Under $\delta=\delta_A$,
$$
\mathbb ET\simeq
\frac{\delta_A}{\sqrt{\bar\pi(1-\bar\pi)(1/n_1+1/n_0)}},
$$
and
$$
\operatorname{Var}T\simeq
\frac{\pi_1(1-\pi_1)/n_1+\pi_0(1-\pi_0)/n_0}
{\bar\pi(1-\bar\pi)(1/n_1+1/n_0)}.
$$
The <central limit theorem> therefore gives an asymptotic normal distribution with this mean and variance.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 207 / 3 / b / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-207/3/b/ii/solution