Normal prediction interval with maximum-likelihood variance ($\bar{X}\pm t_{n-1,1-\alpha /2}\hat{\sigma }\sqrt{(n+1)/(n-1)}$)

For independent normal random variables with unknown common mean and variance, let ${\hat{\sigma }}^2=n^{-1}\sum (X_i-\bar{X}{)}^2$ and $n\ge 2$. 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.