Although separate samples selected from the two populations seems like a natural experimental design for this problem, this design may not produce the best results. In the salary comparison example discussed in the previous section, it would be reasonable to assume that the salaries of individuals in both populations would be related to experience. It may have been the case that the differences we saw in salaries was due to the fact that salary is strongly related to experience and there was a significant difference in experience between the two groups. We cannot eliminate the possible effect of experience and the confusion in interpretation that it causes by our independent sampling design.
By incorporating this additional variable into our experimental design we can reduce variability and potential confusion, thereby increasing the sensitivity of our test. There are two ways this can be accomplished. One way is to include a measure of experience as a second variable that is recorded for each person selected in the two samples. This design involves a type of analysis called Analysis of Covariance which is beyond the scope of this course. The second way to incorporate this additional variable into the analysis is to design a matched pairs sampling process. In this sampling design we randomly select one individual from the first population, determine the experience level of the person selected, then identify from the second population all individuals who have the same experience and randomly select one of those to be matched with the first individual selected. This selecting and matching process is continued to we obtain matched pairs of individuals, matched according to experience. This paired sampling design removes the effect of experience on salaries, and so any differences that remain between the two groups cannot be due to differences in experience.
Other comparison problems involve naturally occurring pairs. A common experimental design to test for the effect of some treatment is to give a pre-treatment test to a group, apply the treatment, and then give a post-treatment test. The test scores are naturally paired - one pre-treatment and one post-treatment score for each individual. In another example, we may wish to obtain a sample of married couples and then compare the scores of the husbands and wives on a questionnaire each takes. The key difference here between paired sampling and independent sampling is that in this case we are randomly selecting a sample of married couples rather than randomly selecting a sample of males who are husbands and separately selecting a sample of females who happen to be wives.
Let denote the pairs of measurements that are obtained from a paired-sample experiment, and suppose that we wish to test the hypotheses
Example Suppose that we implement a matched pairs sampling design for
the comparison of salaries, matching on experience, and find that in a sample
of 18 matched males and females, their salary information is
, , , , .
Note that the individual group standard deviations, , are not part of the paired-sample test, and the sample mean of the pair differences is