Confidence Intervals for Paired Samples: Known Variances

In some studies, observations come in matched pairs: each subject (or experimental unit) contributes two measurements. Common examples include before/after readings on the same patient, twin studies, and the same machine measured under two protocols.

Given $n$ pairs $(X_1, Y_1), (X_2, Y_2), \ldots, (X_n, Y_n),$ we form the paired differences

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

and compute their sample mean

$$\bar{D} = \dfrac{1}{n}\sum\limits_{i=1}^n D_i.$$

When the population standard deviation of the differences $\sigma_D$ is known, a $(1-\alpha)\cdot 100\%$ confidence interval for the mean difference $\mu_D = \mu_X - \mu_Y$ is

$$\bar{D} \,\pm\, z_{\alpha/2}\cdot\dfrac{\sigma_D}{\sqrt{n}}.$$

The critical values for the most common confidence levels are:

$$\begin{array}{c|c} \text{Confidence} & z_{\alpha/2} \\ \hline 90\% & 1.645 \\ 95\% & 1.96 \\ 99\% & 2.576 \end{array}$$

For instance, with $n=9,$ $\bar{D}=4,$ and $\sigma_D=3,$ the margin of error at 95% confidence is

$$1.96\cdot\dfrac{3}{\sqrt{9}}=1.96\cdot 1=1.96,$$

so the interval is $(4-1.96,\,4+1.96)=(2.04,\,5.96).$