= Solution
Let $S_i$ be singleton deliveries among $C_i$ treatment cycles. Under a common-probability <binomial distribution> model, $S_i\sim\operatorname{Bin}(C_i,p)$ independently across years. The pooled <maximum-likelihood estimator> is
$$
\widehat p=\frac{\sum_iS_i}{\sum_iC_i}=\frac{40429}{323362}=0.1250271.
$$
The <Pearson chi-squared test of homogeneity> compares singleton and non-singleton counts across all fourteen years, using
$$
X^2=\sum_{i=1}^{14}\left[\frac{(S_i-C_i\widehat p)^2}{C_i\widehat p}
+\frac{((C_i-S_i)-C_i(1-\widehat p))^2}{C_i(1-\widehat p)}\right]
=\sum_{i=1}^{14}\frac{(S_i-C_i\widehat p)^2}{C_i\widehat p(1-\widehat p)}.
$$
All expected cells are large. One probability has been estimated, so the reference <chi-squared distribution> has 13 degrees of freedom. Numerically,
$$
\boxed{X^2=1188.812,\quad \mathrm{df}=13,\quad p\simeq4.52\times10^{-246}.}
$$
\b[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.
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