## 2.1 Assumptions: Paired samples \(t\)-test

The assumptions of the paired \(t\)-test are exactly the same as those of the one-sample \(t\)-test. They can be summarised as follows:

**Paired samples \(t\)-test Assumptions:**

- The data are numeric
- Observations are independent of one another (that is, the sample is a simple random sample and each individual within the population has an equal chance of being selected)
- The sample mean, \(\overline{X}\), is normally distributed.

When checking assumptions for the paired \(t\)-test, the variable of interest is the ** paired differences** rather than the 'before' and 'after' variables themselves. To illustrate, consider the below snapshot of the

`anorexia`

data set:```
Treat Prewt Postwt paired.differences
1 Cont 80.7 80.2 0.5
2 Cont 89.4 80.1 9.3
3 Cont 91.8 86.4 5.4
4 Cont 74.0 86.3 -12.3
5 Cont 78.1 76.1 2.0
6 Cont 88.3 78.1 10.2
```

The `Prewt`

and `Postwt`

variables represent the pre and post weights of the patients respectively. The `paired.differences`

column is the difference between the post and pre weight for each patient. The paired \(t\)-test is testing whether the average of the ** paired differences** is equal to zero. For this reason, it is this variable that is of interest when checking assumptions.