Write $p=p^*$, $\mu_z=\mathbb E(Y\mid Z=z)$, and $\tau^*=\mu_1-\mu_0$. The summand has finite variance, so the <central limit theorem> gives
$$
\sqrt n(\widetilde\tau-\tau^*)
\xrightarrow{d}N(0,V_{\mathrm{known}}),
$$
where
$$
V_{\mathrm{known}}
=\frac{\mathbb E(Y^2\mid Z=1)}p
+\frac{\mathbb E(Y^2\mid Z=0)}{1-p}
-(\mu_1-\mu_0)^2.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 221 / 2 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-221/2/a/solution