Estimating the Minimum Sample Size in Tests on the Means Of Normal Distributions

Abstract

In testing on the means of a normal distribution with unknown variance, it is well-known that the distribution of the commonly-used statistic (namely, the normalized estimator of the mean) has a non-central t distribution with (n-1) degrees of freedom and non-centrality parameter δ under the alternative hypothesis, where n is the sample size. The non-centrality parameter δ, regarded as a function of the minimum sample size required to achieve a pair of predetermined probabilities of two types of error (α, β), has a limit as this sample size goes to infinity. Exploiting this fact, we propose two estimators of the minimum sample size suitable for the two cases of expected small and large sample sizes. For selected pairs of (α, β), the values of the two estimators are computed and are shown to quite accurate, especially for larger sample sizes. We then modify the estimators for the test on the equality of the means of two normal distributions. The Fisher index of discrimination of the two hypotheses and its limit as a function of the minimum sample size as this size approaches infinity are studied.