= 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 $\widehat\mu_i$. For bootstrap repetition $b$, independently draw
$$
Y_i^{*(b)}\sim\operatorname{Poisson}(\widehat\mu_i),
$$
refit the same Poisson regression to $(X,Y^{*(b)})$, and save its gender estimate $\widehat\beta_1^{*(b)}$. 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.
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