chapter15 - Introduction to Probability and Statistics and...

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Introduction to Probability Introduction to Probability and Statistics and Statistics Thirteenth Edition Thirteenth Edition Chapter 15 Nonparametric Statistics
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What are What are Nonparametric Statistics? Nonparametric Statistics? In all of the preceding chapters we have focused on testing and estimating parameters associated with distributions. In this chapter we will focus on questions such as: Do two distributions have the same center? Do two distributions have the same shape?
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Why Use Why Use Nonparametric Statistics? Nonparametric Statistics? Parametric tests are based upon assumptions that may include the following: The data have the same variance , regardless of the treatments or conditions in the experiment. The data are normally distributed for each of the treatments or conditions in the experiment . What happens when we are not sure that these assumptions have been satisfied?
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How Do Nonparametric Tests How Do Nonparametric Tests Compare Compare with the Usual with the Usual z, z, t t , and , and F F Tests? Tests? Studies have shown that when the usual assumptions are satisfied, nonparametric tests are about 95% efficient when compared to their parametric equivalents. When normality and common variance are not satisfied, the nonparametric procedures can be much more efficient than their parametric equivalents.
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The Wilcoxon Rank Sum Test The Wilcoxon Rank Sum Test Suppose we wish to test the hypothesis that two distributions have the same center. We select two independent random samples from each population. Designate each of the observations from population 1 as an “ A ” and each of the observations from population 2 as a “ B ”. • If H 0 is true, and the two samples have been drawn from the same population, when we rank the values in both samples from small to large, the A’s and B’s should be randomly mixed in the rankings .
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What happens when What happens when H H 0 0 is true? is true? Suppose we had 5 measurements from population 1 and 6 measurements from population 2. If they were drawn from the same population, the rankings might be like this. ABABBABABBA In this case if we summed the ranks of the A measurements and the ranks of the B measurements, the sums would be similar.
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If the observations come from two different populations, perhaps with population 1 lying to the left of population 2, the ranking of the observations might take the following ordering. What happens if What happens if H H 0 0 is not true? is not true? AAABABABBB In this case the sum of the ranks of the B observations would be larger than that for the A observations.
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How to Implement How to Implement Wilcoxon’s Rank Test Wilcoxon’s Rank Test 1 2 1 1 * 1 ) 1 ( T n n n T - + + = Rank the combined sample from smallest to largest.
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chapter15 - Introduction to Probability and Statistics and...

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