Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 4 / 5J / iii / Solution

Treat make as a categorical predictor, converting it to a factor explicitly if necessary:

The models are nested. The first nested-model F-test compares fit2 with the intercept-only fit1. Adding the five quantitative predictors reduces the residual sum of squares from $8584.0$ to $3349.1$, withThere is overwhelming evidence that these quantitative car properties jointly improve the model.

The second test compares fit3 with fit2. Adding make reduces the residual sum of squares further to $840.8$, withThus manufacturer has a highly significant effect even after adjusting for the five quantitative properties, and fit3 is preferred among these nested models.

There are $93$ observations because the intercept-only model has $92=93-1$ residual degrees of freedom. The make term uses $31$ additional degrees of freedom. By the degrees of freedom of a factor predictor, a factor with $m$ represented levels contributes $m-1$ degrees of freedom when an intercept is present. Henceunique manufacturers occur in the dataset.