Confidence Intervals for Paired Samples: Unknown Variances

A paired sample consists of $n$ pairs of observations $(x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n)$, typically arising from two measurements taken on the same subject (e.g., before/after, left/right, treatment/control). To make inferences about the mean difference $\mu_d = \mu_x - \mu_y$, we form the differences

$$d_i = x_i - y_i, \quad i = 1, 2, \ldots, n,$$

and treat $d_1, d_2, \ldots, d_n$ as a single sample from a distribution with mean $\mu_d$ and variance $\sigma_d^2$.

When $\sigma_d^2$ is unknown, we estimate it with the sample variance $s_d^2$ and use the t-distribution with $n - 1$ degrees of freedom. The $100(1-\alpha)\%$ confidence interval for $\mu_d$ is

$$\bar{d} \pm t_{\alpha/2,\, n-1} \cdot \dfrac{s_d}{\sqrt{n}},$$

where

$$\bar{d} = \dfrac{1}{n}\sum\limits_{i=1}^n d_i \qquad \text{and} \qquad s_d = \sqrt{\dfrac{1}{n-1}\sum\limits_{i=1}^n (d_i - \bar{d})^2}$$

are the sample mean and sample standard deviation of the differences.

For instance, if $n = 16$, $\bar{d} = 5$, and $s_d = 4$, then a 95% CI uses $t_{0.025,\, 15} = 2.131$:

$$5 \pm 2.131 \cdot \dfrac{4}{\sqrt{16}} = 5 \pm 2.131 = (2.869,\, 7.131).$$