 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 $b_m$,

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.