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Happiness comes not from material wealth but less desire. 1

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

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Estimation for Point estimator Confidence interval 1. Normal populations with known , or two large samples (n1,n2>30): Z interval 2. Normal populations with unknown t interval – : pooled t interval – : approximate t interval 3. At least one nonnormal population and at least one small sample: out of our scope 5

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6 Two Populations Non-parametric tests

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Sample Size for Estimating Where E is the largest tolerable error and . n is the sample size per sample. 9

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Tests for d0 1. Normal populations with known , or two large samples (n1,n2>30): Z test 2. Normal populations with unknown t test – : pooled t test – : approximate t test 3. At least one nonnormal population and at least one small sample: nonparametric methods 10

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11 Two Populations Non-parametric tests

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Sample Size for Testing When n1=n2=n and the type II error rate must be < if One-tailed tests: Two-tailed tests: 13

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

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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 15

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Paired Samples 16

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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 18

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