C.3 Hypothesis testing

For statistics whose sampling distribution has an approximate normal distribution, the test statistic has the form: \[ \text{test statistic} = \frac{\text{statistic} - \text{parameter}}{\text{s.e.}(\text{statistic})}, \] where \(\text{s.e.}(\text{statistic})\) is the standard error of the statistic. The test-statistic is a \(t\)-score for most hypothesis tests in this book when the sampling distribution is described by a normal distribution, but is a \(z\)-score for a hypothesis test involving one or two proportions.

Notes:

If the test-statistic is a \(z\)-score, the \(P\)-value can be found using tables (Appendix B.1), or approximated using the \(68\)--\(95\)--\(99.7\) rule.
If the test-statistic is a \(t\)-score, the \(P\)-value can be approximated using tables (Appendix B.1), or approximated using the \(68\)--\(95\)--\(99.7\) rule (since \(t\)-scores are similar to \(z\)-scores; Sect. 28.4.
When the sampling distribution for the statistic does not have an approximate normal distribution (e.g., for ORs and correlation coefficients), this formula does not apply and \(P\)-values are taken from software when available.
A hypothesis test about ORs uses a \(\chi^2\) test statistic. For \(2\times 2\) tables only, the \(\chi^2\)-value is equivalent to a \(z\)-score with a value of \(\sqrt{\chi^2}\).