Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 102 / 2 / a / Solution

Let $S_i$ be singleton deliveries among $C_i$ treatment cycles. Under a common-probability binomial distribution model, $S_i\sim \mathrm{Bin}(C_i,p)$ independently across years. The pooled maximum-likelihood estimator isThe Pearson chi-squared test of homogeneity compares singleton and non-singleton counts across all fourteen years, usingAll expected cells are large. One probability has been estimated, so the reference chi-squared distribution has 13 degrees of freedom. Numerically,The constant singleton-per-cycle probability is decisively rejected under the independent-binomial model. The observed proportion rises from $1712/18201=0.0941$ to $3626/23794=0.1524$. Dependence between repeated treatments or patient heterogeneity can invalidate literal binomial precision, but they should be addressed by an explicit variance model rather than ignored.