= Solution
The <quasibinomial regression> retains the <logit link> and mean function $p_i=(1+e^{-x_i^T\beta})^{-1}$ but uses the working <variance function> $\operatorname{Var}(Y_i)=\phi p_i(1-p_i)$. Its <quasi-score equation> is
$$
U(\beta)=\phi^{-1}\sum_i x_i(y_i-p_i)=0,
$$
so multiplying by the common <dispersion parameter> does not change the coefficient estimates. <Iteratively reweighted least squares> therefore gives the same fitted means and coefficients as the binomial model. The usual <Pearson dispersion estimator> is
$$
\widehat\phi=\frac{1}{120-2}\sum_i\frac{(y_i-\widehat p_i)^2}{\widehat p_i(1-\widehat p_i)}.
$$
The estimated coefficient <covariance matrix> is $\widehat\phi(X^TWX)^{-1}$, with $W_{ii}=\widehat p_i(1-\widehat p_i)$. From the intercept <standard errors>, $(1.08921/1.13031)^2\simeq0.92860$; the rounded slope errors give approximately the same factor. The display uses approximate <Student's t-tests> with 118 <residual degrees of freedom> instead of fixed-dispersion normal tests.
For a genuinely individual binary response, the <Bernoulli distribution> identity $Y_i^2=Y_i$ forces $\operatorname{Var}(Y_i)=p_i(1-p_i)$. Thus $\phi\ne1$ is a working <quasi-likelihood> specification, not a different independent binary distribution with that mean. The mild estimated <underdispersion> does not establish an improved model, and the mean predictions are unchanged. In particular, a scalar rescaling does not model correlation among users, and usual likelihood-based <Akaike information criterion> comparisons are unavailable for a family without a specified probability likelihood. \b[The output gives no convincing reason to prefer model3 to model2.]
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