Students need to be able to recognize paired data because even though they may look similar, two-sample tests are inappropriate for paired data. We recall the investigation of the sampling distribution of the mean, repeat that idea with proportions, and discover that, for large enough samples, proportions also have a Normal sampling distribution. We then find results that correspond to all that we have done for means, finding confidence intervals for population proportions and testing hypotheses about their value. Finally, we consider differences between proportions. It is easy to find current examples on the Internet, so this is a good place to have students search for recent surveys and the like and interpret the conclusions, which are usually presented as percentages.
Proportions present one new challenge.
Because the standard deviation of the sampling distribution depends on the mean, we find that hypothesizing a value for the proportion also hypothesizes a value for the standard deviation, leading us to work as if we knew the standard deviation. Conversely, when finding confidence intervals, we must rely on the estimated proportion both in estimating the center of the sampling distribution and in estimating the standard deviation of that distribution, which is now a standard error. Once again, the underlying message is that all of these methods work in essentially the same way. We examine the data in a contingency table and test the difference in the two proportions, finding that we cannot reject the null hypothesis. The dataset returns with data on women added to the table.