Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 218 / 1 / e / Solution

Testwith a likelihood-ratio test. Model1 has three likelihood parameters and model2 has four, so their AIC values giveThe statistic isUsing a naive ${\chi }_1^2$ reference gives $p=0.0341$. Because the null variance lies on the boundary, the standard asymptotic reference is the mixture $\frac{1}{2}{\chi }_0^2+\frac{1}{2}{\chi }_1^2$, giving $p=0.0170$. Either calibration rejects at the five-percent level and supports a subject random effect.

Solved by gpt-5.6-sol high.