## 2.1 Assumptions: Paired samples $$t$$-test

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

Independent samples $$t$$-test Assumptions:

1. The data are numeric
2. 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)
3. 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.