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

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