## 27.9 Statistical validity conditions: One mean

As with any inference procedure, the underlying mathematics requires certain conditions to be met so that the results are statistically valid. For a hypothesis test for one mean, these conditions are the same as for the CI for one mean (Sect. 22.4).

The test will be statistically valid if one of these is true:

1. The sample size is at least 25, or
2. The sample size is smaller than 25 and the population data 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 the 68–95–99.7 rule can be used. 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

Example 27.2 (Statistical validity) The hypothesis test regarding body temperature is statistically valid since the sample size is large ($$n=130$$).

Since the sample size is large, we do not require the data to come from a population with a normal distribution.

Example 27.3 (Driving speeds) In Example 27.1 about mean driving speeds, the sample size was 400, much larger than 25.

The test will be statistically valid.