Setting Up a Test for the Difference Between Two Population Means

A two-sample t test states $H_0: \mu_1 = \mu_2$, equivalently $\mu_1 - \mu_2 = 0$, against an alternative whose direction comes from the research question, with both parameters defined in context. The design question comes first: independent groups take the two-sample procedure, while the same individuals measured twice or matched pairs take the one-sample procedure on differences. The conditions are random, independence between groups, the 10% condition per group, and a shape check per group, with degrees of freedom either technology's value or the conservative $\min(n_1, n_2) - 1$.

The design gets read off the data layout rather than the study: two columns of numbers become two samples even when the same twenty runners produced both, which changes the standard error, the degrees of freedom, and the parameter. The hypotheses get written about $\bar{x}_1$ and $\bar{x}_2$, which are known and differ by 7.2, or assigned the two observed means as values, or set to the observed difference itself. And the alternative's direction gets taken from whichever sample mean came out larger rather than from the question the study was designed to answer.