Abstract We consider the problem of ‘optimally’ allocating a given total number, N, of observations to k≥2 normal populations having unknown means but known variances σ21,σ22,…,σ2k, when it is desired to select the population having the largest mean using a natural single-stage selection procedure based on sample means. Here ‘optimal’ allocation is one that maximizes the infimum of the probability of a correct selection (P(CS)) over the so-called preference zone of the parameter space (Bechhofer (1954)). The solution of this problem enables us to find the smallest possible N and the associated optimal allocation of the sample sizes, viz. n1,n2,…,nk such that Σ ni=N, required to guarantee a specified {δ∗,P∗} probability requirement. We prove that for k≥3, the allocation ni∝σ2i (which is convenient to implement in practice) is locally (and for k=3, numerically checked to be globally) optimal iff P∗≤PL or P∗≥PU, where PL and PU depend on the largest and the smallest relative variances, respect ively. For PL