 Solution

Put $S={\sum }_i(x_i-\bar{x}{)}^2$ and $q=S/{\sigma }_0^2$. Under the null hypothesis, maximizing over the nuisance mean gives ${\hat{\mu }}_0=\bar{x}$. The unrestricted normal mean and variance maximum-likelihood estimators are $\hat{\mu }=\bar{x}$, $\widehat{{\sigma }^2}=S/n$. Thus, for $S>0$,The likelihood-ratio test statistic is consequentlyThe usual model assumes $n\ge 2$. The event $S=0$ has probability zero under a nondegenerate normal distribution; at that observation the unrestricted likelihood is unbounded as the variance tends to zero, and the ratio has limiting value zero. This is the likelihood-ratio test for a normal variance with unknown mean.