The sample size formula in the case of one proportion is \[n=\frac{n\hat{p}(1-\hat{p})}{m^2}\]. In the planning stage for our study, we may not have good knowledge of what to expect for \(\hat{p}\), so we can take a conservative approach to find the minimum sample size needed regardless of what \(\hat{p}\) is. To do this, it turns out that \(\hat{p} = 0.5\) results in the largest sample size. For a 95% confidence interval, this gives us the worst case sample size of

\[n\approx\frac{4*0.5*(1-0.5)}{m^2} = \frac{1}{m^2}\]