= Student t test for a simple-regression intercept
{c}
{title2=$T=(\widehat\alpha-\alpha_0)/(s\sqrt{1/n+\bar x^2/S_{xx}})$}
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 $s^2=\operatorname{SSE}/(n-2)$ 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
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