Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 4 / 17H (Statistics)

LetDifferentiating the Gaussian log-likelihood, equivalently minimizing the residual sum of squares, gives the ordinary least squares estimatorsassuming $S_{xx}>0$.

The centered model has residual sum of squaresIts slope normal equation gives

The intercept normal equation isBecause the centered predictors sum to zero,Meanwhile $\hat{\alpha }=\bar{Y}-\hat{\beta }\bar{x}$, so the two intercept estimates are unequal in general. The parameters themselves satisfy ${\alpha }^{\prime }=\alpha +\beta \bar{x}$.

Averaging the centered model givesThereforeWriting $z_{0.975}={\Phi }^{-1}(0.975)$, an exact 95% confidence interval is

Use the residual variance estimatorUnder the Gaussian linear model, $\bar{Y}$ is independent of $s^2$ andConsequently the Student t confidence interval for a centered regression intercept follows fromIt isThe displayed pivotal quantity lies between its 2.5% and 97.5% quantiles with probability $0.95$, which proves the stated coverage.