29.10 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: