Normal-mean sample size calculation ($n=\lceil {\sigma }^2(z_{1-\alpha }+z_{1-\beta }{)}^2/({\delta }^{\ast }{)}^2\rceil$)

For a one-sided normal-mean test of size $\alpha$ with known standard deviation $\sigma$, a positive alternative ${\delta }^{\ast }$ and target statistical power $1-\beta >\alpha$, the required sample size is the ceiling of ${\sigma }^2(z_{1-\alpha }+z_{1-\beta }{)}^2/({\delta }^{\ast }{)}^2$. This follows by shifting the standard normal quantile in the alternative rejection probability.