Statistical Inference: A Short Course

11.2 Confidence Intervals for the Difference of Means When Sampling from Two Dependent Populations: Paired Comparisons

The inferential techniques developed in the preceding section were based upon random samples taken from two “independent” normal populations. What approach should be taken when the samples are either intrinsically or purposefully designed to be “dependent?” A common source of dependence is when 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, so that the two samples are obviously not independent.

Why is pairing advantageous? A glance back at, say, Equation (11.7) reveals that the precision of the interval estimate of μ_X − μ_Y, for fixed sample sizes n_X and n_Y, varies inversely with , the pooled estimate of the common variance . Here serves as a measure of the unexplained variation among experimental units or subjects that receive similar treatments. One way to possibly reduce this variation, and thus increase the precision of our estimate of μ_X − μ_Y, is to pair the sample observations. In this regard, if the variation in the treatment outcomes between ...