470 CHAPTER 9 Inferences from Two Samples 2. For the entire sample of subjects, find the mean blood pressure before the treatment and then find the mean after the treatment. 3. Obtain a random sample of subjects and use randomness to separate them into one sample given the treatment and another sample given a placebo. An advantage of using the matched pairs from the first approach is that we reduce the extraneous variation, which could easily occur with different independent samples. The strategy for designing an experiment can be generalized by the following principle of good design: When designing an experiment or planning an observational study, using matched pairs is generally better than using two independent samples. Déjà Vu All Over Again The methods of hypothesis testing in this section are the same methods for testing a claim about a population mean (Section 8-3), except that here we use the differences from the matched pairs of sample data. There are no exact procedures for dealing with matched pairs, but the following approximation methods are commonly used. Inferences About Differences from Matched Pairs Objectives 1. Hypothesis Test: Use the differences from matched pairs to test a claim about the mean of the population of all such differences. 2. Confidence Interval: Use the differences from matched pairs to construct a confidence interval estimate of the mean of the population of all such differences. Notation for Matched Pairs d = individual difference between the two values in a single matched pair md = mean value of the differences d for the population of all matched pairs of data d = mean value of the differences d for the paired sample data sd = standard deviation of the differences d for the paired sample data n = number of pairs of sample data Requirements 1. The sample data are matched pairs. 2. The matched pairs are a simple random sample. 3. Either or both of these conditions are satisfied: The number of pairs of sample data is large (n 7 30) or the pairs of values have differences that are from a population having a distribution that is approximately normal. These methods are robust against departures for normality, so the normality requirement is loose. (If the third requirement is not satisfied, alternatives include the resampling methods of bootstrapping and randomization as described in Section 9-5, or the Sign test as described in Section 13-2.) Test Statistic for Matched Pairs (with H0: Md = 0) t = d - md sd2 n KEY ELEMENTS
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