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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ii/paper-2/5k/solution

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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 5K / Solution by

Codex  0 Created 2026-09-23 Updated 2026-09-25

Write

Y_{g j}

for the yield in replicate

j

of genotype

g \in {aa, A a, AA}

. Model 1 is the one-way cell-means model

Y_{g j} = μ_{g} + ε_{g j}, ε_{g j} \sim iid N (0, σ^{2}) .

Thus its three coefficients estimate the three genotype means directly.

Put

D_{i} = 1 {count_{i} \geq 1}

. Model 2 is

Y_{i} = β_{0} + β_{1} D_{i} + ε_{i}, ε_{i} \sim iid N (0, σ^{2}) .

It imposes

μ_{A a} = μ_{AA} = β_{0} + β_{1}

and

μ_{aa} = β_{0}

. The reported finite-sample p-values assume independent normal errors, a common unknown variance in every genotype, fixed full-rank design matrices, and correctness of the relevant mean model. Random assignment of genotypes supports interpreting the fitted differences as treatment effects.

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