Development of a Test Statistic for Testing Equality of Two Means under Unequal Population Variances

Abstract

In this paper, we propose a test statistic for testing equality of two independent sample means for unequal variances. When group variances differ, the pooled sample variance (  ) is inadequate as a single value for the variances. This problem is commonly known as the Behrens – Fisher problem. Instead, the sample harmonic mean of variances (  ) is proposed, examined and found to better represent the unequal variances. The distribution of  which is known to be generalized Beta is further approximated by the chi – square distribution with the degrees of freedom related to that of degrees of freedom of the chi – square distribution of   . Consequently, it is used to replace the pooled sample variance in the resulting proposed t – test. An example of application is provided.

Keywords: Harmonic mean of variances, chi- square distribution, modified t – test statistic