Past exam of the mathematics course of the University of Cambridge / 2007 / ib / Paper 4 / 19C / iv

Under the null hypothesis, the standardized intercept $Z=\sqrt{n}(\hat{\alpha }-{\alpha }_0)/\sigma$ is standard normal, and $V=\mathrm{RSS}/{\sigma }^2$ is independent with chi-squared distribution of $\nu =n-2$ degrees of freedom. Consequently the Student t test for a simple-regression intercept usesChoose a significance level $\gamma \in (0,1)$ before observing the test result. For the two-sided alternative, reject $H_0$ when $|T|>t_{n-2,1-\gamma /2}$, where the subscript denotes the indicated quantile of the Student t-distribution. The null rejection probability is exactly $\gamma$, independent of the nuisance slope and variance. Equivalently report the p-value $2[1-F_{t_{n-2}}(|T_{\mathrm{obs}}|)]$. The test assumes the normal independent-error model and a positive residual variance estimate. In the intercept-only case with $n>1$, use $s^2=\mathrm{RSS}/(n-1)$ and $t_{n-1}$ instead.