= Solution
{c}
The <central limit theorem> gives
$$
\frac Yn\ \mathrel{\dot\sim}\ N\!\left(\frac1\theta,\frac1{n\theta^2}\right),\qquad
\sqrt n\left(\frac{\theta Y}n-1\right)\ \mathrel{\dot\sim}\ N(0,1).
$$
Write $z=z_{1-\alpha/2}$ for the indicated <standard normal quantile>. Inverting the approximate <pivot> inequality and substituting $Y/n=1/\hat\theta$ gives the approximate <confidence interval>
$$
\boxed{\left[\hat\theta\left(1-\frac z{\sqrt n}\right),\quad
\hat\theta\left(1+\frac z{\sqrt n}\right)\right]\cap(0,\infty).}
$$
Its coverage is approximately $1-\alpha$ for large $n$. The intersection respects the positive rate parameter; asymptotically the lower endpoint is positive. This is an asymptotic interval, not an exact finite-sample <confidence interval>.
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