Hypothesis Tests for Two Means: Equal But Unknown Population Variances

Suppose we want to compare the means $\mu_1$ and $\mu_2$ of two normal populations using independent random samples. When the population variances $\sigma_1^2$ and $\sigma_2^2$ are unknown but can be assumed equal, we use the pooled two-sample t-test.

Let the samples have sizes $n_1, n_2,$ means $\bar{x}_1, \bar{x}_2,$ and variances $s_1^2, s_2^2.$ The hypotheses take the form

$$H_0\!:\ \mu_1 - \mu_2 = \delta_0 \qquad \text{vs.} \qquad H_1\!:\ \mu_1 - \mu_2 \neq \delta_0$$

(or a one-sided alternative). The most common case is $\delta_0 = 0,$ corresponding to testing whether the two means are equal.

The test statistic is

$$t = \dfrac{(\bar{x}_1 - \bar{x}_2) - \delta_0}{s_p\,\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}}$$

where $s_p$ is the pooled sample standard deviation (defined in the next cycle). Under $H_0,$ the statistic $t$ follows a $t$-distribution with $n_1 + n_2 - 2$ degrees of freedom.