Keep in mind that the normal distribution can be employed to approximate the binomial distribution in certain situations. Specially, the approximation was considered good if both np and nq were at least 5. Well, now, we are talking regarding two proportions, therefore np and nq should be at least 5 for both samples.
We do not have a way to specifically test the two proportions for values, what we have is the ability to test the difference among 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 as well be computing an average proportion and termed it p-bar. This is the total number of successes divided by total number of trials. The definitions that are essential are shown to the right.
The test statistic consists similar general pattern as before (examined minus expected divided by the standard error). The test statistic employed here is similar to that for the single population proportion, apart from the difference of proportions are used rather than a single proportion, and the value of p-bar is employed rather than p in standard error portion.
As we are using the normal approximation to binomial, the difference of proportions consists of a normal distribution. The test statistic is provided as.
Some of the people will be tempted to try simplifying the denominator of this test statistic wrongly. This can be simplified; however the correct simplification is not to simply the place, the product of p-bar and q-bar over the sum of n's. Keep in mind that to add up fractions, you should have a common denominator, that is why this simplification is wrong.
The right simplification would be to factor a p-bar and q-bar out of two expressions. This is generally the formula given, as it is simpler to compute, however to give it the other way first therefore you could compare it to the other formulas and see how identical they all are.
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