Remember that the normal distribution can be used to approximate the binomial distribution in certain cases. Specifically, the approximation was considered good when np and nq were both at least 5. Well, now, we're talking about two proportions, so np and nq must be at least 5 for both samples.
We don't have a way to specifically test two proportions for values,
what we have is the ability to test the difference between the
proportions. So, much like the test for two means from independent
populations, we will be looking at the difference of the proportions.
We will also be computing an average proportion and calling it p-bar. It is the total number of successes divided by the total number of trials. The definitions which are necessary are shown to the right.
The test statistic has the same general pattern as before (observed minus expected divided by standard error). The test statistic used here is similar to that for a single population proportion, except the difference of proportions are used instead of a single proportion, and the value of p-bar is used instead of p in the standard error portion.
Since we're using the normal approximation to the binomial, the
difference of proportions has a normal distribution. The test
statistic is given.
Some people will be tempted to try to simplify the denominator of this test statistic incorrectly. It can be simplified, but the correct simplification is not to simply place the product of p-bar and q-bar over the sum of the n's. Remember that to add fractions, you must have a common denominator, that is why this simplification is incorrect.
The correct simplification would be to factor a p-bar and q-bar out of the two expressions. This is
usually the formula given, because it is easier to
calculate, but I wanted to give it the other way first
so you could compare it to the other formulas and
see how similar they all are.