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

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.

Allocate the extra orchards evenly, giving three independent orchards to each of the six combinations. Each orchard still has mean variance $8$ under the fitted model. Each pruning marginal now uses six orchards and each spray marginal nine, givingBoth are two-thirds of their original values. The orchard residual statistical degrees of freedom increase from $12-6=6$ to $18-6=12$.

Extra orchards improve independent replication, give a predicted one-third reduction in contrast variances, and improve error estimation. Both this option and using 45 trees require 540 tree measurements, but extra orchards offer the greater predicted statistical gain under the fitted model. They may cost more to recruit and administer; their suitability and comparability also matter.

This produces a split-plot design: assign spray to whole orchards, with six sprayed and six unsprayed, then independently randomize ten trees to each pruning method inside every orchard. Orchards remain experimental units for spray; individual trees become experimental units for pruning. Pruning contrasts and the spray-by-pruning interaction term now lie in the within-block ANOVA stratum, while spray is tested between orchards.

The between-orchard ANOVA stratum has eleven statistical degrees of freedom, split into one for spray and ten for error. The within-orchard ANOVA stratum has 348, split into two for pruning, two for the interaction term and 344 for error. This pooling of within-orchard error is appropriate under the stated compound-symmetry covariance model; additional orchard-specific pruning variation would need its own variance component rather than this simplified error model.

The shared orchard effect cancels in a pruning difference within an orchard. Its variance is $2a/10$, so averaging across twelve orchards givesThe spray contrast still compares means of six orchards per group, each based on 30 trees, so its estimated variance remains $8/3$. For a difference of pruning differences between the two spray groups, each group's pruning difference has estimated variance $2\cdot 180/(10\cdot 6)=6$, and the resulting interaction contrast has estimated variance $12$, compared with $16$ in the original allocation.

Splitting pruning within orchards improves pruning and interaction precision without extra trees, and gives spray a less sparse error estimate; it does not reduce the spray contrast's variance. This option requires tree-level pruning to be practical without interference between neighboring trees. The numerical gains, like those in the other options, assume the current variance components remain applicable next year.