The sample size needed to estimate the difference between two population proportions to within a margin of error E with a confidence level of 1 - α can be found by using the following expression:

Replace n₁ and n₂ by n in the preceding formula (assuming that both samples have the same size) and replace each of p₁, q₁, p₂, and q₂ by 0.5 (because their values are not known). Solving for n results in this expression:

Use this expression to find the size of each sample if you want to estimate the difference between the proportions of men and women who own smartphones. Assume that you want 95% confidence that your error is no more than 0.03.