Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 35 / 3 / e / Solution

Under the point hypothesis $\beta =0$, the test statistic $\sqrt{n}\overline{Y}$ has a standard normal distribution. The observed statistic is three, giving a two-sided p-value $2[1-\Phi (3)]\simeq 0.00270$. This is conventionally strong evidence against that point hypothesis. The narrow model $H_0$ is a continuous prior distribution around zero, rather than a point hypothesis.

For $n_0/n=100$ and $c=100$, Normal-normal conjugacy givesThe narrow-model posterior mean is strongly pulled toward zero and its standard deviation is about $0.0995/\sqrt{n}$. The wide-model posterior mean is about $2.9703/\sqrt{n}$, very close to the observation, with standard deviation about $0.9950/\sqrt{n}$. Each model produces a markedly different posterior, so choosing between them requires their predictive evidence.