This Chapter considers some tests whose test statistic is distributed as a Student’s t. It would seem more natural to present this Chapter after those concerning Gaussian and χ2 distributions, since the Student’s t, as a distribution, is derived from the two distributions above. Nevertheless, it is preferable to first extend the framework considered in the first part of the book (i.e. comparison of two means) to the analogous one where the standard deviations of the two groups are unknown.
Thus, SP estimation is shown here for the tests for comparing the means of two groups whose outcomes have Gaussian distributions with unknown variances, under the conditions of equal and unequal variances. In both cases the test statistic is distributed as a Student’s t. Two tests will therefore be considered, where Gm,λm is a t distribution with a number of dfs depending on m (i.e. f(m)) and noncentrality parameter λm, that is, .
The true distribution tF1 of the variable of interest for the population treated with the new drug is assumed to be the Gaussian N(μ1, σ2), whereas that of the control population is tF2 = N(μ2,σ2). The null hypothesis is H0 : μ1 = μ2, and the one-sided alternative H1 : μ1 > μ2 is considered.
Being the common variance σ2 unknown, it is estimated by the pooled variance estimator: ...