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In an experiment of treatment selections, random samples are drawn from k populations with ordered means. The probability that a sample statistic from the population with the highest mean turns out to be ranked the highest is referred to as the probability of correct selection (PCS). An inequality was proved previously that shows the monotonicity of PCS with respect to change in variance of the samples. In this article, we first present a more general form of the probability inequality to be used to investigate PCS. An extension of the monotonicity of PCS to order statistics is considered. We show that the PCS of the smallest order statistic preserves the monotonicity. Additionally, a normal approximation method is used to further generalize the theory. The general order statistics will not enjoy the same properties, as we reveal the obstacles, and a numerical counter example. [ABSTRACT FROM AUTHOR]

In an experiment of treatment selections, random samples are drawn from k populations with different means. The probability that, among the samples, the sample mean from the population with the highest underlying expected value turns out to be the highest is called the probability of correct selection (PCS). In this article, a probability inequality motivated by the PCS problem, is obtained with respect to change in variances. As an application of the inequality in normal mean selection, it is shown that in a multiple treatment experiment setting, by increasing sample sizes, and thus obtaining a higher precision in the estimation of means, in general (but not always), one will have a higher PCS. However, there are special case exceptions. [ABSTRACT FROM AUTHOR]