= Solution
Write $\widetilde x_*=(1,x_*^T)^T$, $\widehat\vartheta=(\widehat\alpha,\widehat\gamma^T)^T$, and let $\widehat V$ estimate the asymptotic covariance of $\widehat\vartheta$. Then
$$
\widehat\eta_* =\widetilde x_*^T\widehat\vartheta,
\qquad
\widehat s_*^2=\widetilde x_*^T\widehat V\widetilde x_*.
$$
An asymptotic confidence interval for the mean binary response $p_*=\Phi(\eta_*)$ is
$$
\boxed{\left[
\Phi(\widehat\eta_*-z_{1-\alpha/2}\widehat s_*),
\Phi(\widehat\eta_*+z_{1-\alpha/2}\widehat s_*)
\right].}
$$
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