5.3 Paired Samples t-Test

There are two common study designs that employ a paired samples t-test to compare two related groups. One relates the groups as two time points for the same subjects. The second relates the groups as two tests of the same subjects, e.g. comparing reaction time under two lighting conditions.

The paired samples t-test uses the mean of sampled paired differences \(\bar{d}\) as an estimate of the mean of the population paired differences \(\delta\) to evaluate an hypothesized mean \(\delta_0\). Test \(H_0: \delta = \delta_0\) with test statistic \(T = \frac{\bar{d} - \delta_0}{se}\), or define a \((1 - \alpha)\%\) confidence interval as \(\delta = \bar{d} \pm t_{1 - \alpha / 2, n - 1} se\). The paired t-test is really just a one-sample mean t-test operating on variable that is defined as the difference between two variables.

The paired samples t test applies when the sampling distribution of the mean of the population paired differences is normally distributed and there are no significant outliers.