Happiness comes not from material wealth but less desire. 1

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

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

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

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

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

Two-Sample T-Test and CI: C2, C1 Two-sample T for C2 C1 N Mean StDev SE Mean Difference = mu (1) - mu (2) Estimate for difference: % CI for difference: ( , ) T-Test of difference = 0 (vs not =): T-Value = 1.82 P-Value = DF = 6 Both use Pooled StDev = Two-Sample T-Test and CI: C2, C1 Two-sample T for C2 C1 N Mean StDev SE Mean Difference = mu (1) - mu (2) Estimate for difference: % CI for difference: ( , ) T-Test of difference = 0 (vs not =): T-Value = 1.82 P-Value = DF = 5 15

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