= Solution
The <sample mean> $\overline X=T/n$ has <mean> $\theta$ and <variance> $\sigma(\theta)^2=\theta(1-\theta)/n$. The <central limit theorem> gives a <normal approximation> for interior $\theta$ when both expected binomial counts are sufficiently large. Under the <null hypothesis>, a level-$\alpha$ upper-tail cutoff is approximately
$$
\overline X>\theta_0+\sigma(\theta_0)z_\alpha,\qquad\Phi(z_\alpha)=1-\alpha.
$$
To obtain the <power function>, evaluate this same cutoff under the true <statistical parameter> $\theta$, standardizing with $\sigma(\theta)$:
$$
\begin{aligned}
w(\theta)&\approx\Pr_\theta\!\left(\frac{\overline X-\theta}{\sigma(\theta)}>
\frac{\theta_0-\theta+\sigma(\theta_0)z_\alpha}{\sigma(\theta)}\right)\\
&=\boxed{1-\Phi\!\left(\frac{\theta_0-\theta}{\sigma(\theta)}+
\frac{\sigma(\theta_0)}{\sigma(\theta)}z_\alpha\right).}
\end{aligned}
$$
Here $\Phi$ is the <standard normal distribution>'s <cumulative distribution function>. The formula treats the binomial threshold as continuous; exact integer cutoffs and boundary randomization are finite-sample refinements.
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