29.9 Summary

Consider testing a hypothesis about a population mean difference \(\mu_d\), based on the value of the sample mean difference \(\bar{d}\). Under certain statistical validity conditions, the sample mean difference varies with an approximate normal distribution centered around the hypothesised value of \(\mu_d\), with a standard deviation of

\[ \text{s.e.}(\bar{d}) = \frac{s_d}{\sqrt{n}}. \] This distribution describes what values of the sample mean difference could be expected if the value of \(\mu_d\) in the null hypothesis was true. The test statistic is

\[ t = \frac{ \bar{d} - \mu_d}{\text{s.e.}(\bar{d})}, \] where \(\mu_d\) is the hypothesised value in the null hypothesis. The \(t\)-score describes what value of \(\bar{d}\) was observed in the sample, relative to what was expected. The \(t\)-value is like a \(z\)-score, so an approximate \(P\)-value can be estimated using the 68–95–99.7 rule, or is found using software. The \(P\)-values helps determine if the sample evidence is consistent with the assumption, or contradicts the assumption.

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