= Normal-mean likelihood ratio with unknown variance
{title2=$\Lambda=(1+t^2/(n-1))^{-n/2}$}
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.
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