Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2013/iii/paper-35/4/a/solution

Both factors are assigned at orchard level. Therefore the experimental units are the twelve orchards; the trees are observational units within them. The six combinations form a balanced factorial design, replicated twice. The orchard ANOVA stratum has statistical degrees of freedom. Spray uses , pruning uses , and their interaction term uses , leaving six for error.
Dividing each treatment sum of squares in ANOVA by its statistical degrees of freedom and using as the denominator gives all missing entries:
Orchard sourceDegrees of freedomMean squareVariance ratio, one significant figure
Spray19984
Pruning25602
Spray by pruning22020.8
Residual6240Not applicable
The unrounded F-test statistics are , and . The within-orchard tree mean square in ANOVA, 180, is not the treatment error denominator: using it would confuse subsampling with independent replication. The tree ANOVA stratum has statistical degrees of freedom; is the uncorrected total, and the corrected total is 359.

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