Solution. Here \(n = 12\), \(SS_T = 574.67\), and the full model gives \(\hat \sigma ^2 = MS_{\mathrm {Res}} = 26.891/8 = 3.361\).

Two of the entries in full, for the model in \(x_1, x_2\) (\(p = 3\)): \begin {align*} C_3 &= \frac {28.332}{3.361} - 12 + 6 = 8.43 - 6 = 2.43 , \\ \mathrm {AIC}_3 &= 12\ln \frac {28.332}{12} + 6 = 12 \times 0.8591 + 6 = 16.31 . \end {align*}

Every criterion chooses the model in \(x_1\) and \(x_2\). It has the largest \(R^2_{\mathrm {adj}}\) and the smallest \(MS_{\mathrm {Res}}\), \(C_p\), AIC, BIC and PRESS. Its \(C_p = 2.43\) is below \(p = 3\), so there is no sign of bias, while every other model with fewer than four parameters has \(C_p\) far above the line (Figure 13). Adding \(x_3\) raises \(R^2\) only from \(0.951\) to \(0.953\), and makes every other criterion worse; PRESS in particular rises by half, from \(46\) to \(69\). The chosen model is \[ \hat y = 5.84 + 1.98x_1 + 1.30x_2 . \] Each extra bag of fertiliser per hectare goes with about two more bags of maize, at a fixed labour input; distance to market adds nothing once fertiliser and labour are known.