1. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 1 Lecture Slides Elementary Statistics Eleventh Edition and the Triola Statistics Series by Mario F. Triola

2. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 2 Chapter 13 Nonparametric Statistics 13-1Review and Preview 13-2Sign Test 13-3Wilcoxon Signed-Ranks Test for Matched Pairs 13-4Wilcoxon Rank-Sum Test for Two Independent Samples 13-5Kruskal-Wallis Test 13-6Rank Correction 13-7Runs Test for Randomness

3. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 3 Section 13-2 Sign Test

4. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 4 Key Concept The main objective of this section is to understand the sign test procedure, which involves converting data values to plus and minus signs, then testing for disproportionately more of either sign.

5. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 5 Definition 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.

6. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 6 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.

7. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 7 Figure 13-1 Sign Test Procedure

8. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 8 Figure 13-1 Sign Test Procedure

9. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 9 Figure 13-1 Sign Test Procedure

10. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 10 Requirements The sample data have been randomly selected. Note: There is no requirement that the sample data come from a population with a particular distribution, such as a normal distribution.

11. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 11 Notation for Sign Test x = the number of times the less frequent sign occurs n = the total number of positive and negative signs combined

12. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 12 Test Statistic For : x (the number of times the less frequent sign occurs) For :

13. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 13 Critical Values For, critical x values are in Table A-7. For, critical z values are in Table A-2.

14. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 14 Caution When applying the sign test in a one-tailed test, we need to be very careful to avoid making the wrong conclusion when one sign occurs significantly more often than the other, but the sample data contradict the alternative hypothesis. See the following example.

15. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 15 Claims Involving Matched Pairs When using the sign test with data that are matched pairs, we convert the raw data to plus and minus signs as follows: 1.Subtract each value of the second variable from the corresponding value of the first variable. 2.Record only the sign of the difference found in step 1. Exclude ties: that is, any matched pairs in which both values are equal.

16. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 16 Key Concept Underlying This Use of the Sign Test If the two sets of data have equal medians, the number of positive signs should be approximately equal to the number of negative signs.

17. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 17 Example: Table 13-3 includes some of the weights listed in Data Set 3 in Appendix B. Those weights were measured from college students in September and April of their freshman year. Use the sample data in Table 13-3 with a 0.05 significance level to test the claim that there is no difference between the September weights and the April weights. Use the sign test.

18. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 18 : The median of the differences is equal to 0. : The median of the differences is not equal to 0. (in two tails) x = minimum (7, 2) = 2 (From Table 13-3, there are 7 negative signs and 2 positive signs.) Critical value = 1 (From Table A-7 where n = 9 and ) Example:

19. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 19 : The median of the differences is equal to 0. : The median of the differences is not equal to 0. With a test statistic of x = 4 and a critical value of 1, we fail to reject the null hypothesis of no difference. There is not sufficient evidence to warrant rejection of the claim that the median of the differences is equal to 0. Example:

20. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 20 We conclude that the September and April weights appear to be about the same. (If we use the parametric t test for matched pairs (Section 9-4), we conclude that the mean difference is not zero, so the September weights and April weights appear to be different.) The conclusion should be qualified with the limitations noted in the article about the study. Only Rutgers students were used, and study subjects were volunteers instead of being a simple random sample. Example:

21. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 21 Claims Involving Nominal Data The nature of nominal data limits the calculations that are possible, but we can identify the proportion of the sample data that belong to a particular category. Then we can test claims about the corresponding population proportion p.

22. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 22 Example: The Genetics and IVF Institute conducted a clinical trial of its methods for gender selection. As of this writing, 668 of 726 babies born to parents using the XSORT method of gender selection were girls. Use the sign test and a 0.05 significance level to test the claim that this method of gender selection is effective in increasing the likelihood of a baby girl. The procedures are for cases in which. Note that the only requirement is that the sample data are randomly selected. (the proportion of girls is 0.5) (girls are more likely)

23. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 23 Example: Denoting girls by the positive sign (+) and boys by the negative sign (–), we have 668 positive signs and 58 negative signs. Test statistic x = minimum(668, 58) = 58 Test whether 58 boys is low enough to be significant so it is a left-tailed test.

24. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 24 Example: Since, the test statistic is converted to the test statistic x as follows:

25. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 25 Example: With in a left-tailed test, the critical value is. The test statistic is in the critical region bounded by.

26. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 26 Example: We reject the null hypothesis that the proportion of girls is equal to 0.5. There is sufficient evidence to support the claim that girls are more likely with the XSORT method. The XSORT method of gender selection does appear to be effective in increasing the likelihood of a girl.

27. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 27 Claims About the Median of a Single Population The negative and positive signs are based on the claimed value of the median.

28. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 28 Example: Body Temperature Data Set 2 in Appendix B includes measured body temperatures of adults. Use the 106 temperatures listed for 12 A.M. on Day 2 with the sign test to test the claim that the median is less than 98.6ºF. Of the 106 subjects, 68 had temperatures below 98.6ºF, 23 had temperatures above 98.6ºF, and 15 had temperatures equal to 98.6ºF. : Median is equal to 98.6°F. : Median is less than 98.6°F. Since the claim is that the median is less than 98.6°F, the test involves only the left tail.

29. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 29 Example: Body Temperature Discard the 15 zeros. Use ( – ) to denote the 68 temperatures below 98.6°F, and use ( + ) to denote the 23 temperatures above 98.6°F. So n = 91 and x = 23

30. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 30 Example: Body Temperature

31. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 31 Example: Body Temperature We use Table A-2 to get the critical z value of – 1.645. The test statistic of z = –4.61 falls into the critical region. We reject the null hypothesis. We support the claim that the median body temperature of healthy adults is less than 98.6°F.

32. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 32 Example: Body Temperature

33. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 33 Recap In this section we have discussed: Sign tests where data are assigned plus or minus signs and then tested to see if the number of plus and minus signs is equal. Sign tests can be performed on claims involving: Matched pairs Nominal data The median of a single population