The mean of the sampling distribution of the difference between two independent proportions
(p1 - p2) is:
The standard error of p1- p2 is:
The sampling distribution of p1- p2 is approximately normal as long as the proportions are not too close to 1 or 0 and the sample sizes are not too small. As a rule of thumb, if n1 and n2 are both at least 10 and neither is within 0.10 of 0 or 1 then the approximation is satisfactory for most purposes. An alternative rule of thumb is that the approximation is good if both Nπ and N(1 - π) are greater than 10 for both π1 and π2.
To see the application of this sampling distribution, assume that 0.8 of high school graduates but only 0.4 of high school drop outs are able to pass a basic literacy test. If 20 students are sampled from the population of high school graduates and 25 students are sampled from the population of high school drop outs, what is the probability that the proportion of drop outs that pass will be as high as the proportion of graduates?