= Solution
Given numerical values of $\mu$ and $\sigma^2$, maximize the joint log-likelihood over the response rates $0<p_{jk}<1$:
$$
\sum_{j,k}\left[
S_{jk}\log p_{jk}+(n_{jk}-S_{jk})\log(1-p_{jk})
\right]
-\frac1{2\sigma^2}\sum_j
\left\{
\operatorname{logit}(p_{j1})-\operatorname{logit}(p_{j0})-\mu
\right\}^2.
$$
This is a penalized <binomial regression> problem and can be solved by <Newton method> or another numerical optimizer. One may alternate this maximization with the closed-form updates for $\mu$ and $\sigma^2$ from part i until convergence. The selected solution should have a negative-definite Hessian in the fitted log-odds parameters.
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