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notes509fall11sec45 - Sections 4.5-4.7 Two-Sample Problems...

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Sections 4.5-4.7: Two-Sample Problems Paired t-test (Section 4.6) Examples of Paired Differences studies: • Similar subjects are paired off and one of two treatments is given to each subject in the pair. or • We could have two observations on the same subject. The key: With paired data, the pairings cannot be switched around without affecting the analysis. We typically wish to perform inference about the mean difference, denoted δ . Example 1: Twelve cars were equipped with radial tires and driven over a test course. Then the same 12 cars (with the same drivers) were equipped with regular belted tires and driven over the same course. After each run, the cars’ gas economy (in km/l) was measured. Is there evidence that radial tires produce better fuel economy? (Assume normality of data, and use α = .05.)
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Car Gas eco. | 1 2 3 4 5 6 7 8 9 10 11 12 Y 1 (radial)|4.2 4.7 6.6 7.0 6.7 4.5 5.7 6.0 7.4 4.9 6.1 5.2 Y 2 (belted)|4.1 4.9 6.2 6.9 6.8 4.4 5.7 5.8 6.9 4.7 6.0 4.9 Calculate differences: d = Y 1 Y 2 d i : Example 1(a): Find a 95% CI for the mean difference in gas economy between radial tires and belted tires.
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Interpretation: With 95% confidence: Note: With paired data, the two-sample problem really reduces to a one-sample problem on the sample of differences . Paired t-test in R: > radial <- c(4.2, 4.7, 6.6, 7.0, 6.7, 4.5, 5.7, 6.0, 7.4, 4.9, 6.1, 5.2) > belted <- c(4.1, 4.9, 6.2, 6.9, 6.8, 4.4, 5.7, 5.8, 6.9, 4.7, 6.0, 4.9) > t.test(radial, belted, paired = T, alternative = "greater") Paired t-CI in R: > t.test(radial, belted, paired = T, conf.level=0.95)$conf.int
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Two Independent Samples t-test (Section 4.5) Sometimes there’s no natural pairing between samples.
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