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

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- Solution e

Solution

Use the residual variance estimator

s^{2} = \frac{1}{n - 2} \sum_{i = 1}^{n} {Y_{i} - \overset{α}{^}^{'} - \hat{β}^{'} (x_{i} - \overset{x}{ˉ})}^{2} .

Under the Gaussian linear model,

\overset{ˉ}{Y}

is independent of

s^{2}

and

\frac{( n - 2 ) s ^{2}}{σ ^{2}} \sim χ_{n - 2}^{2} .

Consequently the Student t confidence interval for a centered regression intercept follows from

\frac{Y ˉ - α ^{'}}{s / n} \sim t_{n - 2} .

It is

\overset{ˉ}{Y} \pm t_{n - 2, 0.975} \frac{s}{n} .

The displayed pivotal quantity lies between its 2.5% and 97.5% quantiles with probability

0.95

, which proves the stated coverage.

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