 Solution

ID: past-exam-of-the-options-i-course-of-the-university-of-oxford/2004/o/o2/iii/solution

Independence of the slope estimate and residual variance, together with their normal and chi-square laws, gives the Student pivotal quantity

\frac{β - β}{s / S _{xx}} \sim t_{n - 2} .

Thus an exact Gaussian-model two-sided 95% confidence interval is

β \pm t_{n - 2, 0.975} \frac{s}{S _{xx}} .

Using a normal quantile without estimating the variance uncertainty would not give this exact finite-sample coverage.

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