Factorial design 2026-10-07
A factorial design includes every combination of specified factor levels. Balanced replication permits separate estimation of marginal effects and interaction terms. The factor allocation, not the number of subsamples, determines the appropriate error term for each effect.
Past exam of the mathematics course of the University of Cambridge 2013 iii Paper 35 4 a Solution Created 2026-10-03 Updated 2026-10-07
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:
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.