Small sample behaviour of certain tests of the hypothesis of equal means under variance heterogeneity

By using a finite series representation of the distribution of a certain class of statistics, and by utilizing other techniques presented here, two types of statistic are developed for testing the hypothesis of the equality of the means of two normal populations using small sample sizes. One type of statistic, called unilateral, effectively controls the size of the test under variance heterogeneity if it is known a priori that the variance of one specified population is greater than that of the other. The second type of statistic, called bilateral, controls the size when there is no a priori knowledge of the population variances. Tables are included specifying both types of statistic and giving size and power.