= Solution
The <generalized linear mixed model> tries to explain overdispersion by replacing the fixed minority coefficient with a Gaussian <random intercept>. Conditional on the group effect $b_m$,
$$
Y_i\sim\operatorname{Poisson}(\mu_i),
\qquad
\log\mu_i=\beta_0+\beta_1g_i+b_{m_i},
\qquad b_0,b_1\sim N(0,\tau^2).
$$
This is a poor use of a random effect because minority has only two levels. Two realized intercepts contain almost no information about a random-effects distribution or its variance, and the two levels are substantively fixed categories rather than a sample from a population of groups. The model also has worse <Akaike information criterion> than model 1, $1132.8>1122.3$, and its fit does not establish that the original overdispersion has disappeared.
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