3.3 Two sample tests

Another common instance of hypothesis testing is in the comparison of two groups. Given a sample from each of two populations, one thing that we might want to do is to compare the two population means; are they equal or is one significantly larger than the other? We discuss how to do this under the assumption that the populations both have a Normal distribution, with the same variance.

Let $x_1,\mathrm{\dots },x_n$ be realisations of IID random variables with $\mathrm{Normal}({\mu }_X,{\sigma }^2)$ distribution and let $y_1,\mathrm{\dots },y_m$ be realisations of IID random variables with $\mathrm{Normal}({\mu }_Y,{\sigma }^2)$ distribution. How can we test the null hypothesis

against any of the alternatives $H_1:{\mu }_X-{\mu }_Y\ne d$, and $H_1:{\mu }_X-{\mu }_Y>d$? The testing approach depends on whether the samples are paired, or unpaired.