Student t test for a simple-regression intercept ($T=(\hat{\alpha }-{\alpha }_0)/(s\sqrt{1/n+{\bar{x}}^2/S_{xx}})$)

For a Gaussian linear model with an intercept and one explanatory variable, $n>2$ and $S_{xx}>0$, the regression intercept estimator is normal with variance ${\sigma }^2(1/n+{\bar{x}}^2/S_{xx})$. The residual variance estimate $s^2=\mathrm{SSE}/(n-2)$ is independent of the coefficient estimators and $(n-2)s^2/{\sigma }^2$ has a chi-squared distribution with $n-2$ degrees of freedom. Consequently the displayed statistic has a Student t-distribution under $\alpha ={\alpha }_0$, and a two-sided level-$\eta$ test rejects for $|T|>t_{n-2,1-\eta /2}$.

// Target: mathematical-optimization.bigb