= Exact pooled two-sample t statistic
{title2=$T=\frac{\overline X-\overline Y}{S_p\sqrt{1/m+1/n}}$}
For independent normal samples of sizes $m,n$ with a common unknown <variance> and equal means under the null, $Z=(\overline X-\overline Y)/(\sigma\sqrt{1/m+1/n})$ is standard normal. The independent centered sample sums of squares give $V=(m+n-2)S_p^2/\sigma^2\sim\chi^2_{m+n-2}$, independently of $Z$. Thus $Z/\sqrt{V/(m+n-2)}$ has the exact <Student t-distribution>, giving the statistic $(\overline X-\overline Y)/(S_p\sqrt{1/m+1/n})$. A lower one-sided test uses its lower-tail <probability>.
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