Two groups of successes and failures
We now consider data that are obtained as random samples from two populations, with the sampled individuals being categorised into successes and failures.
Since our model involves only two parameters, π1 and π2, the two groups are the same only if π2 - π1 = 0. The value of π2 - π1 is usually unknown but can be estimated by p2 - p1. However p2 - p1 is a random quantity so its variability must be taken into account when interpreting its value.
Example
Note that the questions do not refer to the specific 100 managers in the study. They ask about differences between male and female managers 'in general'.
We are interested in π2 - π1 rather than p2 - p1, so we need to understand the accuracy of our point estimate.