## 27.11 Summary

To test a hypothesis about a population mean $$\mu$$, initially assume the value of $$\mu$$ in the null hypothesis to be true. Then, describe the sampling distribution, which describes what to expect from the sample statistic based on this assumption: under certain statistical validity conditions, the sample mean varies with an approximate normal distribution centered around the hypothesised value of $$\mu$$, with a standard deviation of

$\text{s.e.}(\bar{x}) =\frac{s}{\sqrt{n}}.$ The observations are then summarised, and test statistic computed:

$t = \frac{ \bar{x} - \mu}{\text{s.e.}(\bar{x})},$ where $$\mu$$ is the hypothesised value given in the null hypothesis. The $$t$$-value is like a $$z$$-score, and so an approximate $$P$$-value can be estimated using the 68–95–99.7 rule, or found using software.

The following short video may help explain some of these concepts: