Past exam of the statistics course of the University of Oxford / 2015 / 3 / b / iii

The denominator $\Phi (z)$ is positive, soStrict increase of the standard normal distribution function turns this into $z>{\Phi }^{-1}(10/11)$. Since the coefficient multiplying $\overline{x}$ in $z$ is positive, the rejection rule is precisely $\overline{x}>c$, withThe quantile is approximately $1.33518$. Equivalently, the rule requires posterior probability of $H_0$ below $1/11$, because its prior odds are one. The calculation is an instance of the Bayes factor for the sign of a normal mean.