Student t confidence interval

A Student t confidence interval replaces unknown Gaussian scale by an independent residual estimate.

For two independent normal simple regressions with centered predictors, common unknown error variance and $m>2$ observations in each, let $S_u=\sum u_i^2$, $S_w=\sum w_i^2$. The estimated slope difference is normal with mean $b-d$ and variance ${\sigma }^2(S_u^{-1}+S_w^{-1})$. Pool the two residual sums of squares and use $s^2=RSS/(2m-4)$. Independence from the estimates gives a Student t-distribution pivot, yielding the displayed two-sided interval of confidence $1-\alpha$.

In a two-parameter Gaussian simple linear regression centered at $\bar{x}$, the intercept estimator is $\bar{Y}$. With residual standard deviation $s$,which gives the usual two-sided interval from the corresponding quantiles.