30.10 Summary

To test a hypothesis about a difference between two population means \(\mu_1 - \mu_2\), based on the value of the difference between two sample mean \(\bar{x}_1 - \bar{x}_2\), assume the value of \(\mu_1 - \mu_2\) in the null hypothesis to be true (usually zero). Then, the difference between the sample means varies from sample to sample and, under certain statistical validity conditions, varies with an approximate normal distribution centered around the hypothesised value of \(\mu_1 - \mu_2\), with a standard deviation of \(\text{s.e.}(\bar{x}_1 - \bar{x}_2)\). This distribution describes what values of the sample mean could be expected in the sample if the value of \(\mu\) in the null hypothesis was true. The test statistic is

\[ t = \frac{ (\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\text{s.e.}(\bar{x}_1 - \bar{x}_2)}, \] where \(\mu_1 - \mu_2\) is the hypothesised value given in the null hypothesis. This describes the observations. The \(t\)-value is like a \(z\)-score, and so an approximate \(P\)-value is estimated using the 68–95–99.7 rule, which is how we weigh the evidence to determine if it is consistent with the assumption.

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