= Solution
The beta-binomial regression is
$$
P_j\sim\operatorname{Beta}(\mu_j,\theta),\qquad
C_j\mid P_j\sim\operatorname{Bin}(m_j,P_j),
\qquad
\operatorname{logit}(\mu_j)=z_j^T\beta,
$$
where the beta distribution is parameterized by mean $\mu_j$ and variance parameter $\theta$. Marginally,
$$
\operatorname{Var}(C_j/m_j)
=\frac{\mu_j(1-\mu_j)}{m_j}
\left\{1+(m_j-1)\frac{\theta}{1+\theta}\right\}.
$$
It matches part c when $\rho=\theta/(1+\theta)$, so it is appropriate at the mean-variance level for a common nonnegative intraclass correlation.
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