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Chapter 19

Chapter 19. Two Sample Problems. Conditions. Two populations, independent samples Population means and standard deviations not known Population 1: μ 1 , σ 1 Population 2: μ 2 , σ 2 Use t procedures in inferential methods

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Chapter 19

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  1. Chapter 19 Two Sample Problems Two Sample Problems

  2. Conditions • Two populations, independent samples • Population means and standard deviations not known Population 1: μ1, σ1 Population 2: μ2, σ2 • Use t procedures in inferential methods • In theory, both populations Normal, but in practice, not necessary when n large -- see Ch 18 Two Sample Problems

  3. Sampling Distribution of Mean Difference • Figure shows sampling distribution of xbar1 − xbar2 • Standard error of xbar1 − xbar2 is: m1−m2 Two Sample Problems

  4. Two-Sample t-test • H0: μ1 = μ2 versus Ha: μ1≠μ2 (two-sided) orHa: μ1 > μ2 (one-sided, right) or Ha: μ1 < μ2 (one-sided, left) • Test statistic • Conservative estimate for df = the smaller of (n1−1) or (n2−1) Two Sample Problems

  5. Example: “Smoking Moms” • Does parental smoking damage kid’s lungs? • Forced Vital Capacity (FVC) = physiologic measure of lung capacity (volume of air exhaled in 6 seconds) • Measure FVC in exposed and nonexposed children Two Sample Problems

  6. Hypothesis Test (Smoking Moms) Hypotheses H0: μ1= μ2 versus Ha: μ1 <μ2 (one-sided) or Ha: μ1≠ μ2 (two-sided) Test statistic Two Sample Problems

  7. P-value (Smoking Moms) • Use absolute value of t statistic (3.922) • Table C, row for df = 29 • Largest t* in that row = 3.659 (P = 0.0005) • Therefore: P < 0.0005 (one-sided) P < 0.001 (two-sided) • Highly significant evidence against H0 Two Sample Problems

  8. C Level Conf. Int. for μ1 − μ2 Two Sample Problems

  9. CI for “Smoking Mom’s” Example 95% confidence for example, use t* = 2.045 Two Sample Problems

  10. CI for “Smoking Mom’s” Example 90% confidence for example, use t* = 1.699 Two Sample Problems

  11. Comparing vitamin content of bread, immediately after baking versus 3 days later (the same loaves are used on day one and 3 days later). Comparing vitamin content of bread, immediately after baking versus 3 days later (tests made on independent loaves). Average fuel efficiency for 2005 vehicles is 21 miles per gallon. Is average fuel efficiency higher in the new generation “green vehicles?” Is blood pressure altered by use of an oral contraceptive? Comparing a group of women not using an oral contraceptive with a group taking it. Review insurance records for dollar amount paid after fire damage in houses equipped with a fire extinguisher versus houses without one. Was there a difference in the average dollar amount paid? Which type of test? One-sample? paired samples? two samples? Two Sample Problems

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