15-1 Introduction

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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.

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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.

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15-2 Sign Test
15-2.1 Description of the Test
If the following hypotheses are being tested
The appropriate P-value is
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15-2 Sign Test
15-2.1 Description of the Test
If the following hypotheses are being tested
The appropriate P-value is
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15-2 Sign Test
15-2.1 Description of the Test
If the following hypotheses are being tested
If r lt n/2, then the appropriate P-value is
If r gt n/2, then the appropriate P-value is
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15-2 Sign Test
Example 15-1
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Example 15-1
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15-2 Sign Test
Example 15-1
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15-2 Sign Test
The Normal Approximation
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15-2 Sign Test
Example 15-2
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15-2 Sign Test
Example 15-2
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15-2 Sign Test
15-2.2 Sign Test for Paired Samples
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15-2 Sign Test
Example 15-3
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15-2 Sign Test
Example 15-3
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15-2 Sign Test
Example 15-3
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15-2 Sign Test
15-2.3 Type II Error for the Sign Test
Figure 15-1 Calculation of ? for the sign test.
(a) Normal distributions. (b) Exponential
distributions
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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

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15-3 Wilcoxon Signed-Rank Test
Example 15-4
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Example 15-4
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15-3 Wilcoxon Signed-Rank Test
Example 15-4
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15-3 Wilcoxon Signed-Rank Test
15-3.2 Large-Sample Approximation
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15-3 Wilcoxon Signed-Rank Test
15-3.3 Paired Observations Example 15-5
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15-3 Wilcoxon Signed-Rank Test
15-3.3 Paired Observations Example 15-5
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15-3 Wilcoxon Signed-Rank Test
15-3.3 Paired Observations Example 15-5
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15-4 Wilcoxon Rank-Sum Test
15-4.1 Description of the Test We wish to test
the hypotheses
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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
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15-4 Wilcoxon Rank-Sum Test
Example 15-6
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15-4 Wilcoxon Rank-Sum Test
Example 15-6
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Example 15-6
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15-4 Wilcoxon Rank-Sum Test
Example 15-6
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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
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15-5 Nonparametric Methods in the Analysis of
Variance
The test statistic is
Computational method
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15-5 Nonparametric Methods in the Analysis of
Variance
Example 15-7
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Example 15-7
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