Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 4 / 17H / c / Solution

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 ${\alpha }^{\prime }=\alpha +\beta \bar{x}$.