# Confidence Interval for Paired Means

The method of paired means is used when two sets of data have the same number of elements, and there exists a one-to-one correspondence between the elements of each set. The first step in using this method is to calculate the difference between the two sets of data: $d=y_1-y_2$. This results in one quantitative variable that is then used in the production of a confidence interval and hypothesis test.

In computing a confidence interval the sample must first meet the following conditions:

• Random
• Independent (10%)
• Nearly normal (for the difference, d; the picture to the right demonstrates this)

The calculation of the interval itself, depending on the desired level of confidence, can then be computed by the following formula: $(\bar{d} \pm t^*_{df} (\frac{s_d}{\sqrt{n_d}}))$, where df is the degrees of freedom given by $df=n_d-1$. The t value above is dependent on the degrees of freedom and level of confidence. A statistical computer program or t-table can be used to find this value.

In this case, a confidence interval for the mean difference can tell us, with a certain amount of confidence, the limits between which the true difference should lie. The statistics website stat trek has a good page on computing a confidence interval for paired means going into more detail about each step. The following pdf offers a similar run down on paired means and creating a confidence interval, but also has a quick demo on how to carry out a paired t-test on SPSS. This tutorial has a neat exploration section that allows the user to see what happens to the distribution when changing $\alpha$ and the standard deviation, as well as a short quiz.

The picture above is from this online course offered at Penn State that walks through both a confidence interval and hypothesis test for an example of paired means.