In Lesson 1 we learned about independent samples and paired samples. When we have two independent samples, the observations in the two groups are unrelated to one another and are not matched in any meaningful way. With paired samples, the observations in the two groups are matched in a meaningful way. Most often this occurs when data are collected twice from the same participants, such as in a pre-test / post-test design. But, there could be different participants in each group who are paired together meaningfully, such as brother-sister pairs or husband-wife pairs.
When data are paired, we compute the difference for each case, and then treat those differences as if they are a single measure. When constructing a confidence interval for the difference in paired means, we're really constructing a confidence interval for a single mean, where the single mean is the mean difference.
The population parameter is \(\mu_d\) where \(\mu_d=\mu_1-\mu_2\). The sample statistic is \(\overline x_d\) where \(\overline x_d = \overline x_1 - \overline x_2\).