Statistical Inferences for Two Population Means: Confidence Intervals and Tests

1. Happiness comes not from material wealth but less desire.

2. Inferences Comparing Two Population Means Chapter 6 Confidence intervals Statistical tests Sample size selection

5. Estimation for m1-m2 • Point estimator • Confidence interval • Normal populations with known s1,s2, or two large samples (n1,n2>30): Z interval • Normal populations with unknown s1,s2: t interval • s1=s2: pooled t interval • s1=s2: approximate t interval • At least one nonnormal population and at least one small sample: out of our scope

6. Two Populations Non-parametric tests

9. Sample Size for Estimating m1-m2 Where E is the largest tolerable error and s1=s2=s. n is the sample size per sample.

10. Tests for m1-m2 = d0 • Normal populations with known s1,s2, or two large samples (n1,n2>30): Z test • Normal populations with unknown s1,s2: t test • s1=s2: pooled t test • s1=s2: approximate t test • At least one nonnormal population and at least one small sample: out of our scope

11. Two Populations Non-parametric tests

13. Sample Size for Testing m1-m2 When n1=n2=n and s1=s2=s the type II error rate must be <b if |m1-m2|>=D One-tailed tests: Two-tailed tests:

14. Minitab: stat>>basic statistics>>2 sample t …

15. Two-Sample T-Test and CI: C2, C1 • Two-sample T for C2 • C1 N Mean StDev SE Mean • 1 4 5.88 1.04 0.52 • 2 4 4.22 1.50 0.75 • Difference = mu (1) - mu (2) • Estimate for difference: 1.66442 • 95% CI for difference: (-0.56935, 3.89819) • T-Test of difference = 0 (vs not =): T-Value = 1.82 P-Value = 0.118 DF = 6 • Both use Pooled StDev = 1.2910 • Two-Sample T-Test and CI: C2, C1 • Two-sample T for C2 • C1 N Mean StDev SE Mean • 1 4 5.88 1.04 0.52 • 2 4 4.22 1.50 0.75 • Difference = mu (1) - mu (2) • Estimate for difference: 1.66442 • 95% CI for difference: (-0.68224, 4.01108) • T-Test of difference = 0 (vs not =): T-Value = 1.82 P-Value = 0.128 DF = 5

16. Paired Samples

18. Non-parametric Methods • Independent samples: Wilcoxon Rank Sum Test (also called Manny-Whitney test) • Assumption: distributions of the same shape • Paired samples: Wilcoxon Signed-Rank Test • Assumption: symmetric distribution of the differences • Examples: See Lab 3