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

Write $Y_{gj}$ for the yield in replicate $j$ of genotype $g\in \{aa,Aa,AA\}$. Model 1 is the one-way cell-means modelThus its three coefficients estimate the three genotype means directly.

Put $D_i=1\{{\text{count}}_i\ge 1\}$. Model 2 isIt imposes ${\mu }_{Aa}={\mu }_{AA}={\beta }_0+{\beta }_1$ and ${\mu }_{aa}={\beta }_0$. 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.