 Solution

For $n\ge 2$, use the unbiased residual variance $s^2=\mathrm{RSS}/(n-1)=n{\hat{\sigma }}^2/(n-1)$. Under the null hypothesis, $(\hat{\beta }-{\beta }_0)\sqrt{S_{xx}}/\sigma$ is standard normal and independent of $\mathrm{RSS}/{\sigma }^2\sim {\chi }_{n-1}^2$. HenceFor a chosen significance level $\alpha$, reject the null in favor of the two-sided alternative exactly when $|T|>t_{n-1,1-\alpha /2}$, the stated upper quantile of the Student t-distribution. The exact two-sided p-value is $2P(t_{n-1}\ge |T_{\mathrm{obs}}|)$. This hypothesis test does not substitute the biased MLE for $s^2$ without correcting its factor and does not use a normal critical value when variance is unknown.