Past exam of the statistics course of the University of Oxford / 2014 / 2 / b

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 approximatelyTo obtain the power function, evaluate this same cutoff under the true statistical parameter $\theta$, standardizing with $\sigma (\theta )$: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.

Here $z_{\alpha }=1.96$, $\sigma (1/2)=1/(2\sqrt{n})$ and $\sigma (3/4)=\sqrt{3}/(4\sqrt{n})$. The desired approximate power is at least $0.975=\Phi (1.96)$ exactly when the argument of the upper tail in part (i) is at most $-1.96$:Rearranging and squaring the positive quantities yieldsThis is a normal approximation to a sample size calculation. It is not a guarantee of the exact finite-sample power for an arbitrary unrandomized binomial cutoff; the discrete test size and power can instead be checked using binomial probabilities.