1. 1 1 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Slides by John Loucks St. Edward’s University

2. 2 2 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Chapter 18 Nonparametric Methods n Sign Test n Wilcoxon Signed Rank Test n Mann-Whitney-Wilcoxon Test n Kruskal-Wallis Test n Rank Correlation

3. 3 3 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Nonparametric Methods n Most of the statistical methods referred to as parametric require the use of interval- or ratio-scaled data. n Nonparametric methods are often the only way to analyze categorical ( nominal or ordinal) data and draw statistical conclusions. n Nonparametric methods require no assumptions about the population probability distributions. n Nonparametric methods are often called distribution- free methods.

4. 4 4 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Nonparametric Methods n Whenever the data are quantitative, we will transform the data into categorical data in order to conduct the nonparametric test.

5. 5 5 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Sign Test n The sign test is a versatile method for hypothesis testing that uses the binomial distribution with p =.50 as the sampling distribution. n We present two applications of the sign test:  A hypothesis test about a population median  A matched-sample test about the difference between two populations

6. 6 6 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Hypothesis Test about a Population Median n We can apply the sign test by: Using a plus sign whenever the data in the sample are above the hypothesized value of the median Using a plus sign whenever the data in the sample are above the hypothesized value of the median Using a minus sign whenever the data in the sample are below the hypothesized value of the median Using a minus sign whenever the data in the sample are below the hypothesized value of the median Discarding any data exactly equal to the hypothesized median Discarding any data exactly equal to the hypothesized median

7. 7 7 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Hypothesis Test about a Population Median n The assigning of the plus and minus signs makes the situation into a binomial distribution application. The sample size is the number of trials. The sample size is the number of trials. There are two outcomes possible per trial, a plus sign or a minus sign. There are two outcomes possible per trial, a plus sign or a minus sign. We let p denote the probability of a plus sign. We let p denote the probability of a plus sign. If the population median is in fact a particular value, p should equal.5. If the population median is in fact a particular value, p should equal.5. The trials are independent. The trials are independent.

8. 8 8 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Hypothesis Test about a Population Median: Small-Sample Case n The small-sample case for this sign test should be used whenever n < 20. n The hypotheses are The population median is different than the value assumed. The population median equals the value assumed. n The number of plus signs is our test statistic. n Assuming H 0 is true, the sampling distribution for the test statistic is a binomial distribution with p =.5. H 0 is rejected if the p -value < level of significance, . H 0 is rejected if the p -value < level of significance, .

9. 9 9 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Example: Potato Chip Sales Hypothesis Test about a Population Median: Smaller Sample Size Lawler’s Grocery Store made the decision to carry Cape May Potato Chips based on the manufacturer’s estimate that the median sales should be $450 per week on a per-store basis. Lawler’s Grocery Store made the decision to carry Cape May Potato Chips based on the manufacturer’s estimate that the median sales should be $450 per week on a per-store basis. Lawler’s has been carrying the potato chips for three months. Data showing one-week sales at 10 randomly selected Lawler’s stores are shown on the next slide. Lawler’s has been carrying the potato chips for three months. Data showing one-week sales at 10 randomly selected Lawler’s stores are shown on the next slide.

10. 10 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Hypothesis Test about a Population Median: Smaller Sample Size StoreNumber56193612812WeeklySales$485 562 562 415 415 860 860 426 426Sign+++StoreNumber63398410244WeeklySales$474 662 662 380 380 515 515 721 721Sign++++ n Example: Potato Chip Sales

11. 11 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Hypothesis Test about a Population Median: Smaller Sample Size n Example: Potato Chip Sales Lawler’s management requested the following hypothesis test about the population median weekly sales of Cape May Potato Chips (using  =.10). Lawler’s management requested the following hypothesis test about the population median weekly sales of Cape May Potato Chips (using  =.10). H 0 : Median Sales = $450 H a : Median Sales ≠ $450 H 0 : p =.50 H a : p ≠.50 In terms of the binomial probability p : In terms of the binomial probability p :

12. 12 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Hypothesis Test about a Population Median: Smaller Sample Size n Example: Potato Chip Sales Number of Plus Signs 012345Probability.0010.0098.0098.0439.0439.1172.1172.2051.2051.2461.2461 Number of Plus Signs 678910Probability.2051.1172.1172.0439.0439.0098.0098.0010.0010 Binomial Probabilities with n = 10 and p =.50

13. 13 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Hypothesis Test about a Population Median: Smaller Sample Size n Example: Potato Chip Sales Because the observed number of plus signs is 7, we begin by computing the probability of obtaining 7 or more plus signs. Because the observed number of plus signs is 7, we begin by computing the probability of obtaining 7 or more plus signs. The probability of 7, 8, 9, or 10 plus signs is: The probability of 7, 8, 9, or 10 plus signs is:.1172 +.0439 +.0098 +.0010 =.1719..1172 +.0439 +.0098 +.0010 =.1719. We are using a two-tailed hypothesis test, so: We are using a two-tailed hypothesis test, so: p -value = 2(.1719) =.3438. p -value = 2(.1719) =.3438. With p -value > , (.3438 >.10), we cannot reject H 0. With p -value > , (.3438 >.10), we cannot reject H 0.

14. 14 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Hypothesis Test about a Population Median: Smaller Sample Size n Conclusion Because the p -value > , we cannot reject H 0. There is insufficient evidence in the sample to reject the assumption that the median weekly sales is $450.

15. 15 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Hypothesis Test about a Population Median: Larger Sample Size n With larger sample sizes, we rely on the normal distribution approximation of the binomial distribution to compute the p -value, which makes the computations quicker and easier. Normal Approximation of the Number of Plus Signs when H 0 : p =.50 Mean:  =.50 n Mean:  =.50 n Distribution Form: Approximately normal for n > 20 Distribution Form: Approximately normal for n > 20

16. 16 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Hypothesis Test about a Population Median: Larger Sample Size H 0 : Median Age = 34 years H a : Median Age ≠ 34 years n Example: Trim Fitness Center A hypothesis test is being conducted about the median age of female members of the Trim Fitness Center. In a sample of 40 female members, 25 are older than 34, 14 are younger than 34, and 1 is 34. Is there sufficient evidence to reject H 0 ? Use  =.05.

17. 17 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Hypothesis Test about a Population Median: Larger Sample Size n Example: Trim Fitness Center Letting x denote the number of plus signs, we will use the normal distribution to approximate the binomial probability P ( x < 25). Letting x denote the number of plus signs, we will use the normal distribution to approximate the binomial probability P ( x < 25). Remember that the binomial distribution is discrete and the normal distribution is continuous. Remember that the binomial distribution is discrete and the normal distribution is continuous. To account for this, the binomial probability of 25 is computed by the normal probability interval 24.5 to 25.5. To account for this, the binomial probability of 25 is computed by the normal probability interval 24.5 to 25.5.

18. 18 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. p -Value = 2(1.0000 .9726) =.0548  =.5( n ) =.5(39) = 19.5  =.5( n ) =.5(39) = 19.5 Hypothesis Test about a Population Median: Larger Sample Size n p -Value z = ( x –  )/  = (25.5 – 19.5)/3.1225 = 1.92 n Test Statistic n Mean and Standard Deviation

19. 19 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Hypothesis Test about a Population Median: Larger Sample Size n Rejection Rule n Conclusion Do not reject H 0. The p -value for this two-tail test is.0548. There is insufficient evidence in the sample to conclude that the median age is not 34 for female members of Trim Fitness Center. Using.05 level of significance: Using.05 level of significance: Reject H 0 if p -value <.05 Reject H 0 if p -value <.05

20. 20 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Hypothesis Test with Matched Samples n A common application of the sign test involves using a sample of n potential customers to identify a preference for one of two brands of a product. n The objective is to determine whether there is a difference in preference between the two items being compared. n To record the preference data, we use a plus sign if the individual prefers one brand and a minus sign if the individual prefers the other brand. n Because the data are recorded as plus and minus signs, this test is called the sign test.

21. 21 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Hypothesis Test with Matched Samples: Small-Sample Case n The small-sample case for the sign test should be used whenever n < 20. n The hypotheses are A preference for one brand over the other exists. No preference for one brand over the other exists. n The number of plus signs is our test statistic. n Assuming H 0 is true, the sampling distribution for the test statistic is a binomial distribution with p =.5. H 0 is rejected if the p -value < level of significance, . H 0 is rejected if the p -value < level of significance, .

22. 22 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Hypothesis Test with Matched Samples: Small-Sample Case Maria Gonzales is the supervisor responsible for Maria Gonzales is the supervisor responsible for scheduling telephone operators at a major call center. She is interested in determining whether her operators’ preferences between the day shift (7 a.m. to 3 p.m.) and evening shift (3 p.m. to 11 p.m.) are different. n Example: Major Call Center Maria randomly selected a sample of 16 operators Maria randomly selected a sample of 16 operators who were asked to state a preference for the one of the two work shifts. The data collected from the sample are shown on the next slide.

23. 23 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Hypothesis Test with Matched Samples: Small-Sample Case n Example: Major Call Center Worker12345678 ShiftPreferenceDayEveningEveningEveningDayEveningDay(none) Sign+++Worker 910111213141516 ShiftPreferenceEvening Evening EveningEvening(none)EveningDayEveningEvening Sign+

24. 24 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 4 plus signs 10 negative signs ( n = 14) 4 plus signs 10 negative signs ( n = 14) Hypothesis Test with Matched Samples: Small-Sample Case n Example: Major Call Center Can Maria conclude, using a level of significance of  =.10, that operator preferences are different for the two shifts? Can Maria conclude, using a level of significance of  =.10, that operator preferences are different for the two shifts? A preference for one shift over the other does not exist. A preference for one shift over the other does exist.

25. 25 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Number of Plus Signs 01234567Probability.00006.00085.00085.00555.02222.02222.06110.06110.12219.12219.18329.18329.20947 Number of Plus Signs 8 91011121314Probability.18329.12219.12219.06110.06110.02222.02222.00555.00555.00085.00006 Binomial Probabilities with n = 14 and p =.50 Hypothesis Test with Matched Samples: Small-Sample Case n Example: Major Call Center

26. 26 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Because the observed number of plus signs is 4, we begin by computing the probability of obtaining 4 or less plus signs. Because the observed number of plus signs is 4, we begin by computing the probability of obtaining 4 or less plus signs. The probability of 0, 1, 2, 3, or 4 plus signs is: The probability of 0, 1, 2, 3, or 4 plus signs is:.00006 +.00085 +.00555 +.02222 +.06110 =.08978..00006 +.00085 +.00555 +.02222 +.06110 =.08978. We are using a two-tailed hypothesis test, so: We are using a two-tailed hypothesis test, so: p -value = 2(.08978) =.17956. p -value = 2(.08978) =.17956. With p -value > , (.17956 >.10), we cannot reject H 0. With p -value > , (.17956 >.10), we cannot reject H 0. Hypothesis Test with Matched Samples: Small-Sample Case n Example: Major Call Center

27. 27 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Conclusion Because the p -value > , we cannot reject H 0. There is insufficient evidence in the sample to conclude that a difference in preference exists for the two work shifts. Hypothesis Test with Matched Samples: Small-Sample Case

28. 28 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Using H 0 : p =.5 and n > 20, the sampling distribution for the number of plus signs can be approximated by a normal distribution. n When no preference is stated ( H 0 : p =.5), the sampling distribution will have: n The test statistic is: H 0 is rejected if the p -value < level of significance, . H 0 is rejected if the p -value < level of significance, . Mean:  =.50 n Standard Deviation: ( x is the number of plus signs) of plus signs) Hypothesis Test with Matched Samples: Large-Sample Case

29. 29 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Example: Ketchup Taste Test As part of a market research study, a sample of 80 consumers were asked to taste two brands of ketchup and indicate a preference. Do the data shown on the next slide indicate a significant difference in the consumer preferences for the two brands? Hypothesis Test with Matched Samples: Large-Sample Case

30. 30 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 45 preferred Brand A Ketchup (+ sign recorded) (+ sign recorded) 27 preferred Brand B Ketchup ( _ sign recorded) ( _ sign recorded) 8 had no preference 8 had no preference n Example: Ketchup Taste Test The analysis will be based on a sample size of 45 + 27 = 72. The analysis will be based on a sample size of 45 + 27 = 72. Hypothesis Test with Matched Samples: Large-Sample Case

31. 31 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Hypotheses A preference for one brand over the other exists No preference for one brand over the other exists Hypothesis Test with Matched Samples: Large-Sample Case

32. 32 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Sampling Distribution for Number of Plus Signs  =.5(72) = 36  =.5(72) = 36 Hypothesis Test with Matched Samples: Large-Sample Case

33. 33 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. p -Value = 2(1.0000 -.9875) =.025 n Rejection Rule n p -Value z = ( x –  )/  = (45.5 - 36)/4.243 = 2.24 n Test Statistic Using.05 level of significance: Reject H 0 if p -value <.05 Hypothesis Test with Matched Samples: Large-Sample Case

34. 34 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Conclusion Because the p -value < , we can reject H 0. There is sufficient evidence in the sample to conclude that a difference in preference exists for the two brands of ketchup. Hypothesis Test with Matched Samples: Large-Sample Case

35. 35 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Wilcoxon Signed-Rank Test n The Wilcoxon signed-rank test is a procedure for analyzing data from a matched samples experiment. n The test uses quantitative data but does not require the assumption that the differences between the paired observations are normally distributed. n It only requires the assumption that the differences have a symmetric distribution. n This occurs whenever the shapes of the two populations are the same and the focus is on determining if there is a difference between the two populations’ medians.

36. 36 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Wilcoxon Signed-Rank Test Let T  denote the sum of the negative signed ranks. Let T  denote the sum of the negative signed ranks. n If the medians of the two populations are equal, we would expect the sum of the negative signed ranks and the sum of the positive signed ranks to be approximately the same. n Let T + denote the sum of the positive signed ranks. n We use T + as the test statistic.

37. 37 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Sampling Distribution of T + for the Wilcoxon Signed-Rank Test Mean: Mean: Distribution Form: Approximately normal for n > 10 Distribution Form: Approximately normal for n > 10 Wilcoxon Signed-Rank Test

38. 38 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Example: Express Deliveries Wilcoxon Signed-Rank Test A firm has decided to select one of two express A firm has decided to select one of two express delivery services to provide next-day deliveries to its district offices. To test the delivery times of the two services, the To test the delivery times of the two services, the firm sends two reports to a sample of 10 district offices, with one report carried by one service and the other report carried by the second service. Do the data on the next slide indicate a difference in the two services?

39. 39 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Wilcoxon Signed-Rank Test Seattle Los Angeles Boston Cleveland New York Houston Atlanta St. Louis Milwaukee Denver 32 hrs. 30 19 16 15 18 14 10 7 16 25 hrs. 24 15 15 13 15 15 8 9 11 District Office Office OverNight OverNight NiteFlite

40. 40 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Wilcoxon Signed-Rank Test n Hypotheses H 0 : The difference in the median delivery times of the two services equals 0. the two services equals 0. H a : The difference in the median delivery times of the two services does not equal 0. the two services does not equal 0.

41. 41 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Wilcoxon Signed-Rank Test n Preliminary Steps of the Test Compute the differences between the paired observations. Compute the differences between the paired observations. Discard any differences of zero. Discard any differences of zero. Rank the absolute value of the differences from lowest to highest. Tied differences are assigned the average ranking of their positions. Rank the absolute value of the differences from lowest to highest. Tied differences are assigned the average ranking of their positions. Give the ranks the sign of the original difference in the data. Give the ranks the sign of the original difference in the data. Sum the signed ranks. Sum the signed ranks.... next we will determine whether the sum is significantly different from zero.

42. 42 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Wilcoxon Signed-Rank Test Seattle Los Angeles Boston Cleveland New York Houston Atlanta St. Louis Milwaukee Denver 7 6 4 1 2 3 1111 2 2222 5 District Office Office Differ. Differ. |Diff.| Rank Sign. Rank 10 9 7 1.5 4 6 1.5 4  8 +10 +9 +7 +1.5 +4 +6  1.5 +4  +8 T + = 44.0

43. 43 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Wilcoxon Signed-Rank Test n Test Statistic n p -Value p -Value = 2(1.0000 -.9535) =.093

44. 44 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Conclusion Reject H 0. The p -value for this two-tail test is.093. There is insufficient evidence in the sample to conclude that a difference exists in the median delivery times provided by the two services. Wilcoxon Signed-Rank Test n Rejection Rule Using.05 level of significance, Reject H 0 if p -value <.05

45. 45 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Mann-Whitney-Wilcoxon Test n This test is another nonparametric method for determining whether there is a difference between two populations. n This test is based on two independent samples. n Advantages of this procedure are; n It can be used with either ordinal data or quantitative data. n It does not require the assumption that the populations have a normal distribution.

46. 46 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Mann-Whitney-Wilcoxon Test H a : The two populations are not identical H 0 : The two populations are identical n Instead of testing for the difference between the medians of two populations, this method tests to determine whether the two populations are identical. n The hypotheses are:

47. 47 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Mann-Whitney-Wilcoxon Test n Example: Westin Freezers Manufacturer labels indicate the annual energy cost associated with operating home appliances such as freezers. The energy costs for a sample of 10 Westin freezers and a sample of 10 Easton Freezers are shown on the next slide. Do the data indicate, using  =.05, that a difference exists in the annual energy costs for the two brands of freezers?

48. 48 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Mann-Whitney-Wilcoxon Test $55.10 54.50 54.50 53.20 53.20 53.00 53.00 55.50 55.50 54.90 54.90 55.80 55.80 54.00 54.00 54.20 54.20 55.20 55.20$56.10 54.70 54.70 54.40 54.40 55.40 55.40 54.10 54.10 56.00 56.00 55.50 55.50 55.00 55.00 54.30 54.30 57.00 57.00 Westin Freezers Easton Freezers

49. 49 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Hypotheses Mann-Whitney-Wilcoxon Test H a : Annual energy costs differ for the two brands of freezers. brands of freezers. H 0 : Annual energy costs for Westin freezers and Easton freezers are the same. and Easton freezers are the same.

50. 50 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Mann-Whitney-Wilcoxon Test: Large-Sample Case n First, rank the combined data from the lowest to the highest values, with tied values being assigned the average of the tied rankings. n Then, compute W, the sum of the ranks for the first sample. n Then, compare the observed value of W to the sampling distribution of W for identical populations. The value of the standardized test statistic z will provide the basis for deciding whether to reject H 0.

51. 51 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Mann-Whitney-Wilcoxon Test: Large-Sample Case Approximately normal, provided n 1 > 7 and n 2 > 7 n Sampling Distribution of W with Identical Populations Mean Mean Standard Deviation Standard Deviation Distribution Form Distribution Form  W =  n 1 ( n 1 + n 2 + 1)

52. 52 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Mann-Whitney-Wilcoxon Test $55.10 54.50 54.50 53.20 53.20 53.00 53.00 55.50 55.50 54.90 54.90 55.80 55.80 54.00 54.00 54.20 54.20 55.20 55.20 $56.10 54.70 54.70 54.40 54.40 55.40 55.40 54.10 54.10 56.00 56.00 55.50 55.50 55.00 55.00 54.30 54.30 57.00 57.00 Westin Freezers Easton Freezers Sum of Ranks RankRank 86.5123.5 1 2 12 8 15.5 10 17 3 5 13 19 9 7 14 4 18 15.5 11 6 20

53. 53 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Sampling Distribution of W with Identical Populations  W = ½(10)(21) = 105  W = ½(10)(21) = 105 Mann-Whitney-Wilcoxon Test W

54. 54 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Rejection Rule Using.05 level of significance, Reject H 0 if p -value <.05 n Test Statistic n p -Value p -Value = 2(.0808) =.1616 Mann-Whitney-Wilcoxon Test

55. 55 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Mann-Whitney-Wilcoxon Test n Conclusion Do not reject H 0. The p -value > . There is insufficient evidence in the sample data to conclude that there is a difference in the annual energy cost associated with the two brands of freezers.

56. 56 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Kruskal-Wallis Test n The Mann-Whitney-Wilcoxon test has been extended by Kruskal and Wallis for cases of three or more populations. n The Kruskal-Wallis test can be used with ordinal data as well as with interval or ratio data. n Also, the Kruskal-Wallis test does not require the assumption of normally distributed populations. H a : Not all populations are identical H 0 : All populations are identical

57. 57 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Test Statistic Kruskal-Wallis Test where: k = number of populations k = number of populations n i = number of observations in sample I n i = number of observations in sample I n T =  n i = total number of observations in all samples n T =  n i = total number of observations in all samples R i = sum of the ranks for sample i R i = sum of the ranks for sample i

58. 58 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Kruskal-Wallis Test n When the populations are identical, the sampling distribution of the test statistic H can be approximated by a chi-square distribution with k – 1 degrees of freedom. n This approximation is acceptable if each of the sample sizes n i is > 5. The rejection rule is: Reject H 0 if p -value <  The rejection rule is: Reject H 0 if p -value <  n This test is always expressed as an upper-tailed test.

59. 59 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Kruskal-Wallis Test John Norr, Director of Athletics at Lakewood High John Norr, Director of Athletics at Lakewood High School, is curious about whether a student’s total School, is curious about whether a student’s total number of absences in four years of high school is the number of absences in four years of high school is the same for students participating in no varsity sport, same for students participating in no varsity sport, one varsity sport, and two varsity sports. one varsity sport, and two varsity sports. n Example: Lakewood High School Number of absences data were available for 20 Number of absences data were available for 20 recent graduates and are listed on the next slide. recent graduates and are listed on the next slide. Test whether the three populations are identical in Test whether the three populations are identical in terms of number of absences. Use  =.10. terms of number of absences. Use  =.10.

60. 60 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Kruskal-Wallis Test n Example: Lakewood High School

61. 61 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Kruskal-Wallis Test No Sport Rank 1 Sport Rank 2 Sports Rank1381814126.5 1612126.52219 61191593 272072115 2016.51510.51510.5 1492016.52118 1713104 Total66.577.566 n Example: Lakewood High School

62. 62 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Kruskal-Wallis Test Using test statistic: Reject H 0 if  2 > 4.60517 (2 d.f.) Using p -value: Reject H 0 if p -value <.10 n Rejection Rule k = 3 populations, n 1 = 6, n 2 = 7, n 3 = 7, n T = 20 n Kruskal-Wallis Test Statistic

63. 63 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Kruskal-Wallis Test Do no reject H 0. There is insufficient evidence to conclude that the populations are not identical. conclude that the populations are not identical. ( H =.3532 < 4.60517) ( H =.3532 < 4.60517) n Conclusion

64. 64 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Rank Correlation n The Pearson correlation coefficient, r, is a measure of the linear association between two variables for which interval or ratio data are available. n The Spearman rank-correlation coefficient, r s, is a measure of association between two variables when only ordinal data are available. n Values of r s can range from –1.0 to +1.0, where values near 1.0 indicate a strong positive association between the rankings, and values near 1.0 indicate a strong positive association between the rankings, and values near -1.0 indicate a strong negative association between the rankings. values near -1.0 indicate a strong negative association between the rankings.

65. 65 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Rank Correlation n Spearman Rank-Correlation Coefficient, r s where: n = number of observations being ranked n = number of observations being ranked x i = rank of observation i with respect to the first x i = rank of observation i with respect to the first variable variable y i = rank of observation i with respect to the second y i = rank of observation i with respect to the second variable variable d i = x i - y i d i = x i - y i

66. 66 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Test for Significant Rank Correlation n We may want to use sample results to make an inference about the population rank correlation p s. n To do so, we must test the hypotheses: (No rank correlation exists) (Rank correlation exists)

67. 67 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Rank Correlation Approximately normal, provided n > 10 n Sampling Distribution of r s when p s = 0 Mean Mean Standard Deviation Standard Deviation Distribution Form Distribution Form

68. 68 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Rank Correlation n Example: Crennor Investors Crennor Investors provides a portfolio management service for its clients. Two of Crennor’s analysts ranked ten investments as shown on the next slide. Use rank correlation, with  =.10, to comment on the agreement of the two analysts’ rankings.

69. 69 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Rank Correlation Analyst #2 1 5 6 2 9 7 3 10 4 8 Analyst #1 1 4 9 8 6 3 5 7 2 10 InvestmentA B C D E F G H I J n Example: Crennor Investors (No rank correlation exists) (Rank correlation exists) Analysts’ Rankings Analysts’ Rankings Hypotheses Hypotheses

70. 70 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Rank Correlation ABCDEFGHIJ1498635721015629731048036-3-42-3-22019369164944 Sum =92 Investment Analyst #1 Ranking Analyst #2 Ranking Differ. (Differ.) 2

71. 71 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Sampling Distribution of r s Assuming No Rank Correlation Rank Correlation  r = 0 rsrsrsrs

72. 72 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Test Statistic Rank Correlation z = ( r s -  r )/  r = (.4424 - 0)/.3333 = 1.33 n Rejection Rule With.10 level of significance: Reject H 0 if p -value <.10 n p -Value p -Value = 2(1.0000 -.9082) =.1836

73. 73 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Do no reject H 0. The p -value > . There is not a significant rank correlation. The two analysts are not showing agreement in their ranking of the risk associated with the different investments. Rank Correlation n Conclusion

74. 74 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. End of Chapter 18