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Statistical Tests and Sample Size Selection for Two Population Means

This article explores confidence intervals, point estimators, and statistical tests for comparing two population means. It also covers sample size selection and non-parametric methods for estimation and testing.

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Statistical Tests and Sample Size Selection for Two Population Means

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  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

  3. 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

  4. Two Populations Non-parametric tests

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

  6. 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: nonparametric methods

  7. Two Populations Non-parametric tests

  8. 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:

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

  10. 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

  11. Paired Samples

  12. 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

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