Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 218 / 4 / c / Solution

Conditionally on the school effect, the response variance is ${\hat{\sigma }}^2=1.6531$. Marginally,and two distinct students in the same school have estimated covariance $0.0437$.

There is no single coefficient estimate for the random effect because the 28 school deviations are modeled as draws from a mean-zero distribution. The summary reports the estimated distributional variance; individual empirical Bayes predictions can be extracted separately.