Considerkindependent normal populations with unknown means and a common unit variance and let a better population be defined to be one with a larger absolute value of the mean. The problem of selecting thetbest ofkpopulations is considered; the common number of observations from each population is determined so that the probability of a correct selection is not less than a specified quantity whenever the parameters lie in a pre-assigned parametric sub-space. A procedure for this problem is proposed and tables and bounds for probability of a correct selection are provided; some properties of the proposed procedure are also discussed. The problem of selecting a small non-empty subset containing the best population is also considered; a procedure is proposed and the constant involved in the procedure is determined so that the probability of a correct selection is not less than a specified quantity. The expected size of the selected subset is derived and its supremum obtained. Two secondary problems for determining the common sample size are discussed. The case when the common variance is unknown is also considered. The dual problem of selecting a subset containing the worst population is also treated.