 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/ii/paper-1/5j/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

Write

Y_{i}

for the request count on day

i

. Both fits are Poisson regression models with independent

Y_{i} \sim Poisson (λ_{i}) .

lo g λ_{i} = β_{0} + β_{W} W_{i} + β_{E} E_{i} + β_{B} B_{i} + β_{L} L_{i} + β_{M} M_{i},

where the variables are indicator variables for winter, weekend, bank holiday, and low or medium pollution; non-winter weekdays that are not bank holidays and have high pollution form the baseline. Model 2 imposes

H_{0} : β_{B} = β_{L} = β_{M} = 0,

retaining only winter and weekend effects.

The models are nested, and the likelihood-ratio test statistic is the decrease in residual deviance,

316.39 - 304.97 = 11.42.

There are three restrictions. By Wilks theorem, under

H_{0}

this statistic is asymptotically a chi-squared distribution with three degrees of freedom. Since

11.42 > χ_{3, 0.99}^{2} = 11.34487,

we reject model 2 at the $1%$ level in favour of model 1.

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