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

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

$$t=\frac{\overline{x}-\mu }{\text{s.e.}\left(\overline{x}\right)},$$ 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: