9-3 Matched Pairs 471 Procedures for Inferences with Matched Pairs 1. Verify that the sample data consist of matched pairs, and verify that the requirements in the preceding Key Elements box are satisfied. 2. Find the difference d for each pair of sample values. (Caution: Be sure to subtract in a consistent manner, such as “before – after”). 3. Find the value of d (mean of the differences) and sd (standard deviation of the differences). 4. For hypothesis tests and confidence intervals, use the same t test procedures used for a single population mean (described in Section 8-3). Equivalent Methods Because the hypothesis test and confidence interval in this section use the same distribution and standard error, they are equivalent in the sense that they result in the same conclusions. Consequently, a null hypothesis that the mean difference equals 0 can be tested by determining whether the confidence interval includes 0. P-Values: P-values are automatically provided by technology or the t distribution in Table A-3 can be used. Use the procedure given in Figure 8-3 on page 380. Critical Values: Use Table A-3 (t distribution). For degrees of freedom, use df = n - 1. Confidence Intervals for Matched Pairs d - E 6 md 6 d + E where E = ta>2 sd2 n (Degrees of freedom: df = n - 1.) Are People Honest About their Weight? EXAMPLE 1 It is a common belief that if you ask someone how much they weigh, you tend to get a number that is somewhat lower than the number that you would get by using a scale to actually weigh them. Listed below are measured and reported weights (lb) of random male subjects (from Data Set 4 “Measured and Reported” in Appendix B). Use a 0.05 significance level to test the claim that for males, the measured weights tend to be higher than the reported weights. SOLUTION REQUIREMENT CHECK We address the three requirements listed earlier in the Key Elements box. (1) The data are matched pairs because each pair of values is from the same subject. (2) The pairs of data are randomly selected. We will consider the data to be a simple random sample. (3) Because the number of pairs of data is n = 8, which is not large, we should check for normality of the differences and we should check for outliers. There are no outliers, and a normal quantile plot shows that the points approximate a straight-line pattern with no other pattern, so the differences satisfy the loose requirement of being from a normally distributed population. All requirements are satisfied. TABLE 9-2 Measured and Reported Weights (lb) Subject 1 2 3 4 5 6 7 8 Measured Weight (lb) 152.6 149.3 174.8 119.5 194.9 180.3 215.4 239.6 Reported Weight (lb) 150 148 170 119 185 180 224 239 - Matched Pairs In the late 1950s, Procter & Gamble introduced Crest toothpaste as the first such product with fluoride. To test the effectiveness of Crest in reducing cavities, researchers conducted experiments with several sets of twins. One of the twins in each set was given Crest with fluoride, while the other twin continued to use ordinary toothpaste without fluoride. It was believed that each pair of twins would have similar eating, brushing, and genetic characteristics. Results showed that the twins who used Crest had significantly fewer cavities than those who did not. This use of twins as matched pairs samples allowed the researchers to control many of the different variables affecting cavities. continued
RkJQdWJsaXNoZXIy NjM5ODQ=