29.7 Statistical validity conditions: Mean differences

As with any inferential procedure, these results apply under certain conditions. For a hypothesis test for the mean of paired data, these conditions are the same as for the CI for the mean difference for paired data (Sect. 23.9), and similar to those for one sample mean.

The test above is statistically valid if one of these conditions is true:

The sample size of differences is at least 25; or
The sample size of differences is smaller than 25, and the population of differences has an approximate normal distribution.

The sample size of 25 is a rough figure here, and some books give other values (such as 30). This condition ensures that the distribution of the sample means has an approximate normal distribution so that we can use the 68–95–99.7 rule.

Provided the sample size is larger than about 25, this will be approximately true even if the distribution of the individuals in the population does not have a normal distribution. That is, when \(n>25\) the sample means generally have an approximate normal distribution, even if the data themselves don’t have a normal distribution.

In addition to the statistical validity condition, the test will be

internally valid if the study was well designed; and
externally valid if the sample is a simple random sample and is internally valid.

Example 29.2 (Statistical validity) For the insulation data used above, the sample size is small, so the test will be statistically valid if the differences in the population follow a normal distribution.

We don’t know if they do, though the sample data (Fig. 23.1) don’t identify any obvious doubts. So the test is possibly statistically valid, but we can’t be sure.

Example 29.3 (COVID lockdown) In Example 29.1 concerning COVID lockdowns, the sample size was 213 Spanish health students.

Since the sample size is muich larger than 25,, the test is statistically valid.