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: