Statistical Inference: A Short Course

11.5 Hypothesis Tests for the Difference of Means When Sampling From Two Dependent Populations: Paired Comparisons

A key assumption made in the preceding section was that random samples were taken from two “independent” (normal) populations. However, if the samples from the two populations are paired (each observation in the first sample is related in some particular way to exactly one observation in the second sample), then, as noted in Section 2 above, the samples are not independent. As we shall now see, the analysis of a paired observation experiment reduces to the application of a single-sample approach to hypothesis testing.

Following the approach of Section 2, let us assume that our paired experiment yields n pairs of observations denoted by where X_i is drawn from the first population and Y_i is drawn from the second population, i = 1, . . ., n. For the ith pair , let where D_i can be viewed as the ith observation on the random variable D. Here each provides us with a measure of the difference between, say, the effectiveness of two separate treatments, where the X_i's depict a set ...