The problem of selecting the component having the largest population mean in a k-variate normal population is discussed when all the means, variances and correlations are unknown. The indifference-zone formulation has been adopted, and we propose a two-stage procedure (PS) that guarantees the nominal value (P*) of probability of correct selection under the assumption that all the correlations are non-negative. The procedure PSis the only method thus far proposed that will accomplish the desired objective. It is also of some interest that PSis asymptotically (as δ*↛ 0) "better" than the procedure PRof Dudewicz and Dalal (1975) for k = 2, P*≤.95, while for k≥3 the procedure PRis "better" than PSwhen ρij= 0, i≠j. This comparison is naturally valid when both PRand PSare applicable, and of course there are situations where only PSis applicable.