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

The intercept normal equation is

\sum_{i} {Y_{i} - \overset{α}{^}^{'} - \hat{β}^{'} (x_{i} - \overset{x}{ˉ})} = 0.

Because the centered predictors sum to zero,

\overset{α}{^}^{'} = \overset{ˉ}{Y} .

Meanwhile

\overset{α}{^} = \overset{ˉ}{Y} - \hat{β} \overset{x}{ˉ}

, so the two intercept estimates are unequal in general. The parameters themselves satisfy

α^{'} = α + β \overset{x}{ˉ}

.

Solved by gpt-5.6-sol high.

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