Statistical Tests: Sampling Distributions

What are the main statistical tests and what is the resulting sampling distribution?

Answer

$$Z$$ -test: under $$H_0$$ , the sampling distribution of the (standardized) test statistic follows (approximately) a standard normal distribution, $Z \sim \mathcal{N}(0,1)$ .
$$t$$ -test: under $$H_0$$ , the sampling distribution of the (standardized) test statistic follows (approximately) a $$t$$ -distribution with appropriate degrees of freedom, $t \sim t_{\nu}$ .
$\chi^2$ -test: under $$H_0$$ , the sampling distribution of the test statistic follows (approximately) a chi-square distribution with appropriate degrees of freedom, $\chi^2 \sim \chi^2_{\nu}$ .
$$F$$ -test (ANOVA): under $$H_0$$ , the sampling distribution of the test statistic follows (approximately) an $$F$$ -distribution with appropriate degrees of freedom, $F \sim F_{\nu_1,\nu_2}$ .

Notes and comments

Comment 1: Here we mostly refer to the standardized test statistic.

Comment 2: In many common settings these sampling distributions are exact under additional assumptions (e.g., normal distribution for the population). In particular, in mean-testing problems with a normal population, the sampling distribution is exact: if $\sigma$ is known then the standardized statistic is exactly normal, and if $\sigma$ is unknown then the standardized statistic is exactly Student- $$t$$ .