On Relative Precision in Stratified Sampling for a Mean with Fixed Cost

This paper considers the extrema of the relative precision (R. P.), or the ratio of the variance of optimum allocation to that of proportional allocation in stratified sampling for a mean (= Y ) or total (= Y ) when there is a fixed cost C = c o +σ n h c h over L strata. The upper bound of R. P. is shown to be equal to unity, though the lower bound can be negative while approaching zero if the finite population size N →∞. Numerical results (cf. Table) when the upper bound of R. P. = 1 is attained are included for the cases L ≦4 with various combinations of { W h }, { c h } and a fixed cost C ; these give the appropriate sample sizes (= n p ) determined for proportional sampling in the cases of finite populations of sizes N =5000, 10,000 respectively.