### Section 7.4: Paired Comparisons

#### Key Concepts

Designing an experiment where two measurements may be taken on the same individual is an effective way of limiting the variablity caused by variables which are not of primary interest. In the analysis of such data, the proper methodology is to take differences between the measurements on each individual, and to treat these differences as a single sample.

#### Example

"Breathe-Right" nasal strips were developed to assist people who have severe snoring problems and other breathing disorders. Recently, many athletes have begun to use them as a device for enhancing athletic performance. A team of researchers desired to test whether "Breathe-Right" nasal strips increased oxygenation to the body. One variable of interest is maximal oxygen consumption (VO2 Max). The researchers tested a sample of 16 male athletes on an exercise bicycle, with the strip and without. (There was a time period of 2 days between tests for each athlete.) The data from the experiment is summarized as (Mean +/- SD):
```  Without Breathe-Right     3326 +/- 520
With Breathe-Right        3367 +/- 615
Difference                41.4 +/- 438.3
```
There are many variables that may affect VO2 Max besides whether the "Breathe-Right" nasal strips were worn or not. However, most of these variables (such as age, cardiovascular fitness, etc.) are controlled for by having the same athlete tested twice. Each measurement without the strip is matched to a corresponding measurement on the same individual with the strip. The sixteen individual differences in VO2 Max consist of a single sample of differences.

To test the hypothesis that the nasal strips improve VO2 Max, the test statistic is

```   t = (41.4 - 0) / (438.3 / sqrt{16}) = 0.38
```
In other words, the observed difference in sample means is 0.38 estimated standard errors above the mean, a value that is easily attributable to chance variation.

A one-sided p-value is 0.35. Since we expect to see a result this extreme 35% of the time when the null hypothesis is true (more than once every three experiments), we are not surprised that it occured. Hence, the data is consistent with the hypothesis that "Breathe-Right" nasal strips have no effect on VO2 Max.