Let $X_{1}, \ldots, X_{n}$ be a random sample from a normal distribution with mean $\mu$ and variance $\sigma^{2}$, where $\mu$ and $\sigma^{2}$ are unknown. Derive the form of the size- $\alpha$ generalized likelihood-ratio test of the hypothesis $H_{0}: \mu=\mu_{0}$ against $H_{1}: \mu \neq \mu_{0}$, and show that it is equivalent to the standard $t$-test of size $\alpha$.

[You should state, but need not derive, the distribution of the test statistic.]