1. Slide Slide 1 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Nonparametric Statistics

2. Slide Slide 2 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Definitions  Parametric tests have requirements about the nature or shape of the populations involved.  Nonparametric tests do not require that samples come from populations with normal distributions or have any other particular distributions. Consequently, nonparametric tests are called distribution-free tests. Overview

3. Slide Slide 3 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Advantages of Nonparametric Methods 1. Nonparametric methods can be applied to a wide variety of situations because they do not have the more rigid requirements of the corresponding parametric methods. In particular, nonparametric methods do not require normally distributed populations. 2. Unlike parametric methods, nonparametric methods can often be applied to categorical data, such as the genders of survey respondents. 3. Nonparametric methods usually involve simpler computations than the corresponding parametric methods and are therefore easier to understand and apply.

4. Slide Slide 4 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Disadvantages of Nonparametric Methods 1. Nonparametric methods tend to waste information because exact numerical data are often reduced to a qualitative form. 2. Nonparametric tests are not as efficient as parametric tests, so with a nonparametric test we generally need stronger evidence (such as a larger sample or greater differences) before we reject a null hypothesis.

5. Slide Slide 5 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Efficiency of Nonparametric Methods

6. Slide Slide 6 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Handling Ties in Ranks Find the mean of the ranks involved and assign this mean rank to each of the tied items. Sorted Data 4 5 10 11 12 Rank 1 3 5 6 7.5 Mean is 3. Mean is 7.5. Preliminary Ranking 1 2 3 4 5 6 7 8

7. Slide Slide 7 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Sign Test

8. Slide Slide 8 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Definition Sign Test The sign test is a nonparametric (distribution free) test that uses plus and minus signs to test different claims, including: 1) Claims involving matched pairs of sample data; 2) Claims involving nominal data; 3) Claims about the median of a single population.

9. Slide Slide 9 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Basic Concept of the Sign Test The basic idea underlying the sign test is to analyze the frequencies of the plus and minus signs to determine whether they are significantly different.

10. Slide Slide 10 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Notation for Sign Test x = the number of times the less frequent sign occurs n = the total number of positive and negative signs combined

11. Slide Slide 11 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Test Statistic For n  25 : x (the number of times the less frequent sign occurs) Critical values For n  25, critical x values are in Tables. For n > 25, critical z values are in Tables z = For n > 25 : n ( x + 0.5) – n 2 2

12. Slide Slide 12 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example: Gender Selection Of the 325 babies born to parents using the XSORT method of gender selection, 295 were girls. Use the sign test and a 0.05 significance level to test the claim that this method of gender selection has no effect. The procedures are for cases in which n > 25. Note that the only requirement is that the sample data are randomly selected. H 0 : p = 0.5 (the proportion of girls is 0.5) H 1 : p  0.5

13. Slide Slide 13 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example: Gender Selection Of the 325 babies born to parents using the XSORT method of gender selection, 295 were girls. Use the sign test and a 0.05 significance level to test the claim that this method of gender selection has no effect. Denoting girls by the positive sign (+) and boys by the negative sign (–), we have 295 positive signs and 30 negative signs. Test statistic x = minimum(295, 30) = 30 The test involves two tails.

14. Slide Slide 14 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example: Gender Selection Of the 325 babies born to parents using the XSORT method of gender selection, 295 were girls. Use the sign test and a 0.05 significance level to test the claim that this method of gender selection has no effect. n ( x + 0.5) – z = n 2 2 (30 + 0.5) – z = 325 2 2 = –14.64

15. Slide Slide 15 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example: Gender Selection Of the 325 babies born to parents using the XSORT method of gender selection, 295 were girls. Use the sign test and a 0.05 significance level to test the claim that this method of gender selection has no effect. With  = 0.05 in a two-tailed test, the critical values are z =  1.96. The test statistic z = -14.64 is less than -1.96. We reject the null hypothesis that p = 0.5. There is sufficient evidence to warrant rejection of the claim that the method of gender selection has no effect.

16. Slide Slide 16 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example: Gender Selection Of the 325 babies born to parents using the XSORT method of gender selection, 295 were girls. Use the sign test and a 0.05 significance level to test the claim that this method of gender selection has no effect. Figure 13.2

17. Slide Slide 17 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Wilcoxon Rank-Sum Test for Two Independent Samples

18. Slide Slide 18 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Definition The Wilcoxon rank-sum test is a nonparametric test that uses ranks of sample data from two independent populations. It is used to test the null hypothesis that the two independent samples come from populations with equal medians. H 0 : The two samples come from populations with equal medians. H 1 : The two samples come from populations with different medians.

19. Slide Slide 19 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Basic Concept If two samples are drawn from identical populations and the individual values are all ranked as one combined collection of values, then the high and low ranks should fall evenly between the two samples.

20. Slide Slide 20 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Key Concept This section introduces the Kruskal- Wallis test, which uses ranks of data from three or more independent samples to test the null hypothesis that the samples come from populations with equal medians.

21. Slide Slide 21 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Rank Correlation

22. Slide Slide 22 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Rank Correlation Definition The rank correlation test (or Spearman’s rank correlation test) is a non-parametric test that uses ranks of sample data consisting of matched pairs. It is used to test for an association between two variables, so the null and alternative hypotheses are as follows (where ρ s denotes the rank correlation coefficient for the entire population): H o : ρ s = 0 (There is no correlation between the two variables.) H 1 : ρ s  0 (There is a correlation between the two variables.)

23. Slide Slide 23 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Runs Test for Randomness

24. Slide Slide 24 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Runs Test for Randomness Definitions A run is a sequence of data having the same characteristic; the sequence is preceded and followed by data with a different characteristic or by no data at all. The runs test uses the number of runs in a sequence of sample data to test for randomness in the order of the data.

25. Slide Slide 25 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Fundamental Principles of the Run Test Reject randomness if the number of runs is very low or very high. Example: The sequence of genders FFFFFMMMMM is not random because it has only 2 runs, so the number of runs is very low. Example: The sequence of genders FMFMFMFMFM is not random because there are 10 runs, which is very high. It is important to note that the runs test for randomness is based on the order in which the data occur; it is not based on the frequency of the data.