Dealing with differences of Proportions, p1- p2, under the Null Hypothesis, H0: p1- p2= 0, is advisable, current and conform the rule, to estimate the observed difference as p=(x1+x2)/(n1+n2), because this pooled proportion is simply the total of observed successes divided by the total of trials and no further conditions are added. However, is arguable, though the estimate is fair indeed. Would not be more rational/accurate to weight pw= (n1*x1+n2*x2)/(n1+n2)^2, in order to ascribe more importance to the larger size in comparison with the shorter one, I ask.
By the other hand it seems necessary to emphasize, in view of some statements here occurring, that NHST doesn´t answer, they was not invented to this purpose, though the Null Hypothesis is true or false, but there are a lot of people does not agree and are not only the Psychologists that felt in this fatal misunderstanding. It can be said that Statisticians, but not all Users, are well aware of the ?scandalous? fact that NHST is restricted to provide (or not) sufficient evidence to reject the Null because it is extremely unlike to be true, or we had the bad luck to draw a so odd data from a Population following the Null, that leads the unlucky technician to make a Type I error, persuaded that H0 should be rejected.
Once upon a time . . . a Drug Company stated that a new drug is equally effective to prevent colds for men and woman. Out of 100 women volunteers 38% were infected, out of 200 men, 51%. In consequence we get Z = -2.128, and weighting data Zw= -2.394. Conclusion: It is not dramatic this difference as expected with large sample sizes, the difference woman/man does clearly exist and is a little more evident by the latter method that provides a narrower semi-amplitude for the difference of proportions, 0.054*|Z(alpha/2)|, 88.9% relative to the ordinary 0.061*|Z(alpha/2)|.