Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 35 / 4 / e / Solution

Put $a={\sigma }^2(1-\rho )$ and $c={\sigma }^2\rho$. The tree and orchard residual mean squares in ANOVA have expectations $a$ and $a+30c$. Equating these to the observed values gives the method-of-moments variance component estimatesIn particular ${\hat{\sigma }}^2=\hat{a}+\hat{c}=182$ and the estimated intraclass correlation coefficient is $\hat{\rho }=2/182=1/91$. These are point estimates from the current experiment, rather than guaranteed variance values for the next season.