The mean of the sampling distribution of the difference between two independent proportions
(p_{1} - p_{2}) is:

.

The standard error of p_{1}- p_{2} is:

.

The sampling distribution of p_{1}- p_{2} 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 n_{1} and n_{2} 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 π_{1} and π_{2}.

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?