## 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: