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 $\sqrt{\chi^2}$ .