Report

Chapter 19 Two Sample Problems 2/14/2015 Two Sample Problems 1 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 2/14/2015 Two Sample Problems 2 Sampling Distribution of Mean Difference • Figure shows sampling distribution of xbar1 − xbar2 • Standard error of xbar1 − xbar2 is: 2 1 2 2 s s SE n1 n2 2/14/2015 s12 s22 n1 n2 x1 x 2 m 1−m 2 Two Sample Problems 4 Two-Sample t-test • H0: μ1 = μ2 versus Ha: μ1 ≠ μ2 (two-sided) or Ha: μ1 > μ2 (one-sided, right) or Ha: μ1 < μ2 (one-sided, left) • Test statistic x1 x2 s12 s22 t where SEx1 x2 SEx1 x2 n1 n2 • Conservative estimate for df = the smaller of (n1−1) or (n2−1) 2/14/2015 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 Group xbar s n exhaled in 6 seconds) Exposed 75.5 9.3 30 • Measure FVC in exposed and Non-exp 88.2 15.1 30 nonexposed children 2/14/2015 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 t x1 x2 s12 s22 n1 n2 75.5 88.2 12.7 3.922 3.238 9.32 15.12 30 30 df n 1 30 1 29 2/14/2015 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 2/14/2015 Two Sample Problems 8 C Level Conf. Int. for μ1 − μ2 ( x1 x2 ) t SE x1 x2 where SE x1 x2 2 s1 n1 2 s2 n2 and df the smaller of (n1 1) or (n2 1) 2/14/2015 Two Sample Problems 9 CI for “Smoking Mom’s” Example 95% confidence for example, use t* = 2.045 SE 2 2 s12 s22 9.3 15 .1 3.238 n1 n2 30 30 ( x1 x2 ) (t )( SE) (75 .5 88 .2) (2.045 )(3.238 ) 12 .7 6.6 (19 .3,6.1) 2/14/2015 Two Sample Problems 10 CI for “Smoking Mom’s” Example 90% confidence for example, use t* = 1.699 ( x1 x2 ) t s12 s 22 n1 n2 9.32 15 .12 (75 .5 88 .2) (1.699 ) 30 30 12 .7 5.5 (18 .2,7.2) 2/14/2015 Two Sample Problems 11 Which type of test? One-sample? paired samples? two samples? • Comparing vitamin content of bread, • Is blood pressure altered by use immediately after baking versus 3 of an oral contraceptive? days later (the same loaves are used Comparing a group of women on day one and 3 days later). not using an oral contraceptive • Comparing vitamin content of bread, with a group taking it. immediately after baking versus 3 • Review insurance records for days later (tests made on dollar amount paid after fire independent loaves). damage in houses equipped • Average fuel efficiency for 2005 vehicles is 21 miles per gallon. Is average fuel efficiency higher in the new generation “green vehicles?” with a fire extinguisher versus houses without one. Was there a difference in the average dollar amount paid? 2/14/2015 Two Sample Problems 12