= Normal prediction interval with maximum-likelihood variance
{title2=$\bar X\pm t_{n-1,1-\alpha/2}\widehat\sigma\sqrt{(n+1)/(n-1)}$}
For independent <normal random variables> with unknown common mean and variance, let $\widehat\sigma^2=n^{-1}\sum(X_i-\bar X)^2$ and $n\ge2$. The standardized future difference $X_0-\bar X$ is normal with variance $\sigma^2(1+1/n)$ and is independent of the residual sum of squares. Thus the ratio using the displayed standard error has <Student's t-distribution> with $n-1$ degrees of freedom. This is unconditional repeated-sampling prediction coverage, not a confidence interval for the mean.
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