 Solution

The Quasi-Poisson standard errors are trustworthy only if observations are independent, the log-linear mean is correct, and the variance is proportional to the mean with one common dispersion. Dependence, zero inflation, or covariate-dependent dispersion can invalidate this covariance formula.

A parametric bootstrap under model 1 proceeds as follows. Fit the Poisson model once and retain $X$ and the fitted means ${\hat{\mu }}_i$. For bootstrap repetition $b$, independently drawrefit the same Poisson regression to $(X,Y^{\ast (b)})$, and save its gender estimate ${\hat{\beta }}_1^{\ast (b)}$. The sample standard deviation of these estimates over many repetitions estimates the model-1 standard error. This bootstrap deliberately measures uncertainty under the fitted Poisson model; it does not repair real overdispersion unless the resampling model is enlarged to represent its cause.