Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 13L (Statistical Modelling)

The fitted normal linear model isfor $i=1,\dots ,173$.

Let $X$ be the $173\times 3$ design matrix and write $p=3$. The displayed estimate in row $j$ isWithits standard error isThe $t$-value and two-sided $p$-value arerespectively.

The nested-model $F$-test has hypothesesIts statistic isThe reported $p$-value $5.252\times 10^{-6}$ gives strong evidence against $H_0$: weight and width are jointly useful.

Neither individual coefficient is significant at $5\%$. This is consistent with the predictor correlation $0.877$: strong multicollinearity makes the two partial effects difficult to distinguish and inflates their individual standard errors, even though their joint contribution is clear.

With a well-fitting Gaussian linear model, the residual-versus-fitted plot should be an unstructured horizontal band around zero with roughly constant width. The normal Q-Q plot should lie close to a straight line.

Here the residual spread grows with the fitted value and the response's discreteness produces pronounced bands, indicating nonconstant variance. The Q-Q plot bends strongly upward in the upper tail, indicating right-skewness or heavy positive tails and failure of Gaussian residuals. Thus both homoscedasticity and normality are doubtful.

Two natural improvements are:

The first addresses discreteness and mean-dependent variance; the second addresses systematic lack of fit in the conditional mean.