Past exam of the statistics course of the University of Oxford / 2021 / 2 / a

False. A test size constrains the probability of rejection conditional on the null hypothesis being true:for an exact size-$\alpha$ test of a simple hypothesis. It does not assign a posterior probability to that statistical hypothesis after observing nonrejection. In the frequentist interpretation, the statistical parameter is fixed and the data are random. Nonrejection can occur even when the alternative hypothesis is true; for example, it is particularly likely for an alternative against which the test has low statistical power. A posterior probability would require a prior distribution and a Bayesian model.

True, with the test's rejection rule fixed. If its rejection region in the space of the test statistic is $C$, its statistical power against the specified simple hypothesis $H_1$ isThus the probability distribution of $T$ under the alternative hypothesis determines the statistical power of this test. The null distribution is used to choose a rule with the desired test size, but statistical power is evaluated under the alternative hypothesis. Different rejection rules can have different statistical power even for the same probability distribution of $T$.

False. For the test associated with a p-value, reject at significance level $\alpha$ when $p\le \alpha$. Here $0.1>0.05$, so do not reject at level 0.05. The p-value is neither the posterior probability of the null hypothesis nor the probability of a Type I error conditional on these observed data.