The methods for determining a confidence interval for p above depend upon a good approximation with the Central Limit Theorem. This approximation will be fine if n is relatively large. To consider a confidence interval for p when n is small, note that the binomial random variable is discrete and so expanding the interval by a factor of \(\frac{1}{2n}\) might be in order. Indeed, replace \(z_{\alpha/2}\) by \(t_{\alpha/2}(n-1)\) and continue otherwise.