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It is assumed that there are available k finite populations, each consisting of U primary units, and that each primary unit can be subdivided into T elements. It is further assumed that the populations have a common known variance among primary units, and a common known variance among elements within primary units. The values of the overall population means per element are assumed to be unknown, as is the true pairing of the ranked values of these means with the populations. It is desired to select the population which has the largest overall population mean per element. This selection is to be accomplished by taking a random sample of u ≤ U primary units from each population, and then a random sample of t ≤ T elements from each primary unit. The pair (t,u) is to be chosen in such a way as to guarantee that the probability of a correct selection will be equal to or greater than a specified quantity whenever the true difference between the largest and second largest overall population mean per element is e...

A problem of frequent occurrence in the statistical analysis of experimental data is the estimation of the first two moments of a population with continuous but unknown c.d.f. F(x), based on a random sample (x , * xn) - (x,). Standard estimates for E(x) and Var (x) are x or x ?- t,_as/ n and s , respectively. If the population {x} is non-normal x and S2 are not minimum variance. For the sake of obtaining efficient estimates of E(x) or var (x) the sample (xi) is often transformed by a function F: xi --, y, = F(xi)*, where F is chosen so that (y,) satisfies tests of normality. Estimates for E(y) and var (y) are then computed as y or y ?t tl-,s,/s/ and s2 respectively. (It is assumed that such estimates are efficient; they are, in fact sufficient provided we know that the population {y} is normal). Since these estimates are sufficient, a function of them alone is an efficient estimate of that function's expected value. A rule therefore commonly given for estimating E(x) is E(x) = F-(y) (1) or