= Solution
Let $z=\Phi^{-1}(1-\alpha/2)$ for the standard normal cumulative distribution function. Consistency of the <maximum-likelihood estimator> and <Slutsky's theorem> give the asymptotic standardization
$$
\sqrt{I_n(\widehat\theta)}(\widehat\theta-\theta)\xrightarrow{d}N(0,1).
$$
Its <probability> of lying between $-z$ and $z$ tends to $1-\alpha$. Solving those inequalities for the unknown parameter yields the <Wald confidence interval>
$$
\boxed{\left[\widehat\theta-\frac{z}{\sqrt{I_n(\widehat\theta)}},\quad
\widehat\theta+\frac{z}{\sqrt{I_n(\widehat\theta)}}\right].}
$$
Thus the estimated <standard error> is $I_n(\widehat\theta)^{-1/2}$. Its coverage is approximate and relies on the large-sample regularity assumptions.
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