Sample size calculation: comparing two proportions

We can use the following formula for the sample size n:

n = (Z_α/2+Z_β)² * (p₁(1-p₁)+p₂(1-p₂)) / (p₁-p₂)²,

where Z_α/2 is the critical value of the Normal distribution at α/2 (e.g. for a confidence level of 95%, α is 0.05 and the critical value is 1.96), Z_β is the critical value of the Normal distribution at β (e.g. for a power of 80%, β is 0.2 and the critical value is 0.84) and p₁ and p₂ are the expected sample proportions of the two groups.