(i) Explain what is meant by a uniformly most powerful unbiased test of a null hypothesis against an alternative.

Let $Y_{1}, \ldots, Y_{n}$ be independent, identically distributed $N\left(\mu, \sigma^{2}\right)$ random variables, with $\sigma^{2}$ known. Explain how to construct a uniformly most powerful unbiased size $\alpha$ test of the null hypothesis that $\mu=0$ against the alternative that $\mu \neq 0$.

(ii) Outline briefly the Bayesian approach to hypothesis testing based on Bayes factors.

Let the distribution of $Y_{1}, \ldots, Y_{n}$ be as in (i) above, and suppose we wish to test, as in (i), $\mu=0$ against the alternative $\mu \neq 0$. Suppose we assume a $N\left(0, \tau^{2}\right)$ prior for $\mu$ under the alternative. Find the form of the Bayes factor $B$, and show that, for fixed $n, B$ $\rightarrow \infty$ as $\tau \rightarrow \infty$.