Notice that when computing the confidence intervals above that we choose to just replace some of the p terms with \(\tilde{p}\) so that only one p term was left and could be isolated in the middle. There are other ways to deal with this. The easiest is to take the worst case scenario for the p terms in the denominator above. Indeed, the confidence interval is made wider (and therefore more likely to contain the actual p) if the square root term is as large as possible, using basic calculus it is easy to see that p(1-p) is maximized when p = 1/2. Therefore, a second alternative is to create your confidence interval using

and therefore \(E = \frac{z_{ \alpha/2}}{2\sqrt{n}}\text{.}\) This method should be used only when trying to create the roughest and "safest" interval.