Test
$$
H_0:\tau^2=0
\qquad\text{against}\qquad
H_1:\tau^2>0
$$
with a <likelihood-ratio test>. Model1 has three likelihood parameters and model2 has four, so their AIC values give
$$
-2\ell_1=-42.874-2(3)=-48.874,
\qquad
-2\ell_2=-45.366-2(4)=-53.366.
$$
The statistic is
$$
2(\ell_2-\ell_1)=4.492.
$$
Using 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 $\tfrac12\chi_0^2+\tfrac12\chi_1^2$, 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.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-218/1/e/solution