 7 4 7 2 4 2 4 . 7 3 4 6 1 6 . 1 0 . 0 2 0 9 1 7 7 8 8 9 0 . 6 - 9 9 4 2 0 . 6 41 3 0 4 6 2 4 3 9 . 5 3 9 1 9 4 . 7 0 . 0 2 0 0 8 0 9 9 4 2 0 . 8 - 1 1 9 9 5 0 . 8 10 3 7 3 5 5 3 1 6 . 2 3 6 6 9 0 . 6 0 . 0 2 0 8 2 1 LOG S C A L I N G DATA A N A L Y S I S FOR COMPANY F , KAMLOOPS D I S T R I C T XBAR = 6 5 4 1 9 . 3 P O U N D S ( L B S ) , Y BAR = 1 3 9 0 . 2 C U B I C FEET ( C . F T ) NUMBER OF C L A S S E S =10 C L A S S WIDTH = 1 0 2 6 5 . 1 0 LBS X - C E L L L I M I T S FREQ XVARIANCE YVARIANCE RAT IO ( L B S ) ( L O A D S ) < L B S * * 2 ) I C . F T * * 2 ) ( C . F T / L B ) 1 7 3 0 0 . 0 - 2 7 5 6 5 . 0 9 7 6 2 9 1 7 0 . 6 3 0 5 7 8 . 7 0 . 0 2 4 3 7 6 2 7 5 6 5 . 1 - 3 7 3 3 0 . 1 6 6 5 2 2 5 7 2 . 3 3 6 6 0 8 . 7 0 . 0 2 1 5 4 5 3 7 8 3 0 . 2 - 4 8 0 9 5 . 2 17 4 0 4 1 1 7 7 . 5 2 7 4 2 4 . 9 0 . 0 2 6 4 4 3 4 8 0 9 5 . 3 - 5 8 3 6 0 . 3 4 6 9 2 1 3 0 4 0 . 3 6 7 2 7 5 . 9 0 . 0 2 2 4 8 7 5 8 3 6 0 . 4 - 6 8 6 2 5 . 3 87 7 1 7 3 3 3 3 . 1 3 5 0 0 2 . 9 0 . 0 2 1 3 6 5 6 8 6 2 5 . 4 - 7 8 8 9 0 . 4 45 7 9 2 9 3 6 6 . 5 2 5 9 2 4 . 3 0 . 0 2 0 1 6 4 7 8 3 9 0 . 5 — 8 9 1 5 5 . 4 27 8 7 7 0 6 8 8 . 0 4 0 8 2 3 . 5 0 . 0 2 0 7 2 3 8 9 1 5 5 . 6 - 9 9 4 2 0 . 5 14 1 1 0 4 7 7 1 2 . 0 3 8 1 7 4 . 9 0 . 0 1 8 9 5 6 9 9 4 2 0 . 6 - 1 0 9 6 3 5 . 6 7 1 0 4 6 0 2 8 1 . 9 2723 5 . 5 0 . 0 2 0 7 4 5 1 0 9 6 8 5 . 7 - 1 1 9 9 5 0 . 6 3 2 1 6 3 7 0 7 1 . 3 2 0 6 4 4 . 3 0 . 0 2 0 9 8 3 LOG S C A L I N G DATA A N A L Y S I S FOR COMPANY F i KAMLOOPS D I S T R I C T XBAR = 6 5 4 1 9 . 3 P O U N D S ( L B S ) , YBAR = 1 3 9 0 . 2 C U B I C F E E T ( C . F T ) NUMBER OF C L A S S E S =15 C L A S S WIDTH = 6 3 4 3 . 4 0 LBS X - C E L L L I M I T S FREQ X V A R I A N C E Y V A R I A N C E R A T I O ( L B S ) ( LOADS) ( L B S * * 2 ) < C . F T * * 2 ) ( C . F T / L B ) 1 7 3 0 0 . 0 — 2 4 1 4 3 . 3 8 7 4 0 3 4 5 4 . 2 3 4 0 3 7 . 7 0 . 0 2 5 2 1 2 2 4 1 4 3 . 4 — 3 0 9 8 6 . 7 2 1 7 9 9 9 8 4 0 . 0 6 0 . 5 0 . 0 1 6 7 5 6 3 0 9 3 6 . 8 - 3 7 8 3 0 . 1 5 4 2 2 9 1 6 8 , 0 2 3 7 8 2 . 7 0 . 0 2 2 6 3 9 3 7 8 3 0 . 2 - 4 4 6 7 3 . 5 13 2 2 9 0 5 0 9 . 3 1 6 5 7 3 . 6 0 . 0 2 6 4 1 0 4 4 6 7 3 . 6 - 5 1 5 1 6 . 9 15 3 4 3 7 0 7 5 . 4 4 4 1 1 3 . 7 0 . 0 2 2 4 1 7 5 1 5 1 7 . 0 - 5 8 3 6 0 . 3 35 3 4 4 3 3 6 0 . 9 6 7 2 6 3 . 3 0 . 0 2 2 8 9 9 5 8 3 6 0 . 4 - 6 5 2 0 3 . 7 60 2 9 3 9 1 1 1 . 7 3 3 7 9 7 . 4 0 . 0 2 1 5 5 3 6 5 2 0 3 . 8 - 7 2 0 4 7 . 1 45 4 1 1 9 8 0 8 . 3 2962 4 . 7 0 . 0 2 0 6 0 3 7 2 0 4 7 . 2 - 7 8 8 9 0 . 4 27 4 6 3 1 9 5 3 . 2 1 4 6 2 5 . 9 0 . 0 2 0 2 1 6 7 8 8 9 0 . 6 - 8 5 7 3 3 . 8 18 3 6 1 0 5 7 1 . 2 2 5 9 3 5 . 3 0 . 0 2 0 1 9 0 8 5 7 3 3 . 9 - 9 2 5 7 7 . 2 15 2 5 4 7 4 4 7 . 1 5 1 6 8 7 . 8 0 . 0 2 0 5 1 3 9 2 5 7 7 . 3 - 9 9 4 2 0 . 6 8 2 4 0 0 3 7 0 . 0 1 8 9 9 1 . 7 0 . 0 1 9 1 1 9 9 9 4 2 0 . 7 - 1 0 6 2 6 3 . 9 6 5 7 3 4 3 9 6 . 3 3 0 3 3 9 . 1 0 . 0 2 0 7 6 5 1 0 6 2 6 4 . 1 - 1 1 3 1 0 7 . 3 3 1 9 5 7 3 6 4 . 0 5 9 9 4 . 3 0 . 0 2 0 7 6 2 1 1 3 1 0 7 . 4 - 1 1 9 9 5 0 . 7 1 0 . 0 0 . 0 0 . 0 2 1 2 7 6 APPENDIX XVII L O G S C A L I N G D A T A A N A L Y S I S FOR COMPANY G» P R I N C E R U P 1 XBAR = 53242.1 P O U N D S * L B S ) , Y B A R = 1070,6 C U b I C F i NUMB cP OF C L A S S E S = 5 C L A S S W I D T H =10020.20 L B S X - C E L L L I M I T S ( L B S ) 30350.0 -^0380.2 -50410.4 -60440. 6 -70470.8 -40380. 1 50410.3 60440,5 70470. 6 80500.8 F R E Q ( L O A O S ) 2 46 364 1 9 1 X V A R I A N C E ( L S S * * 2 ) 3 845 6254.0 3361577a 8 7523059,3 6098531.6 11749346.0 LOG S C A L I N G DATA A N A L Y S I S FOR COMPANY Y V A R I A N C E ( C o f T* •*2) 1 7 2 9 3 . 0 1 6 3 x 9 , 5 1 2 4 3 3 . 5 1 2 9 0 5 . 3 2 6 3 7 0 . j G, P R I N C E R U P E 178. {INTERIOR. E T { C . F T ) R A T I O { C . F T / L B ) 0.019692 0.01 9307 0.018535 0.018025 0.016724 XBAR = NUMBER n c2 ? ? : L P ° U N D S ( L B S ) ' Y B A R = 1070.6 C U B I C OF C L A S S c S =10 C L A S S W I D T H = 5015.10 L3> ERT ( I N T E R I O R ) E E T ( C . F T ) X - C E L L L I M I T S ( L B S ) F REQ ( L O A D S ) X V A R I A N C E ( L B S * * 2 ) Y V A R I A N C E ( C . F T * * 2 ) R A T I O { C . F T / L B ) 30350.0 - 35365. 0 1 0. 0 0.0 90729.3 2179413.4 1713766,7 2070519.9 1684324. 5 1980179.2 1365644.3 12496384.0 0,0 35365,1 - 40380.1 I 4 42 117 247 148 43 6 2 0.019473 40330.2 -45395.3 -50410.4 -55425.5 -60440.6 -65455.7 -70470.3 -75485.8 -45395. 2 50410.3 55425,4 60440.5 65455.6 70470.6 75485.7 80500.8 0.0 33609.7 149x3.9 10373.4 11456,9 120x8.1 15568.1 7381,1 16562. 0 0.019862 0.020031 0.01 9243 0.018725 0.018454 0.018239 0.017344 0.016069 0.018526 LOG SCALING DATA ANALYSIS FOR COMPANY G, PRINCH RJPERT ( U T F R I O R ) - C E L L L I M I T S { L B S ) 30350.0 33693.4 37036.8 40380.2 43723.6 47067.0 50 410 53753. 8 57097.2 60440. 63 734. 0 67127 70470 73314.1 77157.5 4 -6 -, 4 , 8 33693.3 37036.7 40380. 1 43723,5 47066.9 50410.3 53753, 57097. 60440, 63783.9 67127.3 70470,6 73814.0 77157.4 80500.8 7 1 • 5 FREQ ( L O A D S ) 1 0 1 0 11 3 5 76 104 184 125 49 1 7 6 1 1 X V A R I A N C E ( L B S * * 2 ) 0. 0 0.0 0.0 o . o 900305.7 1219240. 0 859228.1 974261.4 937615. 8 730032.5 936021.3 1005678.5 1365644.3 0. 0 0. 0 Y V A R I A N C E (C.FT**2> 0.0 0.0 0.0 0.0 21095.7 15282.5 7 76 6.3 13853.1 9601.5 11447.2 15100.3 14811.1 7381.1 0.0 0.0 R A T I O ( C . F T / L B ) 0,019473 0.0 0.019862 0.0 0.020069 0.019084 0.018352 0.013275 0.0185 5 8 0.013311 0.017541 0.017461 0.016069 0.017934 0.019081 179. APPENDIX XVIII LOG SCALING DATA ANALYSIS FOR COMPANY H, V A N C O U V E R (COAST) XBAR = . 95297.7 POUNDS(LBS), YBAR = 1 7 4 3 . 3 CUBIC F E E T ( C . F T ) NUMBER OF CLASSES = 5 CLASS WIDTH =21160.20 LBS X-CELL LIMITS FREQ XVARIANCE YVARIANCE RATIO ( LBS ) (LOADS) (LBS**2 ) (C.FT**2) (C.FT/LB) 39700oO - 60850.1 12 45533172. 1 10452.6 0.019679 6 0 8 6 0 o 2 - 32020o3 41 36510589.7 3417 8.6 0.013994 82020.4 - .103180,4 195 31270878. 6 27314.9 0.013302 LG3180.6 - 124340.6 91 28754637.2 29365.1 0.017982 124340.3 - 145500.8 9 58126335.1 59524.3 0.018413 LOG SCALING DATA ANALYSIS FOR COMPANY H, COAST (VAN. ISLAND) XBAR = 95297.7 P0UNDS(L3S), Y BAR = 1743.8 CUBIC FE ETCC. FT) NUMBER OF CLASSES =10 CLASS WIDTH =10580.10 LBS X-CELL LIMITS FREQ XVARIANCE YVARIANCE RATIO ( LBS) 1 LOADS) (L6S**2) (C.FT**2) ( C . F T / L B ) 39700.0 - 50280.0 3 14763261.3 11162.3 0.021521 50280. 1 - 60860.1 9 13355829.3 56b3.5 0,019206 60860.2 - 71440.1 12 9647863. 2 2122.9 0.018689 71440.3 - 82020.2 2 9 11801289.9 30964.5 0.019102 82020.3 - 92600.3 78 8380591.6 16841.1 0.018496 926C0.4 - 103180.3 117 8702473.6 25530.5 0.018185 103180.4 - 113760.4 67 9659009.5 . 23477.3 0.013022 113760,5 - 124340.4 24 8579053.9 27202.4 0.017831 124340.6 - 134920.5 5 4975851.0 39931.7 0.013069 134920.6 - 145500.6 4 23135754.3 20513.0 0.018804 LOG SCALING DATA ANALYSIS FOR COMPANY H f COAST IVAN. ISLAND) XBAR = 95297.7 POUNDS(LBS), YBAR = 1743.8 CUBIC F E E T ( C . F T ) NUMBER OF CLASSES =15 CLASS WIDTH = 7053.40 LBS X-CEL L LIMITS FREQ XVARIANCE YVARIANCE RATIO (LBS) (LOADS) (LBS**2) (C.FT**2) (C.FT/LB) 39700. 0 — 46753.3 2 2644856.0 242.0 0.021151 46753.4 - 53 806.7 4 5095554.3 143 6.7 0.020322 53806.8 - 60360.1 6 5393591.5 5922.7 0.018959 60860. 2 - 67913.4 9 4378044.4 1359.7 0.013842 67913.6 - 74966.8 14 3166778. 7 49190.1 0.019710 74966.9 - 82020.2 18 4903261.3 183S7.9 0.018544 82020,3 - 39073.6 51 3266312.4 17432.3 0.013681 89073.7 - 96126.9 66 3018694.1 22233.2 0.018350 96127. 1 - 103180.3 78 3396948.1 25139.0 0.018048 L03180.4 - 110233.7 52 3573865.6 22195.1 0.013035 L10233.8 - 117287.1 30 3681036.8 21886.6 0.017983 L17237.2 - 124340.4 9 4736839.9 36130.0 0.017709 L24340.6 - 1313 93. 8 5 4975851.0 39931.7 0.018069 L3139.3.9 - 138447. 2 1 0.0 0.0 0.019223 133447. 3 - 145500.6 3 10201845.3 23565.3 0.018673 APPENDIX XIX 180. LOG SCALING DATA ANALYSIS FOR COMPANY I, PRINCE RUPERT (COAST) XBAR = 96942.2 POUNDS(LBS), YBAR = 1861 o0 CUBIC FEET(Co FT) NUMBER OF CLASSES = 5 CLASS WIDTH =36330.20 LBS X-CELL. LIMITS FREQ XVARIANCE YVARIANCE RAT 10 ( LBS) (LOADS) (LBS**2) (C.FT**2) (C.FT/LB) 21200oO - 53080. 1 68 113788435.4 56135.1 0.018911 58080.2 - 94960.3 107 127653884.0 88166.0 0.020252 94960.4 - 131840.4 151 116421269.2 82764.0 0.019486 131840.6 - 168720.6 69 33156154.2 148546o0 0.018043 168720.8 - 205600.8 3 55739733.3 79902.3 0.017677 LOG SCALING DATA ANALYSIS FOR COMPANY I, COAST (VAN. ISLAND) XBAR = 96942o2 POUNDS(LBS), YBAR = 1361.0 CUBIC FEET ( C FT) NUMBER OF CLASSES =10 CLASS WIDTH =18440.10 LBS X-CELL LIMITS FREQ XVARIANCE YVARIANCE RATIO (LBS) (LOADS) (LBS**2) iC.FT**2) (C.FT/LB) 21200.0 - 39640.0 33 22514395.3 12676.4 0.019019 39640.1 - 58030.1 3 5 28543680.3 33647*3 . 0.018846 58080.2 - 76520.1 67 18075257.7 29691.1 0.020030 76520.3 - 94960.2 40 25307543.0 4735 8.1 0.020532 94960.3 - 113400.3 75 30920037.2 62603.3 0.020060 113400.4 - 131840.3 76 22858230.9 7256 3.9 0.019006 131840.4 - 150280.4 46 24373 73 0.4 103340.6 0.018088 150230.5 - 168720.4 23 18096220.3 194134.8 0.017962 163720.6 - 187160.5 0 0. 0 0.0 0.0 L37160.6 - 205600.6 3 55739733.3 79902.3 0.017677 LOG SCALING DATA ANALYSIS FOR COMPANY I , COAST IVAN. ISLAND) XBAR = 96942.2 POUNDS(LBS), YBAR = 1861.0 CUBIC FE ET(C.FT) NUMBER OF CLASSES =15 CLASS WIDTH =12293.40 LBS X-CELL LIMITS FREQ XVARIANCE YVARIANCE RATIO ( LBS) (LOADS) (LSS**2) (C.FT**2) (C.FT/LB) 21200.0 - 33493.3 18 11826375.4 15680.7 0.021194 33493.4 - 45786.7 22 9490783.5 11753.1 0.016951 45736.8 - 53080.1 28 9 772486.3 26226.3 0.019223 58080.2 - 70373.4 56 9629019.0 2 7 6 0 1 . 7 0.020085 70373.6 - 82666.8 18 13123891.9 77863.7 0.020536 82666.9 - 94960.2 33 11772248.6 31357.4 0.020326 94960.3 - 107253.6 50 14193490.0 48343.9 0.020013 107253. 7 - 119546.9 47 14090182.5 71647.1 0.019694 119547.1 - 131840.3 54 11031889.3 65 9x7.9 0.018924 131840.4 - 144133.7 35 13235788.0 120003. 7 0.018373 144133.8 - 156427.1 25 16151381.3 13280 0.7 0.017373 156427.2 - 168720.4 9 7043384.0 155350.6 0.018631 168720.6 - 181013.8 0 0.0 0.0 0.0 131013.9 - 193307.2 1 0. 0 0.0 0.016943 193307.3 - 205600.6 2 34440248.0 53133.0 0.018024 181. APPENDIX XX LOG SCALING DATA ANALYSIS FOR COMPANY J, KAMLOOPS DISTRICT XBAR = 76415o9 POUNDS(LBS), YBAR = 1585,1 CUBIC FEE T{Co FT) NUMBER OF CLASSES CLASS WIDTH =15358.20.LBS X-C; LL LIMITS (LBS) FREQ ( LOADS ) XVARIANCE (LBS**2) YVARIANCE (C.FT**2) (C RATIO • FT/LB) 39660.0 55013. 2 70376.4 85734, 6 101092.8 55018. 1 70376,3 85734,4 101092,6 116450.8 26 11732 549.2 101 14330636.4 80 20671545.8 83 24061395.8 17 18990380.9 953 8,6 14748.5 20705,5 28083.3 16018.1 0.020393 0,020079 0,020349 0.021142 0.021153 LOG SCALING DATA ANALYSIS FOR COMPANY J , KAMLOOPS DISTRICT XBAR = 76415.9 POUNDSiLBS), YBAR = 1535.1 CUBIC FEET (C. FT) NUMBER OF CLASS ES =10 CLASS WIDTH = 7679,10 LBS X-CELL LIMITS FREQ XVARIANCE YVARIANCE RATIO ( LBS) (LOADS) (L3S**2) (C.FT**2) (C.FT/LB) 39660.0 - 47339.0 3 13894339.3 641.3 0.020794 4733 9.1 - 55018.1 23 4741130.1 3446.5 0.020349 55018.2 - 62697.2 64 3751255. 8 11405.2 0,019983 62697.3 - 70376.3 37 6403063.0 7376.1 0.020230 70376.4 - 73055.3 3 9 5319254. 5 15989.1 0,020814 73055.4 - 85734.4 41 4873960.6 11647.5 0.020879 8 57 34. 5 - 93413. 4 51 415 0803.4 15204.3 0.021041 93413.6 - 101092,5 37 4388152. 3 20192.2 0,021269 101092.6 - 108771.6 13 5237996.3 13984.4 0.021241 108771. 7 - 116450.6 4 7977466. 7 9803.0 0.020386 LOG SCALING DATA ANALYSIS FOR COMPANY J j KAMLOOPS DISTR ICT XBAR = 76415.9 POUNDSILBS). YBAR = 1585.1 CUBIC FEET(C.FT) NUMBER OF CLASSES =15 CLASS WIDTH = 5119.40 LBS X-CELL LIMITS FREQ XVARIANCE YVARIANCE RATIO ( LBS ) (LOADS) (LBS**2) (C.FT**2> (C.FT/LB) 39660.0 - 44779.3 2 12300512.0 162.0 0.021262 44779.4 - 49898, 7 7 772553.1 1141.3 0,019940 49893,8 - 55013. 1 17 2479438.1 8273.8 0.020483 55018.2 - 60137. 5 27 1607123.1 5055.4 0.020139 60137.6 - 65256.9 51 1472725, 8 13787.6 0.020032 65257,0 - 70376,3 23 1919982.5 6372.9 0.020114 70376.4 - 75495. 6 25 2110287.3 6099.5 0.020778 75495,8 - 80615.0 26 2058261,7 22 776.5 0.021047 80615.1 - 85734.4 29 2084225.1 10702.4 0.020737 85734.5 - 90853.3 39 1702980.1 14419.7 0.02L090 90853.9 - 95973,1 20 2999497.3 10829.3 0.020725 95973.3 - 101092. 5 29 2439094.8 13323.5 0.021476 101092.6 - 106211, 9 10 2097524.4 10215.8 0.021063 106212,0 - 111331.3 5 3562476.0 3708,5 0.021384 111331.4 - 116450. 6 2 10120064.0 16744.5 0.020985 182 . APPENDIX XXI LOG SCALING DATA ANALYSIS FOR COMPANY K , KAMLOOPS DISTRICT X3AR = 63680.0 POUNDS(LBS), Y3AR = 1271.0 CUBIC FEET(C.FT) NUMBER OF CLASSES = 5 CLASS WIDTH = 7936.20 LBS X-CELL LIMITS FREQ XVARIANCE YVARIANCE RATIO { LBS ) (LOADS) (LBS**2) (C.-FT**2) (C.FT/LB) 37840.0 - 45776.1 1 0.0 0.0 0.024524 45776.2 - 53712. 3 6 4966756.3 487 3.9 0.021126 53712.4 - 61648. 5 47 3671760.3 6963.6 0.019931 61648.6 - 69584.6 121 3443975.3 17360.3 0.019970 69584.8 - 77520.8 16 4539319. 5 596 9.2 0.019493 LOG SCALING DATA ANALYSIS FOR COMPANY K t KAMLOOPS DISTRICT XBAR . = 63630.0 POUNDS(LB'S) t YBAR = 12 71.0 CUBIC FEET(C.FT) NUMBER OF l CLASSES =10 CLASS WIDTH = 3968.10 LBS X-CELL LIMITS FREQ XVARIANCE YVARIANCE RATIO (LBS) (LOADS) (LBS#*2) (C.FT**2) (C.FT/LB) 37840.0 - 41808. 0 1 0. 0 0.0 0.024524 41308.1 - 45 776.1 0 0.0 0.0 0.0 45776.2 - 49744.2 2 1692672.0 112.5 0.023563 49744.3 - 53712.3 4 2112592.0 3618.9 0.019939 53712.4 - 57680.4 8 1714464.0 2366.7 0.020034 57630.5 - 61648. 5 39 1081516.9 697 7.7 0.019911 61648.6 - 65616.5 69 1189305.7 29012.4 0.020249 65616.6 - 69534.6 52 812189.5 3079.1 0.019619 69534.7 - 73552.6 14 1550193.1 2043.5 0.019418 73552.8 - 77520.7 2 4087224.0 23322.0 0.Q19989 LOG SCALING DATA ANALYSIS FOR COMPANY K, KAMLOOPS DISTRICT XBAR = 63680.0 POUNDS(LBS), YBAR = 1271.0 CUBIC FEET(C.FT) NUMBER OF CLASSES =15 CLASS WIDTH = 2645.40 LBS X-CELL LIMITS FREQ XVARIANCE YVARIANCE RATIO { LBS) (LOADS) (L8S**2) (C,FT**2) (C.FT/LB) 37340.0 - 4048 5.3 1 0. 0 0.0 0.024524 40485.4 - 43130.7 0 0.0 0.0 0.0 43130.8 - 45776.1 0 0. 0 0.0 0.0 45776.2 - 48421.5 1 0. 0 0.0 0.024190 43421.6 - 51066. 9 3 689597.3 693 2.3 0.020712 51067.0 - 53712.3 2 561784.0 1922.0 0.020346 53712.4 - 56357.7 4 441652.0 245 6.7 0.020315 56357. 8 - 59003.1 9 412681.8 2402.1 0.019643 59003.2 - 61648. 5 34 483476.8 7050.1 0.019962 61648.6 - 64293.9 44 595757.5 3805.0 0.020002 64294.0 - 66939.3 53 549656.1 34494.5 0.020037 66939.4 - 69534.6 24 363642.9 3313. 7 0.019775 69534. 3 - 72230.0 10 322058.7 2036.3 0.019472 72230.1 - 74875.4 5 744752.0 1127.5 0.019182 74375.5 - 77520.8 i Oo 0 0.0 0.021156 183. APPENDIX XXII LOG SCALING DATA ANALYSIS FOR COMPANY L, CARIBOO DISTRICT XBAR = 66015.3 POUNDS(LBS), YBAR = 1280.7 CUBIC FEE T(Co FT) NUMBER OF CLASSES = 5 CLASS WIDTH =16784.20 L3S X-CELL LIMITS ( LBS) 5980oO -22764.2 -39548.4 -56332.6 -73116. 3 -22764.1 39548.3 56332.5 73116.6 89900.8 FREQ ( LOADS) 3 5 15 181 45 XVARIANCE (LBS**2) 15322533.3 823024.0 10319974. 1 16756243.9 13571515.9 YVARIANCE (C.FT**2) 457 9.0 251.2 9373.5 11333.0 1057 9.3 RATIO (C.FT/LB) 0.018984 0.019100 0.019616 0.019463 0.019146 LOG SCALING DATA ANALYSIS FOR COMPANY L, CARIBOU DISTRICT XBAR = 66015.3 POUNOS(LBS), Y3AR = 1230.7 CUBIC FEET{C» FT) NUMBER OF CLASSES =10 CLASS WIDTH = 8392.10 LBS X-CELL LIMITS FREQ XVARIANCS YVARIANCE RATIO (LBS) (LOADS) (L8S**2) (C.FT**2) (C.FT/LB) 5980.0 — 14372.0 3 15322533.3 4579.0 0.013984 14372. 1 - 22764.1 0 0.0 0.0 0.0 22764. 2 - 31156.2 5 323024.0 251.2 0.019100 31156.3 - 39548.3 0 0.0 . 0.0 0.0 39543.4 - 47940.4 2 304120. 0 11704.5 0.019647 47940. 5 - 56332.5 13 6400932.0 7643.6 0.019612 56332.6 - 64724.6 62 5540817.7 7632.7 0.019367 64724.7 - 73116.6 119. 5325935.7 5884.7 0.019503 73116.3 - 81508.7 40 3749809.8 9304.3 0.019202 81508. 8 89900.8 5 7382508.0 1364.3 0.018749 LOG SCALING DATA ANALYSIS FOR COMPANY L, CARIBOU DISTRICT XBAR = 66015.3 POUNDS(LBS), YBAR = 1280.7 CUBIC FE ET(C.FT) NUMBER OF CLASSES =15 CLASS WIDTH = 5 5 94. 73 LBS X-CELL LIMITS FREQ XVARIANCE YVARIANCE RATIO . (LBS) (LOADS) (L8S**2) (C.FT**2) (C.FT/LB) 5930. 0 — 11574.6 2 180000.0 32.0 0.019586 11574.7 - 17169.4 1 0. 0 0.0 0.018405 17169.5 - 22764.1 0 0.0 0.0 0.0 22764. 2 - 28353.8 3 488893.3 380.3 0.019498 28353.9 - 33953.6 2 105656.0 242.0 0.018530 33953.7 - 39548.3 0 0.0 0.0 0.0 39548. 4 - 45143.0 C 0.0 0.0 0.0 45143.1 - 50737.7 5 3880960. 0 20864.7 0.020925 50737. 8 - 56332.5 10 4623829.3 5239.5 0.019026 56332.6 - 61927.2 27 2270934.3 5 76 5 . 7 0o019672 61927.3 - 67521.9 32 2847760.2 9251.3 0.019564 67522.0 73116.6 72 2720195.6 5306.1 0.019290 73116.7 - 78711.3 3 4 1718640.1 7229.3 0.019199 78711.4 - 84305.9 8 3615477.7 15170.3 0.019275 84306. 1 - 89900. 6 3 5141461.3 1261.0 0.018312 APPENDIX XXIII 184. LOG SCALING DATA ANALYSIS FOR COMPANY M» PRINCE GEORGE DISTRICT XBAR = 92269.2 POUNDS(LBS), YBAR = 1852o8 CUBIC FEET(C» FT) NUMBER OF CLASSES = 5 CLASS WIDTH =30990.20 LBS X-CELL LIMITS (LBS) 21500. 0 52490.2 83480.4 114470.6 145460.3 52490. 1 83480.3 114470.4 145460. 6 176450.8 FREQ (LOADS) 17 123 32 36 30 XVARIANC E (LSS**2) 43295516.8 3 3924925.2 134411197.7 63418213.9 83862817. 5 YVARIANCE (C.FT*=*2) 29019.0 18190.1 5992 8.3 32923.3 61070.1 RAT 10 (Co FT/LB) 0.019760 0.020162 0.020211 0.020016 0.01 9939 LOG SCALING DATA ANALYSIS FOR COMPANY M, PRINCE GEORGE DISTRICT XBAR = 92269.2 POUNDS(LBS), YBAR = 1852.8 CUBIC FEET(C.FT) NUMBER OF CLASSES =10 CLASS WIDTH =15495.10 LBS X-CELL LIMITS FREQ XVARIANCE YVARIANCE RATIO ( LBS) (LOADS) (L8S**2) (C.FT**2) (C.FT/LB) 21500.0 - 36995.0 16 6987768.3 2471.1 0.019344 36995.1 - 52490.1 1 0. 0 0.0 0.023117 52490.2 - 67985.1 47 5142592.3 860 7.6 0,020243 67935.3 - 33480.2 76 17268911.4 11409,9 0.020118 83480.3 - 98975.3 13 27918271.8 10356.3 0.020347 98975.4 - 114470.3 19 23423721.4 25780.0 0.020136 114470.4 - 129965.4 12 19779886.5 19817.3 0.019989 129965.5 - 145460.4 24 16167553.7 11243,5 0.020028 145460.6 - 160955. 5 16 20593758. 1 21931.3 0.019800 160955.6 - 176450.6 14 18316646.7 35350.6 0.020083 LOG SCALING DATA ANALYSIS FOR COMPANY -M, PRINCE GEORGE DISTRICT XBAR = 92269.2 POUNDS(LBS)t YBAR = 1852.8 CUBIC FEET(C.FT) NUMBER OF CLASSES =15 CLASS WIDTH =10330,07 LBS X-CELL LIMITS FREQ XVARIANCE YVARIANCE RATIO ( LBS) (LOADS) (LBS**2) (C,FT**2) (C.FT/LB) 21500.0 - 31830.0 15 33673 93. 8 1644,4 0.019418 31330.1 - 42160.0 1 0.0 0.0 0.018506 42160.1 - 52490. 1 1 0. 0 0.0 0.023117 52490.2 - 62820.2 14 655084.5 345 8.2 0.020133 62820.3 - 73150,2 72 7909925.7 9477.2 0.020269 73150.3 - 83480.3 37 7986067,2 10091.7 0.019972 83430.4 - 93 810.3 10 6250183.8 10800.4 0.020636 93810.4 - 104140.4 8 11380031.9 21005.1 0.020135 104140.5 - 114470.4 14 8164923.0 26433.4 0.020017 114470.6 - 124800.5 6 14508338.7 3979.8 0.019895 124800.6 - 13 5130.6 14 13508861.8 21322.7 0.020314 135130.7 - 145460.6 16 7487999. 7 12382.4 0.019812 145460.8 - 155 790.7 10 8210306.7 15433,1 0.019832 155790.8 - 166120.8 9 10736801.8 25792.2 0.019309 166120.9 - 176450.8 11 12306470.4 34291.4 0.020125 APPENDIX XXIV 185. ' LOG SCALING DATA ANALYS IS FOR COMPANIES A, D » F, J , K t L »M — INTERIOR XBAR = 71539x,3 POUNDS(LBS), YBAR = 143 7.4 CUBIC FE ET ( C.FT ) NUMBER OF CL ASSES = 5 CLASS WIDTH 34094.20 LBS X-C EL L LIMITS FREQ XVARIANCE YVARIANCE RATIO (LBS) ( LOADS) (L3S**2) (C.FT**2) (C.FT/LB) 59 30o 0 - 40074.1 42 5339537lo4 36531.4 0.020972 40074. 2 - 74168.3 1199 41435414.5 24381.9 0.020003 74163. 4 - 103262.4 407 74602967. 5 63 85 9. 8 0.020228 108262. 6 - 142356.6 53 119733023.2 52070.5 0.020120 142356.3 - 176450.8 33 98371941.0 61862ol 0.019965 LOG SCALING DATA ANALYS IS FOR COMPANIES A, D t F t J , K , L t M — INTERIOR X3AR = 71539.3 POUNDS (LBS) , YBAR = 14 3 7.4 CUBIC FE ET(C.FT ) NUMBER OF CLASSES =10 CLASS WIDTH = 17047. 10 LBS X-C£LL LIMITS FREQ . XVARIANCE YVARIANCE RATIO (LBS ) (LOADS) (L3S**2) (C. FT**2) (C.FT/LB) 5930. 0 - 2 3 02 7. C 8 35652859.4 15313.2 0.019221 23027.1 - 40074.1 34 25627835. 0 2 3774.6 0.021197 40074.2 - 57121.2 153 18342639.7 32347.3 0.021420 57121.3 74163.3 1046 18582535.9 17951.5 0.019341 74168.4 - 91215.3 3 23 22125825. 1 33025.0 0.020043 91215. 4 - 108262.4 84 16955338.0 30035.0 0.020309 108262.5 - 125309.4 27 19840408. 6 2495 4.5 0.020303 125309.6 - 142356.5 26 23166560.5 15590.4 0.019958 142 3 56. 6 - 159403.6 16 20390246.6 13700.7 0.019324 159403.7 - 176450.6 17 27041390.2 34911.4 0.02 0033 LOG SCALING DATA ANALYSIS FOR COMPANIES A , D » F t J , K t L» M — INTERIOR XBAR = 71589.3 PQUNDS(LBS), YBAR = 1437.4 CUBIC FE E T C C . F T ) NUMBER OF CLASSES =15 CLASS WIDTH •= 11364.73 LBS X-C£LL LIMITS FREQ XVARIANCE YVARIANCE RATIO ( LBS) (LOADS) (LBS**2) (C.FT**2) (C.FT / L B ) 5980. 0 - 17344.6 4 22071439. 0 377 6.3 0.019441 17344. 7 - 28709.4 27 5721903. 2 9623.9 0.020338 28709.5 - 40074.1 11 11031678.8 27632.2 0.021373 40074.2 - 51438.8 68 8380463.5 22391.2 •0.021943 51433.9 - 62803.6 381 8324511.9 19923.1 0.020382 62803. 7 - 74163.3 750 9563453.1 - 15480.2 0.019713 7416 3. 4 - 85532.9 257 9497321.7 21305.6 0.019841 35533.1 - 96897.6 96 11234800.0 22755. 5 0.02066S 96 89 7. 3 - 103262.3 54 8133237. 5 26413.4 0.020969 108262.4 - 119627.0 23 6936214. 5 13149. 3 0.020243 11962 7. 1 - 130991.7 11 7-338566.0 16959.3 0.020275 130991.8 - 142356.4 19 7201559. 4 6215.3 0.019914 142356.5 - 153721.1 10 8298067.2 1367 7. 2 0.019969 153721. 2 - 165C85. 3 12 14135994.6 2139 2.9 0.019302 165035. 9 - 1 76450.4 11 1 1187700. 3 31174.0 0.020125 APPENDIX XXIV (cont'd.) 186. LOG SCALING DATA ANALYSIS FOR COMPANIES 'A ,D ,F ,J.K,L.H- — INTERIOR XBAR = 71539,3 POUNDS(LBS) , YBAR = 1437,4 CUBIC FEST(Co FTJ NUMBER OF CLASSES =20 CLASS WIDTH = 8523,55 LBS X-CELL LIMITS FREQ XVARIANCE YVARIANCE RATIO ( L BS ) (LOADS) {LQS'**2) (C.FT-*2) (C.FT/LB) 593G, 0 - 14 503,4 3 10215022,2 305 2.7 0,018934 14503,5 - 23027.0 5 3393257. 6 4593o4 0.019282 23027, I - 31550.5 25 3992165,0 6799,0 0.020699 31550,6 - 40074.1 9 6175073.8 1776 1.1 0.022131 40074,2 - 43597.6 3 5 4037077.3 28052.9 0.023493 43597, 7 - 57121,2 11 8 6737518,5 32331.3 0.020900 57121,3 - 65644.7 506 5024414.1 17724.1 0.020035 65644o 8 - 74168.2 540 5735376.7 11735.5 0.019637 74168. 3 - 82691.7 215 5205158.6 17085,3 0,019726 82691.8 - 91215.2 103 5993377. 7 24052.0 0.020612 91215.3 - 99738.7 49 5911461.2 13926o6 0.020631 99738, 3 - 108262. 2 35 5308559.3 26168.3 0,0 21043 108262.3 - 116735.7 20 3859440.0 19239.2 0o020275 116785, 3 - 125309.2 7 9636009.1 1433 4.9 0,020373 125309.3 - 133332.7 9 64643 50.6 25933.7 0.020263 133332.8 - 142356.2 17 5302203.2 5593 a2 0.019304 142356. 3 - 150879.7 9 56646 23.4 10425.1 0.019903 150879.8 - 159403.2 7 2366923.4 778 7.9 0.019723 159403.3 - 167926,7 9 8145079.1 1735 9.1 0.019924 167926,8 - 176450.2 8 6443527.0 25099*9 0.020253 LOG SCALING DATA ANALYSIS FOR COMPANIES A,D,F,J,K,L,M —INTERIOR XBAR = 71589.3 POUNDS(LBS)t Y BAR = 1437.4 CUBIC FEETlC.FT) NUMBER OF CLASSES =25 CLASS WIDTH = 6313.34 LBS X-CEL L LIMITS FREQ XVARIANCE YVARIANCE RATIO (LBS) {LOADS) (L3S**2) (C.FT**2i (C.FT/LB ) 5930.0 12793.7 90000. 0 I 6. 0 0.0195 86 12793. 3 - 19617.6 3 4620628.9 1954.9 0.018448 19617,7 - 26436.4 19 2243393.9- 11013.7 0.021645 26436.5 - 33 255.3 11 4072526.8 lo43.4 0.018778 33255. 4 - 4G074.1 7 2217768.5 9259.3 0.022989 40074.2 - 46892.9 24 2103644. 0 29318.5 0.024378 46893.0 — 53711.3 67 3503507.8 29049.4 0.020919 53711.9 - 6 0 53 0. 6 188 357 2948.6 13395.1 0.020445 60530,7 — 67349.4 520 2923634. 4 16129.5 0.019981 67349. 5 - 74168.2 400 3786521.2 1155 6.4 0.019585 74163,3 - 80987.0 189 3630533.7 15149.5 0.019679 30937.1 - 87805.8 99 4284475.9 24580.4 0.020507 87805, 9 - 94624. 6 49 3505795.2 22S00.9 0.020520 94624.8 - 101443,4 49 3879616. 3 2773 8.6 0.020785 101443.6 - 108262,3 21 3321850.3 10399.4 0.021021 103262. 4 - 1 15031.1 18 2779 896,3 13337.3 0.020252 115081.2 - 121899.9 6 2320575.6 11424.9 0.020391 121900.0 - 128713.7 8 1732453.8 13354.7 0.020020 128713,8 - 135 537.5 9 3371740,4 720 3.4 0,02 0402 135537.6 - 142356.3 12 2428020.9 625 9. 5 0.019702 142356. 4 - 149175. 1 7 4922422,5 3673o6 0,020217 149175.3 - 15 5993.9 6 4850313. 2 1899 9.5 0.019597 15 59 94. 1 - 162812,8 6 3148058.6 19994,1 0,019750 162312,9 - 169631.6 7 2890456.5 15185.0 0,019794 1696i1„7 — 176450,4 7 4167397.6 2033 7.4 0,020360 A P P E N D I X X X V B . C . F O R E S T S E R V I C E A C T , P A R T V I I I P A R T V I I I T I M B E R - S C A L I N G 4. In this Part, unless the context otherwise, requires, " licensed scaler " includes any person licensed as a scaler or ap-pointed as an acting-sealer under this Part; " Official Scaler " includes any person appointed as an Official Scaler or Acting Official Scaler under this Part; "scale ," "scaled," and "-scaling " refer.. to.„thc_raeasurcnient of ^irnber..by_.a..liccnsed-.or_official scaler in accordance with lhe_ provisions of th i sAct j ind the regulations. R .S . 1948, c. 128, sT 6 1 . " !3/i:/7-t 188 1960 FOREST CHAP. 153 Duty of lumbermen lo have tim-ber scaled by licensed scaler. Appointment of acting-sealer. Power to designate scaler east of Cascade Mountains. Sawing of unsealed timber prohibited. Export of unsealed timber or timber on which royalty is unpaid prohibited. Penalty. Seizure. 65. (1) Where timber is cut in any operation on land within the jurisdiction of the Legislature, the timber shall be scaled forthwith by a licensed scaler, who shall be employed and paid by the person carrying on the operation, unless otherwise provided by this Act; and upon failure of that person to comply with the provisions of this subsection, the Deputy Minister may designate and direct a licensed scaler to scale the timber; and the cost of such scaling is payable to the Crown by that person. (2) Where it is shown to the satisfaction of the Deputy Minister that the services of a person licensed as a scaler under this Part cannot be obtained by the personln charge of The operation?, the Deputy Minis-ter may appoint an unlicensed person as an acting-sealer to perform the duties of a licensed scaler temporarily until a person licenseb*~as-a-scaler under thiFI*art is available. (3) Notwithstanding anything in this section contained, the Deputy Minister may designate and direct a licensed scaler to scale the timber cut on land within the jurisdiction of the Legislature situate east of the Cascade Mountains, and the costs of such scaling are recoverable by the Crown from the person carrying on the operation at the discretion of the Minister. R.S. 1948, c. 128, s. 62; 1974, c. 36, s. 17 (proc. eff. Oct. 25, 1974). 66. No person shall saw or otherwise manufacture any timber until it has been scaled, and every person contravening this section is guilty of an offence and liable, on summary conviction, to a penalty not exceed-ing five hundred dollars, and, in addition to the penalty so imposed, to have all timber in respect of which the offence was committed and lumber and other manufactured wood products produced from the timber seized and forfeited to the Crown, either wholly or in part, as the Minister may direct. R.S. 1948, c. 128, s. 63; 1953 (2nd Sess.), c. 9, s. 5. 6 7 . (1) No person shall export or remove from the Province any timber or lumber in respect of which any royalty, tax, or revenue is payable to Her Majesty in right of the Province, unless a permit is obtained from an officer of the Forest Service certifying that the timber has been scaled, and all royalty, taxes, and revenue so payable in respect thereof have been paid. (2) Every contravention of this section renders the offender liable to forfeit and pay to Her Majesty the sum of one thousand dollars, to be recovered, with all costs as between solicitor and client, in an action brought in the name of Her Majesty in any Court of competent juris-diction. (3) (a) The Minister, or any person authorized by him, may do all things necessary to prevent a contravention of this section and to secure compliance therewith, and may for such purpose take, seize, and hold all timber which is, or is suspected to be, in course of transit out of the 1503 13/12/74 CHAP. 153 FOREST 9 ELIZ. 2 Province in contravention of this section, and may also take, seize, and hold every vehicle that is transporting such timber or every boat that is towing any such timber. (b) If the Minister decides that it is not the intention of the holder, owner, or person in possession of the timber to use it in the Province, or to manufacture it or cause it to be manufactured into sawn lumber or other manufactured wood product in the Province, or to dispose of the timber to others who will use the same in the Province, or have the same so manufactured in the Province, the Minister may sell or cause to be sold such timber and vehicle or boat by public auction, and the proceeds of the sale shall be the property of Her Majesty, and shall form part of the Consolidated Revenue Fund. (c) In case the boat escapes after having been so seized, or in case its seizure is avoided by the removal of the boat outside the waters of the Province, it may at any time afterwards be seized, if found in any of the waters of the Province, and sold as above provided. R.S. 1948, c. 128, s. 64; 1950, c. 22, s. 13; 1960, c. 16, 8. 5. Timber-scaling. 68. (1) Timber shall be scaled in accordance with the regulations in cubic feet or such other measure determined by the Minister. (2) The British Columbia Cubic Scale as established from time to time by the Minister shall be the official scale and shall be used and applied in the scaling of timber. (3) The Minister may, from time to time, fix or determine the basis of cubic measure according to the British Columbia Cubic Scale and that basis shall be applicable to any other measurement pertaining to the scaling of timber and referred to in this Act or in any statutory provision, document, or instrument. 1972, c. 22, s. 9. Examination of applicants for scaler's licence. 6 9 . (1) The Lieutenant-Governor in Council shall appoint a Board of Examiners, one or more members of which shall examine and test the ability and knowledge of every person applying to be licensed as a scaler under this Part. (2) Each candidate for examination shall make application to the Forest Service in the manner prescribed by the regulations. (3) Examinations shall be held at such times and places and shall be conducted in such manner and on such subjects as are prescribed by the regulations. R.S. 1948, c. 128, s. 66; 1974, c. 36, s. 18 (proc. eft. Oct. 25, 1974). Issuance of scalers' licences. 1504 7 0 . The Board of Examiners shall issue to each candidate who passes the prescribed examinations in a satisfactory manner, and who is judged by the Board to be trustworthy and of good character, a licence entitling him to act as a scaler of timber under this Part; but no_person shall be licensed as.a scaler unless he is_ajCanadian citizen residing in the P r o v ^ i n c e T R . S . 1948, c. 128, s. 67; 1974, c. 367s. 19 (proc. eff. Oct. 25, 1974)7 13/12/74 1 1960 FORESl CHAP. 153 Oath ol office. Duties of licensed scalert. Arbitration in case of disputes. Com. Official'! salary. 13/12/74 71. Every licensed scaler, before entering on his duties, shall make and file in the Forest Service an affidavit in the form following:— 1, A.B., do solemnly swear that I will, while acting as scaler, without fear, favour, or affection, and to the best of my ability and judgment, classify correctly, scale and measure, according to law, all timber and products of the forest which I may be employed to scale or measure, and that I will make true returns of the same to the District Forester or any other officer of the Forest Service as required. Sworn before me at the of , in the Province of | British Columbia, this day of .19 . I A B -C. D., J.P R.S. 1948, c. 128, s. 68; 1974, c. 36, s. 20 (proc. eff. Oct. 25, 1974). 72. (1) lt is the duty of licensed scalers to measure fairly and cor-rectly, to the best of their skill, knowledge, and ability, and to classify all timber cut on land within the jurisdiction of the Legislature, and to enter in a record, in a form approved by the Forest Service, what they believe to be the proper contents and classifications of the timber. (2) After measuring and classifying any timber, each licensed scaler, at such times as the Forest Service requires, shall (a) transmit to the Forest Service a correct copy of the record of its measurement and classification as entered in his record; (Z>) when called upon to do so, submit his record to any duly authorized officer of the Forest Service; (c) give all information asked for in his power to give; and (d) shall furnish any statements or copies of statements which the Forest Service may from time to time require. R.S. 1948, c. 128, s. 69; 1974, c. 36, s. 21 (proc. eff. Oct. 25, 1974). 73. (1) Upon request of any party to the dispute, any pjs.trictBEor==» ester, Super-visor~of-Scalers. oi othex-duly authorizedjjfficer of the Forest Service shall act as arbitrator in anyjdisp.u,tejthal^ or a purchaser and a licensed scaler as to thcmeasjirement-or-classifica^ j i o n Qf_any_timbj^~ang3nsTaw^  jvithout appeal. (2) Where a District Forester, Supervisor of Scalers, or other officer acts as arbitrator under this section, he shall collect from_ih§:_person requesting his services the costs of measurement and classification, and, shall transmit the amount collected to the Forest Service; but where he finds the disputed measurement or classification made by the licensed scaler to be inaccurate, no charge shall be collected. (3) Where the salary of the official who so acted_as__arbitrator is payable from th^ScalmgJFund. thejuriounJLreceived by the ForesJLServ^ ice under~triis~section shall be, paid into the Scaling Fund, but^therwise-^ shall be accounted for as Provincial revenue. R.S. 1948, c. 128, s. 70. 1505 CHAP. 1 5 3 FOREST 9 ELIZ. 2 Suspension and cancellation of licences. 7 4 . (1) The Chief Forester may suspend the licence of any licensed scaler who is not perfonning properly the duties of his office or who is not complying with the regulations. (2) The Minister, on the recommendation of the Chief Forester, may cancel the licence of a licensed scaler. R.S. 1948, c. 128, s. 71; 1962, c. 24, s. 7. Scale by Offi-cial Scaler as basis of sales of timber ln certain districts. Exception. 7 5 . (1) In the district comprising that part of the Province situate west of the Cascade Mountains, and in such other districts of the Prov-ince as the Lieutenant-Governor in Council, by Proclamation, may from time to time prescribe, it shall be a term of every sale or other dealing with timber that the sale or other dealing shall be based on the scale of the timber made under this Act by an Official Scaler, and any person who buys, sells, or otherwise deals with timber contrary to the provisions of this section is liable, on summary conviction, to a penalty not exceeding five hundred dollars. (2) This section does not apply to (a) timber which has previously been scaled under this Act by an Official Scaler, unless the timber is lumber reject logs or logs from pulp leases or licences on which additional charges are due to the Crown, or unless such logs are in the possession of or have been salvaged by a log salvage permittee or beach-comber; (6) timber cut on landsl^other than CrownJandsx which carrying reseryations-of .royalty or~otrief"liKi^harges in favour_.of_the_ Crown, if the Forest Service has exempted such_timber_from, the provisjons_pXsubssfition.Cl)• R.S. 1948, c. 128, s. 72; 1956, c. 20, s. 3. Official seat-ing service. Appointment of Official Scs'ers. 7 6 . (1) The Forest Service shall maintain a sufficient staff of Official Scalers for each district, with such equipment as in the opinion of the Deputy Minister is necessary. (2) O^fficiatScalers shall be appointed pursuant to the provisions of the Public Service A ct, and shall hold office during pleasure. A l l Official Scalers shall for all purposes be deemed members of the Public Service within the meaning of the Public Service Act, notwithstanding the fact that their salaries may be paid from t h e i J c a h ^ J F j n J ^ g f ^ e ^ s t r i c t - a n d not by way of yearly salary voted by the Legislative Assembly. R.S. 1948, c. 128, s. 73. Acting Official Scalers. 7 7 . N o person shall be appointed or act as an Official Scaler unless he holds a licence as a scaler under this Part, but where the Deputy Minister considers that, on account of the location of the timber to be scaled, the cost of sending an Official Scaler to carry out the provisions of this Act would be excessive, he may appoint an unlicensed person as an Acting Official Scaler to perform the duties of an Official Scaler, and the appoint-1506 f\PPeTOk»>< XXV C< r^>rV©l) 192 1960 F O R E S T CHAP. 153 Scaling Fund. Scaling fees. Advances to Scaling Fund. Delivery of copy of scale. Rescale. Crown lien for fees. n/i?/74 ment may be for a specified time, or for specified work, or until ine appointment is cancelled by the Deputy Minister. R.S. 1948, c. 128, s. 74. 7 S . (1) There shall be established and continue to be maintained in the Provincial Treasury a fund for the providing of adequate service for the scaling of timber in each district constituted by or under this Part, io be known as the " Scaling Fund " of the district, into which shall be paid all moneys received under this A c t for the scaling of timber in the district, whether for fees or expenses. AJLjsahurjejjjejcj^ incurred in mamtammg_the.scaling.service in.each.districLarc-chargeable to the Scafingjjundj3i_the djstrict,.and_aieJpayable_directry from that fund upon vouchers_approved-by the_Muiister,„ or m a y b e j p j i d jn_the__ first instance from any moneys appropriated by the Legislative Assembly for that purpose, and subsequently charged to the Scaling Fund. (2) The cost of maintaining the scaling service in each district shall be estimated by the Forest Service, and the Minister may from time to time make rules prescribing the fees and expenses to be charged by the Forest Service for the scaling of timber; and the Minister shall fix such fees as in his opinion will be sufficient to meet the actual cost of main-taining the scaling service. (3) If at any time the amount at the credit of the Scaling Fund is insufficient to meet the salaries and expenses chargeable thereto, the Minister of Finance shaUjidvance to the Scaling Fund out of the C o n -solidated Revenue Fund.sjich sums as may_be.necessary. A l l moneys so advanced shall be repaid to the Consolidated Revenue Fund out of moneys collected by the Forest Service under this section. R.S. 1948, c. 128, s. 75; 1949, c. 24, s. 4. 7 9 . Upon demand of the vendor or owner of any timber scaled by an Official Scaler, and upon payment of the fees and charges payable therefor, the Forest Service shall deliver to the vendor or owner a certi-fied copy of the scale of the timber. R.S. 1948, c. 128, s. 76. S O . In the event of a vendor or purchaser objecting to any Official Scaler, or to his scaling, then, on application to the District Forester or Supervisor of Scalers, another Official Scaler may be selected to scale the timber in question; and the person requiring such services shall pay the additional fees and expenses occasioned by the substitution of another Official Scaler; but where an original scale made by an Official Scaler is, in the opinion of the District Forester or Supervisor of Scalers, proved inaccurate, no charge other than the correct fees and expenses for scaling the timber shall be imposed. R.S. 1948, c. 128, s. 77. 8 1 . (1) The Crown has a lien for the amount of all costs, fees, and expenses chargeable by the Forest Service or payable to the Crown under this Part, and for all expenses of seizure, detention, or sale incurred in enforcing the lien, 1507 CHAP. 153 F O R E S T 9 ELIZ. 2 (a) upon the timber in respect of which the costs, fees, or expenses are chargeable or payable; and (b) upon all sawmills or other factories, and lands^ appurtenant thereto, in or for which the timber has been or is being manu-factured, used, or consumed; and (c) upon machinery, equipment, and material that were used in the logging operation, or that belong to the person from whom the costs, fees, and expenses are due; the lien to constitute a charge to the like extent, and to confer the same rights, and to be registrable and enforceable in the same manner as the liens created by the provisions of this Act for the recovery of royalty, including an absolute, unconditional power to sell. fbiVbyCa0ctioa. Such costs, fees, and expenses are also a debt due to the Crown recoverable by action in any Court of competent jurisdiction from the person against whom they are chargeable or by whom they are payable. R.S. 1948, c. 128, s. 78. ""proof * n a n v prosecution or action brought against any person for any contravention of this Act, the burden of proving that the requirements and provisions of this Part have been complied with is upon the defendant. R.S. 1948, c. 128, s. 79. P A R T IX TIMBER-MARKING S 3 . [Repealed. 1966, c. 18, s. 11.] b«mea f to i m " Subject to subsection (3), every person engaged in the nwregfi'tCTed D u s u i e s s of cutting and removing timber shall, before cutting any timber, timber marks, apply for and obtain from the Forest Service such number of registered timber marks as the Deputy Minister considers necessary to distinguish clearly the classes of timber subject to different conditions of tenure, royalty, stumpage, tax, or manufacture which the person proposes to cut. (2) Subject to subsection (5), the holder of a registered timber mark prescribed under this section for the marking of any timber which is cut by him shall, before removing the timber so cut from the land on which it is cut, put that timber mark in a conspicuous place on each piece of the timber. (3) The Deputy Minister may grant such exemptions from the provisions of subsection (1) as he considers advisable. (4) Subject to subsection ( 5 ) , no person shall float or raft any timber on the salt or fresh waters of the Province unless each piece of the timber bears the registered timber mark in a conspicuous place so that it is readily discernible when the piece of timber is floated. 1508 13/12/74 A XJ- X . I J i.-t LJ J - IX J T W Y V A. J 94 BRITISH. COLUMBIA SCALING REGULATIONS B . C . Reg. 11 /59 . . P R O V I N C E O F D R 1 T I S H C O L U M B I A F O R E S T A C T REGULATION M A D E AND APPROVED BY ORDER IN COUNCIL N O . 2867 ON D E C E M B E R 1.9, 1958, AND AS A M E N D E D BY B . C . R E G . 6 4 / 6 4 S C A L I N G R E G U L A T I O N Division (1).—General 1.01 A l l timber for which a royalty rate is designated in cubic feet under Part VII of the Forest Act shall be scaled in cubic feet according to the British Columbia Cubic Scale whenever the person for whom the scale is made fails or does not elect or is otherwise not entitled to have the scale made in board-foot or other measure. 1.02 A l ! forest products on which royalty has been reserved or on which Crown revenues are payable shall be scaled, measured, or counted forthwith after such forest product has been cut subject, to section 10 of this regulation, and returns thereof shall be made on the forms furnished or approved by the Forest Service for that purpose. 1.03 Tine Chief Forester may specify the mechanics and course of procedure and methods for scaling, measuring, or counting all forest products. 1.04 Where standards have been set up, aii scaling-sticks or other devices used for scaling or measuring timber, wood, or bark or forest products, to comply with the provisions of the Forest Act, shall be of standards approved by the Chief Forester. Division (2).—Scaling Procedure 2.01 A licensed scaler, in making a scale as required by the Forest Act, shall make only such deductions from the scale of any t.imber for defects as may be authorized by the Chief Forester. 2.02 The Chief Forester may specify the merchantable length or the maxi-mum or minimum sizes, or otherwise limit the dimensions, of logs or timber for the purpose of scaling same. 2.03 A licensed scaler shall scale each piece of timber or unit, or parcel of wood, individually, and forthwith record the scale thereof in the record of scale furnished by the Forest Service and kept for that purpose as designated by the Chief Forester. The scaler shall mark each, piece of timber or unit, or parcel of wood, at the time of making sucli scale, in a manner to indicate that a scale has been made. 2.04 'Hie Chief Forester may require any owner of timber or person carrying on a lumbering operation to a (lord io a Supervisor of Scalers or any Forest Officer designated by the Chief Forester provisions for proper arrangements and facilities deemed necessary by the Chief Forester to enable such Supervisor of Scalers or Forest Officer to investigate into the scaling or procedure of scaling on any lumbering operation. APPENDIX ./XXVI ( Coi>±c£) Division (3).—Rescale 3.01 Whenever the Chief Forester has reason to believe that a scale sub-mitted is incorrect or does not represent the true scale of the contents of any timber, he may have the scale checked, and in the event the first scale is deemed to be incorrect or not a true scale of the contents of such timber, then he may determine the cost of such check scale and the responsibility therefor, and all charges for royalty and (or) stumpage shall be based on such check scale. 3.02 Wherever a rescale of timber is required by a person in respect of timber previously scaled, the Chief Forester may determine the conditions under which such rescale shall be made. Division (4).—Acting-sealer 4.01 Where an unlicensed person is appointed as an acting-sealer, that ap-pointment is subject to such terms and conditions as the Deputy Minister of Forests may prescribe. The fee for such appointment is the amount as approved by the Deputy Minister of Forests. Division (5).—Scale by Purchaser of Crown Timber 5.01 Timber cut from Crown lands shall not be scaled by the purchaser thereof for the purpose of determining Crown revenue, but shall be scaled by a licensed scaler or an acting-sealer other than the purchaser; provided, however, the District Forester of the district where such timber is situate may grant an exemption from this requirement. Division (6).—Examination of Candidates for a Scaling Licence 6.01 A candidate for an examination for a licence to scale makes the appli-cation therefor in writing to the Chief Forester, or the District Forester, or the Superintendent of Scaling, or the Inspector of Scalers in the district in which the candidate wishes to scale. Such candidate shall be 18 years of age or over and a British subject at the time the licence is issued and a resident of British Columbia. 6.02 Examination of candidates for a licence to scale shall be held at such time or times during each year and at such places within the Province as the Board of Examiners decides. 6.03 A candidate for examination for a licence to scale shall qualify on such subjects as the Board deems necessary to test the ability and knowledge of the candidate to carry out the duties of a licensed scaler. 6.04 Examinations may be conducted by the Board, or by one or more members of the Board under the direction of the Board, on such subjects as deemed necessary by the Board. Such examinations may be oral, written, or practical tests, all or any of them. 6.05 The Board may require any person to whom a licence has been issued to be re-examined on written notice from the Board. Division (7).—Publication of Scale Rules 7.01 No person shall publish any book or other publication or manufacture any device purporting to set out the official British Columbia Log Rule, or the official British Columbia Cubic Scale, or the official method of scaling any forest products in British Columbia, unless permission therefor has been obtained in writing from the Minister of Lands, Forests, and Water Resources. Division (8).—Interference with Scaler while Making a Scale 8.01 No person shall in any way interfere, impede, or prevent a licensed scaler or person authorized to act as a licensed scaler from making any scale he is required to make under the provisions of the Forest Act. Division (9).—Returns to Be Made by a Scaler 9.01 No licensed scaler or person authorized to act as a licensed scaler shall make and subscribe his name to a return of scale of timber required by the Forest Act, or this regulation, unless such timber has in fact been scaled by him. Any-licensed scaler or person authorized to act as a licensed scaler who makes a return of scale of timber as aforesaid and signs the record of scale as having been scaled by him and who has not in fact scaled such timber is guilty of an offence against the Forest Act, and his licence may be suspended or cancelled forthwith. Division (10 ).—Place of Scaling 10.01 Where any timber or product of the forest is required to be scaled, measured, or counted under the provisions of the Forest Act or of this regulation, the Chief Forester may designate the place or places where such scale, measurement, or counting shall take place. Division (11).—Responsibility for Scale of Drift-timber 11.01 Where timber is found or gathered up as drift-timber, it is the duty of such person finding or gathering up such timber to have the same scaled forth-with, except in an official log salvage district, where the licensee of the receiving station is responsible for obtaining the official scale of timber received by it. Division (12).—Scale of Timber Previously Scaled Mixed with Unsealed Timber 12.01 Where timber has been previously scaled and a further scale is required of some or all of such timber, either together with other unsealed timber or pre-viously scaled timber, the person requiring such further scale shall furnish to the Chief Forester or the Superintendent of Scaling all particulars of the previous scale or scales to enable the Chief Forester to determine whether all charges owing or due to the Crown on such timber have been paid and (or) whether further charges are due and payable. Division (13).—Responsibility of Tow-boats to Have Copy of Scale at Time of Towing 13.01 The operator of every tow-boat shall, before commencing the towing of any boom of logs, obtain and have with him during the towing a description of the boom by number of sections, timber marks, species, and number of pieces, inclusive of boom-sticks and swifters, or the scale account or a copy of the scale of the logs so towed signed by a licensed scaler, and shall on demand of any Forest Officer produce such description or account or copy of scale. Division (14 ).—Definition of Chief Forester 14.01 " Chief Forester," wherever mentioned in this regulation, shall include all persons authorized by the Chief Forester. Printed by A . S U T T O N , Printer to the Queen's Most Excellent Majesty in right of the Province of British Columbia. 1 9 6 4 GOVERNMENT OF BRITISH COLUMBIA FOREST SERVICE 197 APPENDIX XXVII PRINCE RUPERT V8J 1B9-. JUNE 17, 1976 Our F i l e : 03926 TO: A l l Established Licensees in the Prince Rupert Forest D i s t r i c t . Dear S i r ( s ) : As of Apr i l 1, 1976, the scaling programme has been financed by direct appropriation voted by the Legislature. The-costs of scaling w i l l s t i l l be charged back to the industry, and where timber is sold for stumpage the costs of scaling w i l l be allowed in the appraisal of timber. In order to simplify the administrative procedures, scaling fees w i l l be charged on a 'per cunit 1 basis, effective July 1, 1976. This rate w i l l be a l l inclusive, and no extra charges for shift d i f f e r e n t i a l , overtime, travel time and expenses w i l l be made. The per cunit rate w i l l be multiplied by the total volume figure to calculate the total charge. The rates, effective July 1, 1976, w i l l be as follows: | Weight Scaling F.S. Weighmasters Non F.S. Weighmasters 45c per cunit 25 c " " 100% stick scaling | F.S. Scalers 1 Non F.S. Scalers 1 1. Volumes not computed • | 2. Volumes computed 70c " 20c " 10c " I Minor product scaling 2 No charge j Scaling trespass-volumes j 1 J Where requests are made to provide an additional service beyond chat required for normal Forest Service purposes, and such services can be provided, an additional charge may be made to cover the additional costs. Such requests should be submitted to the Dist r i c t Forester, Prince Rupert, well in advance of need. The minimum time required is deemed to be six weeks. Where, on short notice, industry requests a Scaling Station be manned beyond the previously agreed to hours, or where industry requests the services of a scaler where there is insufficient work for a full-time scaler,' the charge rate may be $15.00 per hour, plus a l l expenses incurred. The f i n a l adjudication w i l l be made by the Dist r i c t Forester. 2 

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