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Nonparametric Methods for Hypothesis Testing and Confidence Intervals

Learn about nonparametric and distribution-free methods for hypothesis testing and constructing confidence intervals without making assumptions about the underlying population distribution.

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Nonparametric Methods for Hypothesis Testing and Confidence Intervals

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  1. 15-1 Introduction • Most of the hypothesis-testing and confidence interval procedures discussed in previous chapters are based on the assumption that we are working with random samples from normal populations. • These procedures are often called parametric methods • In this chapter, nonparametric and distribution free methods will be discussed. • We usually make no assumptions about the distribution of the underlying population.

  2. 15-2 Sign Test 15-2.1 Description of the Test • The sign test is used to test hypotheses about the median of a continuous distribution. • Let R+ represent the number of differences • that are positive.

  3. 15-2 Sign Test 15-2.1 Description of the Test If the following hypotheses are being tested: The appropriate P-value is

  4. 15-2 Sign Test 15-2.1 Description of the Test If the following hypotheses are being tested: The appropriate P-value is

  5. 15-2 Sign Test 15-2.1 Description of the Test If the following hypotheses are being tested: If r+ < n/2, then the appropriate P-value is If r+ > n/2, then the appropriate P-value is

  6. 15-2 Sign Test Example 15-1

  7. Example 15-1

  8. 15-2 Sign Test Example 15-1

  9. 15-2 Sign Test The Normal Approximation

  10. 15-2 Sign Test Example 15-2

  11. 15-2 Sign Test Example 15-2

  12. 15-2 Sign Test 15-2.2 Sign Test for Paired Samples

  13. 15-2 Sign Test Example 15-3

  14. 15-2 Sign Test Example 15-3

  15. 15-2 Sign Test Example 15-3

  16. 15-2 Sign Test 15-2.3 Type II Error for the Sign Test Figure 15-1Calculation of  for the sign test. (a) Normal distributions. (b) Exponential distributions

  17. 15-3 Wilcoxon Signed-Rank Test • The Wilcoxon signed-rank test applies to the case of symmetric continuous distributions. • Under this assumption, the mean equals the median. • The null hypothesis is H0:  = 0

  18. 15-3 Wilcoxon Signed-Rank Test Example 15-4

  19. Example 15-4

  20. 15-3 Wilcoxon Signed-Rank Test Example 15-4

  21. 15-3 Wilcoxon Signed-Rank Test 15-3.2 Large-Sample Approximation

  22. 15-3 Wilcoxon Signed-Rank Test 15-3.3 Paired Observations Example 15-5

  23. 15-3 Wilcoxon Signed-Rank Test 15-3.3 Paired Observations Example 15-5

  24. 15-3 Wilcoxon Signed-Rank Test 15-3.3 Paired Observations Example 15-5

  25. 15-4 Wilcoxon Rank-Sum Test 15-4.1 Description of the Test We wish to test the hypotheses

  26. 15-4 Wilcoxon Rank-Sum Test 15-4.1 Description of the Test Test procedure: Arrange all n1 + n2 observations in ascending order of magnitude and assign ranks. Let W1 be the sum of the ranks in the smaller sample. Let W2 be the sum of the ranks in the other sample. Then: W2 = [(n1 + n2)(n1 + n2 + 1)]/2 – W1

  27. 15-4 Wilcoxon Rank-Sum Test Example 15-6

  28. 15-4 Wilcoxon Rank-Sum Test Example 15-6

  29. Example 15-6

  30. 15-4 Wilcoxon Rank-Sum Test Example 15-6

  31. 15-5 Nonparametric Methods in the Analysis of Variance The single-factor analysis of variance model for comparing a population means is The hypothesis of interest is

  32. 15-5 Nonparametric Methods in the Analysis of Variance The test statistic is Computational method

  33. 15-5 Nonparametric Methods in the Analysis of Variance Example 15-7

  34. Example 15-7

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