Paired t-Test

Used when two related measurements are taken on the same sample population. Evaluates whether the mean difference between the paired observations is significantly different.

Let each pair have values $(X_1, X_2)$. Suppose $d_i = X_{1i} - X_{2i}$.

$$\bar{d} = \frac{1}{n} \sum_{i=1}^{n} d_i \quad \text{and} \quad s_d = \sqrt{\frac{\sum (d_i - \bar{d})^2}{n - 1}}$$

Hypotheses Setup

$$H_0 : \mu_d = d_0 \,\, \text{(no mean difference)}$$ $$H_1 : \mu_d < d_0 \text{ or } \mu_d > d_0 \text{ or } \mu_d \ne d_0$$

Test Statistic

Under $H_0$ with $\nu = n - 1$:

$$t = \frac{\bar{d} - d_0}{s_d / \sqrt{n}} \sim t_\nu$$