Hypothesis Tests for Two Means: Paired-Sample T-Test

In a paired-sample t-test, we have $n$ paired observations $(X_1, Y_1), \ldots, (X_n, Y_n)$ and we want to test a hypothesis about the population mean of the differences $\mu_d,$ but the population standard deviation $\sigma_d$ is unknown.

The setup mirrors the paired-sample z-test, with two changes: we use the sample standard deviation $s_d$ in place of $\sigma_d,$ and we compare to a $t$-distribution with $n-1$ degrees of freedom instead of the standard normal.

We form the differences $D_i = X_i - Y_i$ and compute

$$\bar{d} = \dfrac{1}{n}\sum\limits_{i=1}^n D_i, \qquad s_d = \sqrt{\dfrac{1}{n-1}\sum\limits_{i=1}^n (D_i - \bar{d})^2}.$$

The test statistic for $H_0: \mu_d = \mu_{d,0}$ is

$$t = \dfrac{\bar{d} - \mu_{d,0}}{s_d / \sqrt{n}},$$

which follows a $t$-distribution with $n-1$ degrees of freedom under $H_0.$

For instance, if $n=4$ paired observations yield $\bar{d} = 5$ and $s_d = 4,$ then to test $H_0: \mu_d = 0$ we have

$$t = \dfrac{5 - 0}{4/\sqrt{4}} = \dfrac{5}{2} = 2.5, \qquad \text{df} = 3.$$