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

Under the fitted compound-symmetry covariance model, an orchard mean based on $k$ trees has varianceWith the current estimates this falls from $v_{30}=2+180/30=8$ to $v_{45}=2+180/45=6$. The numbers of orchards contributing to each marginal remain unchanged, so both marginal treatment-contrast variances fall by a factor $6/8=3/4$: pruning differences have estimated variance $3$ and the spray difference has estimated variance $2$.

Adding trees gives a predicted 25% reduction in these variances, but no extra independent orchard replication. It requires 50% more tree measurements and leaves six orchard residual statistical degrees of freedom. Increasing $k$ reduces the $a/k$ term but cannot remove the shared orchard component $c$, so gains eventually diminish.