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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2013/iii/paper-30/1/f/solution

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Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 30 / 1 / f / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-07

A natural next fit is quadratic regression, which remains a normal linear model in its unknown regression coefficients:

Y_{i} = β_{0} + β_{1} x_{i} + β_{2} x_{i}^{2} + ε_{i}, ε_{i} \sim iid N (0, σ^{2}) .

(1)

The new design matrix has rows

(1, x_{i}, x_{i}^{2})

and must have matrix rank three; three distinct predictor values suffice. The curved regression residual pattern suggests trying a positive quadratic term, but its sign and adequacy should be checked after fitting. Inspect the new regression residuals to see whether the systematic curvature has disappeared.

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