Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 37 / 2 / d

Let $h(q)=1$ for $q=0,3,6$ and $h(9)=2$. The new normal linear model allows an age-by-policy interaction:The corner-point constraints are ${\alpha }_{21}={\delta }_{3rd}={\gamma }_F={\xi }_1=0$ and ${\kappa }_{21,p}={\kappa }_{a,3rd}=0$. Compared with the six-parameter reduced additive model, there are two additional interaction coefficients. Its F-test givesThus the interaction model significantly improves the reduced additive model. Estimated premiums decrease with age. Men have an estimated premium $94.375$ higher than women at fixed other characteristics, and 9 points adds $41.708$ relative to 0, 3, or 6. Comprehensive cover costs more than third-party cover, but its increment depends on age: $175.375$ at 21, $148.500$ at 30, and $79.500$ at 40. These are model-based comparisons, including extrapolations into the unobserved age-gender combinations.

For a 40-year-old woman with comprehensive cover and 6 points, the gender and points contrasts vanish. The fitted premium is