 Normal-mean likelihood ratio with unknown variance

For an independent sample from a normal distribution, maximization of the likelihood over the unknown variance gives the likelihood ratio for a fixed mean against an unrestricted mean as displayed, where $t=\sqrt{n}(\bar{X}-{\mu }_0)/S$ and $S^2={\sum }_i(X_i-\bar{X}{)}^2/(n-1)$. It decreases with $|t|$. Under the null, the normal sample mean is independent of its centered sum of squares, so $t$ has the exact Student t-distribution with $n-1$ degrees of freedom. The resulting two-sided test requires no asymptotic chi-squared approximation.