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

A paired sample consists of two measurements taken on the same individual or matched unit. Typical examples include before/after observations on the same subject, twin studies, and left-vs-right comparisons.

The key idea is to collapse the two samples into a single sample of differences:

$$D_i = X_i - Y_i, \qquad i = 1, 2, \ldots, n.$$

The two-sample paired problem then becomes a one-sample problem on $D$.

To test $H_0 : \mu_D = \mu_0$ when the population standard deviation $\sigma_D$ is known, the paired-sample Z statistic is

$$Z = \dfrac{\bar D - \mu_0}{\sigma_D / \sqrt{n}},$$

where $\bar D = \dfrac{1}{n}\sum\limits_{i=1}^n D_i$ is the sample mean of the differences. Under $H_0$, $Z \sim N(0,1)$.

For example, suppose $n=4$ paired differences are $D = 1, 3, 4, 0$, with $\sigma_D = 2$, and we test $H_0: \mu_D = 0$. Then $\bar D = 8/4 = 2$ and

$$Z = \dfrac{2 - 0}{2/\sqrt{4}} = \dfrac{2}{1} = 2.$$