Chapter 19 Two Sample Problems 12242021 Two Sample

Chapter 19 Two Sample Problems 12/24/2021 Two Sample Problems 1

Conditions • Two populations, independent samples • Population means and standard deviations not known Population 1: μ 1, σ1 Population 2: μ 2, σ2 • Use t procedures in inferential methods • In theory, both populations Normal, but in practice, not necessary when n large -see Ch 18 12/24/2021 Two Sample Problems 2

Sampling Distribution of Mean Difference • Figure shows sampling distribution of xbar 1 − xbar 2 • Standard error of xbar 1 − xbar 2 is: m 1 − m 2 12/24/2021 Two Sample Problems 4

Two-Sample t-test • H 0: μ 1 = μ 2 versus Ha: μ 1 ≠ μ 2 (two-sided) or Ha: μ 1 > μ 2 (one-sided, right) or Ha: μ 1 < μ 2 (one-sided, left) • Test statistic • Conservative estimate for df = the smaller of (n 1− 1) or (n 2− 1) 12/24/2021 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 12/24/2021 Two Sample Problems 6

Hypothesis Test (Smoking Moms) Hypotheses H 0: μ 1 = μ 2 versus Ha: μ 1 < μ 2 (one-sided) or Ha: μ 1 ≠ μ 2 (two-sided) Test statistic 12/24/2021 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 H 0 12/24/2021 Two Sample Problems 8

C Level Conf. Int. for μ 1 − μ 2 12/24/2021 Two Sample Problems 9

CI for “Smoking Mom’s” Example 95% confidence for example, use t* = 2. 045 12/24/2021 Two Sample Problems 10

CI for “Smoking Mom’s” Example 90% confidence for example, use t* = 1. 699 12/24/2021 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? 12/24/2021 Two Sample Problems 12