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AN EXPERT SAMPLE ALLOCATION PROGRAM James W. Mer&erson INTRODUcrION A univariate optimum allocation where formula for a L stratified sampling design with simple random sampling within strata may produce Dh > Nb in some strata. CochraD (1977) gives a procedure for determining the optimum allocation for this situation when equal costs among - number of strata - popUlation size of stratum h strata are - standard deviation of data in stratum h assumed and a fixed desired precision level is specified. When costs among strata differ this procedure is not applicable. An explicit procedure for this case does available in sampling textbooks. - average data collection cost in stratum h not seem to be L N When costs differ greatly among strat8$ and especially when low costs are associated with strata having high variances and high costs are associated with strata having low variances. the total sample size may exceed the sum of the number of population units. An example is presented involving such a situation. Also. a general procedure for determining allocations when some initial n h are greater than the corresponding N is presented. The procedure is applicable for aetermining optimum allocations for a stratified sampling design. with simple random sampling within strata. for a fixed desired level of precision when costs among strata are unequal. The procedure has been implemented using Pascal. Input standard deviations are computed using SAS. - ~ N h=1 h D - desired size of s s - standard error of the estimate of the population mean - mean of data in stratum h Applying (2.1) n = 307. The allocations to each stratum is computed using the formula: (2.2) • FXAMPLE Given a desired coefficient of variation (CV) equal to 0.02. consider determining the optimum allocation using the following data: ttl h y 100 ~.!l~ 25 3.54 20.0 = 17. and n3 = 282. , ___ 1___1.!:1.L_~.Q,.!l_ The formula for computing the total sample is as follows: 8. tt2 This allocation is impossible since n 3 ) N • 3 Also. n > N implies that attempts to follow,a procedure similar to the procedure in Cochran (1977) is fruitless. Cochran's procedure is outlined in Appendix A. l--iiio---wo--"3-:33-1o.02 = Since we have n3 > N3 • set ng = N~. The formula for computing the varia~ce of the estimate of the population mean is as follows: size 772 REFERENCES Clearly, the contribution to the standard error of the estimate of the population mean from stratum 3 will be zero. Strata 1 and 2 will account for all of the sample variance-. Having set ttg = N3 • let's compute a sub-total sample size for strata 1 and 2. Since strata 1 and 2 will ac~ount for all of the variance. the quantity N2D2 in (2.1) remains the same. The formula for the sub-total sample size for strata 1 and 2 is: Cochran. William (1977). Sampling New York: John Wiley and Sons. Teohniques. Hansen. Morris. Hurwitz. William and Madow. William (1953), Sample Survey Methods and Theory. New York: John Wiley and Sons. Jessen. Raymond (1978). Statistical Survey Techniques. New York: 70hn Wiley and Sons. Raj. Des (1968). McGraw-Hill. n Applying (2.4) n': 43. The allocation to Sampling Theory. New York: Sukhatme. Pandurang, Sukhatme. Balkrishna. Sukhatme. Shashikala and Asot. C. (1984). Sampling Theory of Surveys with Applications. Ames. Iowa: Iowa State University Press. each stratum is computed using the formula: "h , 1/2 2 1/2 In ~sh/(ch) 1/1 ~ ~sh/(ch) 1 h=l n 14 and n n 1 , 8110e8 tion 29. The revised 2 APPENDIX A : Cochran's Procedure Cochran's procedure for allocations requiring more than 100 per cent sampling (equal costs among strata) is outlined: is , 14, tt2 = 29 and n3 = 100. 1 1. I f GENERAL PROCEDURE When computing the allocation for sampling design with simple > Nh If n h 2. Compute set a , "h stratified random sampling for nh = (n - all n 8 N s (C )1/2)( l N s /(Ch)1/2) h d h h h ____!u.~h_____ _ where I is the index set {1.2 ••.•• L} and Q is a set containing each h such that ~ > Nh The set d is the difference of sets I and Q. 3. Compute ~= , 1/2 In Nhsh/(ch) 1/1 had. n , tt • tt3' 1 2 The ~ Nhsh/(c h ) h • d optimum 1/2 allocation for 1 is then ~. 773 where and such that n h > Nh • Q is I is the index set a set containing each h The set d is the difference The optimum allocation is then , n l • n 2 • na' 2 N D2 + l NhS~ h • d all h • d of sets I and Q. l h n h = Nh 1 Nk)(Nhsh)/( 1 Nhs h ) , kaQ h e d {1.2 ••••• L} , ( , set 2. Compute within strata, one or more nh may initially be greater than the corresponning Nh • When this sit_uation occurs the following procedure may be applied: 1. > Nh nh •••• nL"