Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 32 / 1 / b / Solution

Let $z_p={\Phi }^{-1}(p)$ denote a quantile of the standard normal distribution. Reject when $W_n>z_{1-\alpha }$. The statistical power at the specified positive effect isEquating this to $1-\beta$ and using $z_{\beta }=-z_{1-\beta }$ gives $\sqrt{n}{\delta }^{\ast }/\sigma =z_{1-\alpha }+z_{1-\beta }$. Thus, for the usual target $1-\beta >\alpha$,Rounding up ensures at least the target statistical power. This normal-mean sample size calculation assumes a positive integer sample size; if a requested power is at most $\alpha$, every positive sample size already exceeds that target for ${\delta }^{\ast }>0$, and one should not square a negative quantile sum to impose an unnecessary lower bound.