= Solution
Let $H_i$ be the number healthy among $m_i=25$ patients and $Y_i=H_i/m_i$. The fitted <grouped-binomial logistic regression> is
$$
H_i\sim\operatorname{Binomial}(m_i,p_i),
\qquad
\operatorname{logit}(p_i)=\beta_0+\beta_S\mathbf1_{\{\mathrm{sex}=M\}}+\beta_T\mathbf1_{\{\mathrm{treatment}=1\}}.
$$
For $y=h/m$ its mass has <exponential dispersion family> form
$$
\Pr(Y=y)=\exp\left\{
\frac{y\theta-b(\theta)}{\phi}+c(y,\phi)
\right\},
$$
where
$$
\theta=\log\frac p{1-p},\qquad b(\theta)=\log(1+e^\theta),
\qquad\phi=\frac1m.
$$
Thus $\mathbb EY=p=b'(\theta)$ and
$$
\operatorname{Var}(Y)=\phi V(p),
\qquad\boxed{V(p)=p(1-p).}
$$
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