= Solution
A <linear combination> of <independent> normal <random variables> is normal, with <variances> adding when the coefficients are $1,-1$. Applying the joint central limit approximation from (i) gives
$$
\boxed{\hat p-\hat q\approx N\left(p-q,\frac{p(1-p)}{2M}+\frac{q(1-q)}{2N}\right).}
$$
Equivalently the difference minus $p-q$, divided by the displayed <standard deviation>, converges in distribution to a <standard normal> under the same regular large-sample regime.
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