Suppose that $X_{1}, \ldots, X_{n}$ are independent normally distributed random variables, each with mean $\mu$ and variance 1 , and consider testing $H_{0}: \mu=0$ against $H_{1}: \mu=1$. Explain what is meant by the critical region, the size and the power of a test.

For $0<\alpha<1$, derive the test that is most powerful among all tests of size at most $\alpha$. Obtain an expression for the power of your test in terms of the standard normal distribution function $\Phi(\cdot)$.

[Results from the course may be used without proof provided they are clearly stated.]