Chapter 11 Nonparametric Tests LarsonFarber 4 th ed

Chapter 11 Nonparametric Tests Larson/Farber 4 th ed 1

Chapter Outline • • • 11. 1 The Sign Test 11. 2 The Wilcoxon Tests 11. 3 The Kruskal-Wallis Test 11. 4 Rank Correlation 11. 5 The Runs Test Larson/Farber 4 th ed 2

Section 11. 1 The Sign Test Larson/Farber 4 th ed 3

Section 11. 1 Objectives • Use the sign test to test a population median • Use the paired-sample sign test to test the difference between two population medians (dependent samples) Larson/Farber 4 th ed 4

Nonparametric Test Nonparametric test • A hypothesis test that does not require any specific conditions concerning the shape of the population or the value of any population parameters. • Generally easier to perform than parametric tests. • Usually less efficient than parametric tests (stronger evidence is required to reject the null hypothesis). Larson/Farber 4 th ed 5

Sign Test for a Population Median Sign Test • A nonparametric test that can be used to test a population median against a hypothesized value k. • Left-tailed test: H 0: median k and Ha: median < k • Right-tailed test: H 0: median k and Ha: median > k • Two-tailed test: H 0: median = k and Ha: median k Larson/Farber 4 th ed 6

Sign Test for a Population Median • To use the sign test, each entry is compared with the hypothesized median k. § If the entry is below the median, a sign is assigned. § If the entry is above the median, a + sign is assigned. § If the entry is equal to the median, 0 is assigned. • Compare the number of + and – signs. Larson/Farber 4 th ed 7

Sign Test for a Population Median Test Statistic for the Sign Test • When n 25, the test statistic x for the sign test is the smaller number of + or signs. • When n > 25, the test statistic for the sign test is where x is the smaller number of + or signs and n is the sample size (the total number of + or signs). Larson/Farber 4 th ed 8

Performing a Sign Test for a Population Median In Words In Symbols 1. State the claim. Identify the null and alternative hypotheses. State H 0 and Ha. 2. Specify the level of significance. Identify . 3. Determine the sample size n by assigning + signs and – signs to the sample data. 4. Determine the critical value. Larson/Farber 4 th ed n = total number of + and – signs If n 25, use Table 8. If n > 25, use Table 4. 9

Performing a Sign Test for a Population Median In Words 5. Calculate the test statistic. 6. Make a decision to reject or fail to reject the null hypothesis. 7. Interpret the decision in the context of the original claim. Larson/Farber 4 th ed In Symbols If n 25, use x. If n > 25, use If the test statistic is less than or equal to the critical value, reject H 0. Otherwise, fail to reject H 0. 10

Example: Using the Sign Test A bank manager claims that the median number of customers per day is no more than 750. A teller doubts the accuracy of this claim. The number of bank customers per day for 16 randomly selected days are listed below. At α = 0. 05, can the teller reject the bank manager’s claim? 775 765 801 742 754 753 739 751 745 750 777 769 756 760 782 789 Larson/Farber 4 th ed 11

Solution: Using the Sign Test • H 0: median ≤ 750 • Ha: median > 750 • Compare each data entry with the hypothesized median 750 775 754 745 756 765 753 750 760 801 739 777 782 742 751 769 789 + + – + + + 0 + + – + + + • There are 3 – signs and 12 + signs n = 12 + 3 = 15 Larson/Farber 4 th ed 12

Solution: Using the Sign Test • α = 0. 05 • Critical Value: Use Table 8 (n ≤ 25) Critical value is 3 Larson/Farber 4 th ed 13

Solution: Using the Sign Test • Test Statistic: x = 3 (n ≤ 25; use smaller number of + or – signs) • Decision: Reject H 0 At the 5% level of significance, the teller can reject the bank manager’s claim that the median number of customers per day is no more than 750. Larson/Farber 4 th ed 14

Example: Using the Sign Test A car dealership claims to give customers a median trade-in offer of at least $6000. A random sample of 103 transactions revealed that the trade-in offer for 60 automobiles was less than $6000 and the trade-in offer for 40 automobiles was more than $6000. At α = 0. 01, can you reject the dealership’s claim? Larson/Farber 4 th ed 15

Solution: Using the Sign Test • • H 0: median ≥ 6000 Ha: median < 6000 α = 0. 01 Critical value: n > 25 • Test Statistic: There are 60 – signs and 40 + signs. n = 60 + 40 = 100 x = 40 0. 01 -2. 33 -1. 9 Larson/Farber 4 th ed 0 z • Decision: Fail to Reject H 0 At the 1% level of significance you cannot reject the dealership’s claim. 16

The Paired-Sample Sign Test Paired-sample sign test • Used to test the difference between two population medians when the populations are not normally distributed. • For the paired-sample sign test to be used, the following must be true. 1. A sample must be randomly selected from each population. 2. The samples must be dependent (paired). • The difference between corresponding data entries is found and the sign of the difference is recorded. Larson/Farber 4 th ed 17

Performing The Paired-Sample Sign Test In Words In Symbols 1. State the claim. Identify the null and alternative hypotheses. State H 0 and Ha. 2. Specify the level of significance. Identify . 3. Determine the sample size n by finding the difference for each data pair. Assign a + sign for a positive difference, a – sign for a negative difference, and a 0 for no difference. Larson/Farber 4 th ed n = total number of + and – signs 18

Performing The Paired-Sample Sign Test In Words In Symbols 4. Determine the critical value. Use Table 8 in Appendix B. 5. Find the test statistic. x = lesser number of + and – signs 6. Make a decision to reject or fail to reject the null hypothesis. If the test statistic is less than or equal to the critical value, reject H 0. Otherwise, fail to reject H 0. 7. Interpret the decision in the context of the original claim. Larson/Farber 4 th ed 19

Example: Paired-Sample Sign Test A psychologist claims that the number of repeat offenders will decrease if first-time offenders complete a particular rehabilitation course. You randomly select 10 prisons and record the number of repeat offenders during a two-year period. Then, after first-time offenders complete the course, you record the number of repeat offenders at each prison for another two-year period. The results are shown on the next slide. At α = 0. 025, can you support the psychologist’s claim? Larson/Farber 4 th ed 20

Example: Paired-Sample Sign Test Prison 1 2 3 4 5 6 7 8 9 10 Before 21 34 9 45 30 54 37 36 33 40 After 19 22 16 31 21 30 22 18 17 21 Sign + + – + + + + Solution: • H 0: The number of repeat offenders will not decrease. • Ha: The number of repeat offenders will decrease. • Determine the sign of the difference between the “before” and “after” data. Larson/Farber 4 th ed 21

Solution: Paired-Sample Sign Test Sign + + – + + + + • α = 0. 025 (one-tailed) • n = 1 + 9 = 10 • Critical value: Critical value is 1 Larson/Farber 4 th ed 22

Solution: Paired-Sample Sign Test Sign + + – + + + + • Test Statistic: x = 1 (the smaller number of + or – signs) • Decision: Reject H 0 At the 2. 5% level of significance, you can support the psychologist’s claim that the number of repeat offenders will decrease. Larson/Farber 4 th ed 23

Section 11. 1 Summary • Used the sign test to test a population median • Used the paired-sample sign test to test the difference between two population medians (dependent samples) Larson/Farber 4 th ed 24

Section 11. 2 The Wilcoxon Tests Larson/Farber 4 th ed 25

Section 11. 2 Objectives • Use the Wilcoxon signed-rank test to determine if two dependent samples are selected from populations having the same distribution • Use the Wilcoxon rank sum test to determine if two independent samples are selected from populations having the same distribution. Larson/Farber 4 th ed 26

The Wilcoxon Signed-Rank Test • A nonparametric test that can be used to determine whether two dependent samples were selected from populations having the same distribution. • Unlike the sign test, it considers the magnitude, or size, of the data entries. Larson/Farber 4 th ed 27

Performing The Wilcoxon Signed-Rank Test In Words In Symbols 1. State the claim. Identify the null and alternative hypotheses. State H 0 and Ha. 2. Specify the level of significance. Identify . 3. Determine the sample size n, which is the number of pairs of data for which the difference is not 0. 4. Determine the critical value. Larson/Farber 4 th ed Use Table 9 in Appendix B. 28

Performing The Wilcoxon Signed-Rank Test In Words 5. Calculate the test statistic ws. a. Complete a table using the headers listed at the right. b. Find the sum of the positive ranks and the sum of the negative ranks. c. Select the smaller of absolute values of the sums. Larson/Farber 4 th ed In Symbols Headers: Sample 1, Sample 2, Difference, Absolute value, Rank, and Signed rank takes on the same sign as its corresponding difference. 29

Performing The Wilcoxon Signed-Rank Test In Words 6. Make a decision to reject or fail to reject the null hypothesis. 7. Interpret the decision in the context of the original claim. Larson/Farber 4 th ed In Symbols If ws is less than or equal to the critical value, reject H 0. Otherwise, fail to reject H 0. 30

Example: Wilcoxon Signed-Rank Test A sports psychologist believes that listening to music affects the length of athletes’ workout sessions. The length of time (in minutes) of 10 athletes’ workout sessions, while listening to music and while not listening to music, are shown in the table. At α = 0. 05, can you support the sports psychologist’s claim? With music 45 38 28 39 41 47 62 54 33 44 Without 38 40 33 36 42 41 54 47 28 35 music Larson/Farber 4 th ed 31

Solution: Wilcoxon Signed-Rank Test • H 0: There is no difference in the length of the athletes’ workout sessions. • Ha: There is a difference in the length of the athletes’ workout sessions. • α = 0. 05 (two-tailed test) • n = 10 (the difference between each data pair is not 0) Larson/Farber 4 th ed 32

Solution: Wilcoxon Signed-Rank Test • Critical Value Table 9 Critical value is 8 Larson/Farber 4 th ed 33

Solution: Wilcoxon Signed-Rank Test • Test Statistic: With music Without music Absolute value Rank 45 38 7 7 7. 5 7 38 40 -2 2 28 33 -5 5 2 4. 5 -2 -4. 5 39 36 3 3 41 42 -1 1 3 -1 47 41 6 6 62 54 8 8 9 9 54 47 7 7 33 28 5 5 7. 5 44 35 9 9 10 10 Larson/Farber 4 th ed Difference Signed rank 34

Solution: Wilcoxon Signed-Rank Test • Test Statistic: The sum of the negative ranks is: -1 + (-2) + (-4. 5) = -7. 5 The sum of the positive ranks is: (+3) + (+4. 5) + (+6) + (+7. 5) + (+9) + (+10) = 47. 5 ws = 7. 5 (the smaller of the absolute value of these two sums: |-7. 5| < |47. 5|) Larson/Farber 4 th ed 35

Solution: Wilcoxon Signed-Rank Test • Decision: Reject H 0 At the 5% level of significance, you have enough evidence to support the claim that music makes a difference in the length of athletes’ workout sessions. Larson/Farber 4 th ed 36

The Wilcoxon Rank Sum Test • A nonparametric test that can be used to determine whether two independent samples were selected from populations having the same distribution. • A requirement for the Wilcoxon rank sum test is that the sample size of both samples must be at least 10. • n 1 represents the size of the smaller sample and n 2 represents the size of the larger sample. • When calculating the sum of the ranks R, use the ranks for the smaller of the two samples. Larson/Farber 4 th ed 37

Test Statistic for The Wilcoxon Rank Sum Test • Given two independent samples, the test statistic z for the Wilcoxon rank sum test is where R = sum of the ranks for the smaller sample, and Larson/Farber 4 th ed 38

Performing The Wilcoxon Rank Sum Test In Words In Symbols 1. State the claim. Identify the null and alternative hypotheses. State H 0 and Ha. 2. Specify the level of significance. Identify . 3. Determine the critical value(s). 4. Determine the sample sizes. Larson/Farber 4 th ed Use Table 4 in Appendix B. n 1 ≤ n 2 39

Performing The Wilcoxon Rank Sum Test In Words 5. Find the sum of the ranks for the smaller sample. In Symbols R a. List the combined data in ascending order. b. Rank the combined data. c. Add the sum of the ranks for the smaller sample. Larson/Farber 4 th ed 40

Performing The Wilcoxon Rank Sum Test In Words In Symbols 6. Calculate the test statistic. 7. Make a decision to reject or fail to reject the null hypothesis. If z is in the rejection region, reject H 0. Otherwise, fail to reject H 0. 8. Interpret the decision in the context of the original claim. Larson/Farber 4 th ed 41

Example: Wilcoxon Rank Sum Test The table shows the earnings (in thousands of dollars) of a random sample of 10 male and 12 female pharmaceutical sales representatives. At α = 0. 10, can you conclude that there is a difference between the males’ and females’ earnings? Male Female Larson/Farber 4 th ed 58 73 94 81 78 74 66 75 97 79 66 57 81 73 65 78 71 67 64 77 80 70 42

Solution: Wilcoxon Rank Sum Test • H 0: There is no difference between the males’ and the females’ earnings. • Ha: There is a difference between the males’ and the females’ earnings. • α = 0. 10 (two-tailed test) • Rejection Region: 0. 05 -1. 645 0 Larson/Farber 4 th ed 0. 05 1. 645 Z 43

Solution: Wilcoxon Rank Sum Test To find the values of R, μR, and R, construct a table that shows the combined data in ascending order and the corresponding ranks. Ordered data 57 58 64 65 66 66 67 70 71 73 73 Larson/Farber 4 th ed Sample Rank F M F F F F 1 2 3 4 5. 5 7 8 9 10. 5 Ordered data 74 75 77 78 78 79 80 81 81 94 97 Sample Rank M M F M F M M 12 13 14 15. 5 17 18 19. 5 21 22 44

Solution: Wilcoxon Rank Sum Test • Because the smaller sample is the sample of males, R is the sum of the male rankings. R = 2 + 5. 5 + 10. 5 + 12 + 13 + 15. 5 + 17 + 19. 5 +21 + 22 = 138 • Using n 1 = 10 and n 2 = 12, we can find μR, and R. Larson/Farber 4 th ed 45

Solution: Wilcoxon Rank Sum Test • H 0: no difference in earnings. • Ha: difference in earnings. • α = 0. 10 • Test Statistic • Rejection Region: • Decision: Fail to reject H 0 At the 10% level of significance, you cannot conclude that there is a difference between the males’ and females’ earnings. 0. 05 -1. 645 0 0. 05 1. 645 1. 52 Larson/Farber 4 th ed Z 46

Section 11. 2 Summary • Used the Wilcoxon signed-rank test to determine if two dependent samples are selected from populations having the same distribution • Used the Wilcoxon rank sum test to determine if two independent samples are selected from populations having the same distribution. Larson/Farber 4 th ed 47

Section 11. 3 The Kruskal-Wallis Test Larson/Farber 4 th ed 48

Section 11. 3 Objectives • Use the Kruskal-Wallis test to determine whether three or more samples were selected from populations having the same distribution. Larson/Farber 4 th ed 49

The Kruskal-Wallis Test Kruskal-Wallis test • A nonparametric test that can be used to determine whether three or more independent samples were selected from populations having the same distribution. • The null and alternative hypotheses for the Kruskal. Wallis test are as follows. H 0: There is no difference in the distribution of the populations. Ha: There is a difference in the distribution of the populations. Larson/Farber 4 th ed 50

The Kruskal-Wallis Test • Two conditions for using the Kruskal-Wallis test are • Each sample must be randomly selected • The size of each sample must be at least 5. • If these conditions are met, the test is approximated by a chi-square distribution with k – 1 degrees of freedom where k is the number of samples. Larson/Farber 4 th ed 51

The Kruskal-Wallis Test Statistic for the Kruskal-Wallis Test • Given three or more independent samples, the test statistic H for the Kruskal-Wallis test is where k represent the number of samples, ni is the size of the ith sample, N is the sum of the sample sizes, Ri is the sum of the ranks of the ith sample. Larson/Farber 4 th ed 52

Performing a Kruskal-Wallis Test In Words 1. State the claim. Identify the null and alternative hypotheses. 2. Specify the level of significance. 3. Identify the degrees of freedom 4. Determine the critical value and the rejection region. Larson/Farber 4 th ed In Symbols State H 0 and Ha. Identify . d. f. = k – 1 Use Table 6 in Appendix B. 53

Performing a Kruskal-Wallis Test In Words In Symbols 5. Find the sum of the ranks for each sample. a. List the combined data in ascending order. b. Rank the combined data. 6. Calculate the test statistic. Larson/Farber 4 th ed 54

Performing a Kruskal-Wallis Test In Words 7. Make a decision to reject or fail to reject the null hypothesis. In Symbols If H is in the rejection region, reject H 0. Otherwise, fail to reject H 0. 8. Interpret the decision in the context of the original claim. Larson/Farber 4 th ed 55

Example: Kruskal-Wallis Test You want to compare the hourly pay rates of actuaries who work in California, Indiana, and Maryland. To do so, you randomly select several actuaries in each state and record their hourly pay rate. The hourly pay rates are shown on the next slide. At α = 0. 01, can you conclude that the distributions of actuaries’ hourly pay rates in these three states are different? (Adapted from U. S. Bureau of Labor Statistics) Larson/Farber 4 th ed 56

Example: Kruskal-Wallis Test Sample Hourly Pay Rates CA (Sample 1) IN (Sample 2) MD (Sample 3) 40. 50 33. 45 49. 68 44. 98 40. 12 44. 94 47. 78 38. 65 48. 80 43. 20 35. 98 49. 20 37. 10 35. 97 40. 37 49. 88 4570 48. 79 42. 05 53. 82 52. 94 35. 97 45. 35 41. 70 38. 25 53. 25 43. 85 Larson/Farber 4 th ed 43. 57 57

Solution: Kruskal-Wallis Test • H 0: There is no difference in the hourly pay rates in the three states. • Ha: There is a difference in the hourly pay rates in the three states. • α = 0. 01 • d. f. = k – 1 = 3 – 1 = 2 • Rejection Region: 0. 01 0 Larson/Farber 4 th ed 9. 210 χ2 58

Solution: Kruskal-Wallis Test The table shows the combined data listed in ascending order and the corresponding ranks. Ordered Data Sample Rank 33. 45 IN 1 41. 70 CA 11 47. 78 CA 21 35. 97 IN 2. 5 42. 05 IN 12. 5 48. 79 MD 22 35. 97 IN 2. 5 42. 05 CA 12. 5 48. 80 MD 23 35. 98 IN 4 43. 20 CA 14 49. 20 MD 24 37. 10 CA 5 43. 57 MD 15 49. 68 MD 25 38. 25 IN 6 43, 85 CA 16 49. 88 CA 26 38. 65 IN 7 44. 94 MD 17 52. 94 CA 27 40. 12 IN 8 44. 98 CA 18 53. 25 MD 28 40. 37 MD 9 45. 35 MD 19 53. 82 MD 29 40. 50 CA 10 45. 70 IN 20 Larson/Farber 4 th ed 59

Solution: Kruskal-Wallis Test The sum of the ranks for each sample is as follows. R 1 = 5 + 10 + 11 + 12. 5 + 14 + 16 + 18 + 21 + 26 + 27 = 160. 5 R 2 = 1 + 2. 5 +4 + 6 + 7 + 8 + 12. 5 + 20 = 63. 5 R 3 = 9 + 15 + 17 + 19 + 22 + 23 + 24 + 25 + 28 + 29 = 211 Larson/Farber 4 th ed 60

Solution: Kruskal-Wallis Test • H 0: no difference in hourly pay rates • Test Statistic H ≈ 13. 12 • Ha: difference in hourly pay rates. • Decision: Reject H 0 At the 1% level of significance, you can conclude that there is a difference in actuaries’ hourly pay rates in California, Indiana, and Maryland. • α = 0. 01 • d. f. = 3 – 1 = 2 • Rejection Region: 0. 01 0 Larson/Farber 4 th ed 9. 210 13. 12 χ2 61

Section 11. 3 Summary • Used the Kruskal-Wallis test to determine whether three or more samples were selected from populations having the same distribution. Larson/Farber 4 th ed 62

Section 11. 4 Rank Correlation Larson/Farber 4 th ed 63

Section 11. 4 Objectives • Use the Spearman rank correlation coefficient to determine whether the correlation between two variables is significant. Larson/Farber 4 th ed 64

The Spearman Rank Correlation Coefficient • A measure of the strength of the relationship between two variables. • Nonparametric equivalent to the Pearson correlation coefficient. • Calculated using the ranks of paired sample data entries. • Denoted rs Larson/Farber 4 th ed 65

The Spearman Rank Correlation Coefficient Spearman rank correlation coefficient rs • The formula for the Spearman rank correlation coefficient is where n is the number of paired data entries d is the difference between the ranks of a paired data entry. Larson/Farber 4 th ed 66

The Spearman Rank Correlation Coefficient • The values of rs range from -1 to 1, inclusive. § If the ranks of corresponding data pairs are identical, rs is equal to +1. § If the ranks are in “reverse” order, rs is equal to -1. § If there is no relationship, rs is equal to 0. • To determine whether the correlation between variables is significant, you can perform a hypothesis test for the population correlation coefficient ρs. Larson/Farber 4 th ed 67

The Spearman Rank Correlation Coefficient • The null and alternative hypotheses for this test are as follows. H 0: ρs = 0 (There is no correlation between the variables. ) Ha: ρs 0 (There is a significant correlation between the variables. ) Larson/Farber 4 th ed 68

Testing the Significance of the Spearman Rank Correlation Coefficient In Words 1. State the null and alternative hypotheses. 2. Specify the level of significance. 3. Determine the critical value. In Symbols State H 0 and Ha. Identify . Use Table 10 in Appendix B. 4. Find the test statistic. Larson/Farber 4 th ed 69

Testing the Significance of the Spearman Rank Correlation Coefficient In Words 5. Make a decision to reject or fail to reject the null hypothesis. In Symbols If |rs| is greater than the critical value, reject H 0. Otherwise, fail to reject H 0. 6. Interpret the decision in the context of the original claim. Larson/Farber 4 th ed 70

Example: The Spearman Rank Correlation Coefficient The table shows the prices (in dollars per 100 pounds) received by U. S. farmers for beef and lamb from 1999 to 2005. At α = 0. 05, can you conclude that there is a correlation between the beef and lamb prices? (Source: U. S. Department of Agriculture) Larson/Farber 4 th ed Year 1999 2000 2001 Beef Lamb 63. 4 74. 5 68. 6 79. 8 71. 3 66. 9 2002 2003 2004 2005 66. 5 79. 7 85. 8 89. 7 74. 1 94. 4 101. 0 110. 0 71

Solution: The Spearman Rank Correlation Coefficient • H 0: ρs = 0 (no correlation between beef and lamb prices) • Ha: ρs ≠ 0 (correlation between beef and lamb prices) • α = 0. 05 • n=7 • Critical value: Table 10 The critical value is 0. 786 Larson/Farber 4 th ed 72

Solution: The Spearman Rank Correlation Coefficient Beef 63. 4 68. 6 71. 3 66. 5 79. 7 85. 8 89. 7 Rank Lamb Rank 74. 5 1 3 79. 8 3 4 66. 9 4 1 2 2 74. 1 5 5 94. 4 6 6 101. 0 7 7 110. 0 d d 2 -2 -1 3 0 0 4 1 9 0 0 Σd 2 = 14 Larson/Farber 4 th ed 73

Solution: The Spearman Rank Correlation Coefficient • H 0: ρ s = 0 • H a: ρ s ≠ 0 • α = 0. 05 • n=7 • Critical value: The critical value is 0. 786 Larson/Farber 4 th ed • Test Statistic • Decision: Fail to Reject H 0 At the 5% level of significance, you cannot conclude that there is a significant correlation between beef and lamb prices between 1999 and 2005. 74

Section 11. 4 Summary • Used the Spearman rank correlation coefficient to determine whether the correlation between two variables is significant. Larson/Farber 4 th ed 75

Section 11. 5 The Runs Test Larson/Farber 4 th ed 76

Section 11. 5 Objectives • Use the runs test to determine whether a data set is random. Larson/Farber 4 th ed 77

The Runs Test for Randomness • A run is a sequence of data having the same characteristic. • Each run is preceded by and followed by data with a different characteristic or by no data at all. • The number of data in a run is called the length of the run. Larson/Farber 4 th ed 78

Example: Finding the Number of Runs A liquid-dispensing machine has been designed to fill one-liter bottles. A quality control inspector decides whether each bottle is filled to an acceptable level and passes inspection (P) or fails inspection (F). Determine the number of runs for the sequence and find the length of each run. PPFFFFPPPPPP Solution: Length of run: Larson/Farber 4 th ed There are 5 runs. PP 2 FFFF 4 P FFF 1 3 PPPPPP 6 79

Runs Test for Randomness • A nonparametric test that can be used to determine whether a sequence of sample data is random. • The null and alternative hypotheses for this test are as follows. H 0: The sequence of data is random. Ha: The sequence of data is not random. Larson/Farber 4 th ed 80

Test Statistic for the Runs Test • When n 1 20 and n 2 20, the test statistic for the runs test is G, the number of runs. • When n 1 > 20 or n 2 > 20, the test statistic for the runs test is where Larson/Farber 4 th ed 81

Performing a Runs Test for Randomness In Words In Symbols 1. State the claim. Identify null and alternative hypotheses. State H 0 and Ha. 2. Specify the level of significance. (Use α = 0. 05 for the runs test. ) Identify . 3. Determine the number of data that have each characteristic and the Determine n 1, n 2, and G. number of runs. Larson/Farber 4 th ed 82

Performing a Runs Test for Randomness In Words In Symbols 4. Determine the critical values. If n 1 20 and n 2 20, use Table 12. If n 1 > 20 or n 2 > 20, use Table 4. 5. Calculate the test statistic. If n 1 20 and n 2 20, use G. If n 1 > 20 or n 2 > 20, use Larson/Farber 4 th ed 83

Performing a Runs Test for Randomness In Words 6. Make a decision to reject or fail to reject the null hypothesis. In Symbols If G the lower critical value, or if G the upper critical value, reject H 0. Otherwise, fail to reject H 0. 7. Interpret the decision in the context of the original claim. Larson/Farber 4 th ed 84

Example: Using the Runs Test A foreman for a construction company records injuries reported by workers during his shift. The following sequence shows whether any injuries were reported during each month in a recent year. I represents a month in which at least one injury was reported and N represents a month in which no injuries were reported. At α = 0. 05, can you conclude that the occurrence of injuries each month is not random? I I N N N Larson/Farber 4 th ed 85

Solution: Using the Runs Test • H 0: The occurrence of injuries is random. • Ha: The occurrence of injuries is not random. • • • I I N N N n 1 = number of Is = 5 n 2 = number of Ns = 7 G = number of runs = 6 α = 0. 05 Critical value: Because n 1 ≤ 20, n 2 ≤ 20, use Table 12 Larson/Farber 4 th ed 86

Solution: Using the Runs Test Value of n 1 • Critical value: • n 1 = 5 n 2 = 7 Larson/Farber 4 th ed G=6 The lower critical value is 3 and the upper critical value is 11. 87

Solution: Using the Runs Test • H 0: random • Ha: not random • • n 1 = 5 n 2 = 7 G= 6 α = 0. 05 Critical value: lower critical value = 3 upper critical value =11 • Test Statistic: • Decision: Fail to Reject H 0 At the 5% level of significance, you do not have enough evidence to support the claim that the occurrence of injuries is not random. So, it appears that the injuries reported by workers during the foreman’s shift occur randomly. G=6 Larson/Farber 4 th ed 88

Example: Using the Runs Test You want to determine whether the selection of recently hired employees in a large company is random with respect to gender. The genders of 36 recently hired employees are shown below. At α = 0. 05, can you conclude that the selection is not random? MMFFFFMMMMMM FFFFFMMMMMMM FFFMMMMFMMFM Larson/Farber 4 th ed 89

Solution: Using the Runs Test • H 0: The selection of employees is random. • Ha: The selection of employees is not random. MM FFFF MMMMMM FFFFF MMMMMMM FFF MMMM F M • n 1 = number of Fs = 14 • n 2 = number of Ms = 22 • G = number of runs = 11 Larson/Farber 4 th ed 90

Solution: Using the Runs Test • H 0: random • Ha: not random • • • Test Statistic: n 1 = 14 n 2 = 22 G = 11 α = 0. 05 Critical value: • Decision: n 2 > 20, use Table 4 0. 025 -1. 96 Larson/Farber 4 th ed 0. 025 0 1. 96 Z 91

Solution: Using the Runs Test Larson/Farber 4 th ed 92

Solution: Using the Runs Test • H 0: random • Ha: not random • • • Test Statistic: z = -2. 53 n 1 = 14 n 2 = 22 G = 11 α = 0. 05 Critical value: 0. 025 -1. 96 -2. 53 Larson/Farber 4 th ed • Decision: Reject H 0 You have enough evidence at the 5% level of significance to support the claim that the selection of employees with respect to gender is not random. 0. 025 0 1. 96 Z 93

Section 11. 5 Summary • Used the runs test to determine whether a data set is random. Larson/Farber 4 th ed 94