1. Chapter 11
Nonparametric Tests
Larson/Farber 4th ed

2. 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 4th ed

3. Section 11.1
The Sign Test
Larson/Farber 4th ed

4. 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 4th ed

5. 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 4th ed

6. Sign Test for a Population Median
A nonparametric test that can be used to test a population median against a hypothesized value k.
Left-tailed test:
H0: median  k and Ha: median < k
Right-tailed test:
H0: median  k and Ha: median > k
Two-tailed test:
H0: median = k and Ha: median  k
Larson/Farber 4th ed

7. 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 4th ed

8. 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 4th ed

9. Performing a Sign Test for a Population Median
In Words In Symbols
State the claim. Identify the null and alternative hypotheses.
Specify the level of significance.
Determine the sample size n by assigning + signs and – signs to the sample data.
Determine the critical value.
State H0 and Ha.
Identify .
n = total number of + and – signs
If n  25, use Table 8. If n > 25, use Table 4.
Larson/Farber 4th ed

10. Performing a Sign Test for a Population Median
In Words In Symbols
Calculate the test statistic.
If n  25, use x. If n > 25, use
Make a decision to reject or fail to reject the null hypothesis.
Interpret the decision in the context of the original claim.
If the test statistic is less than or equal to the critical value, reject H0. Otherwise, fail to reject H0.
Larson/Farber 4th ed

11. 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?
Larson/Farber 4th ed

12. Solution: Using the Sign Test
Ha:
median ≤ 750
median > 750
Compare each data entry with the hypothesized median 750
+ + + –
+ + – +
– 0 + +
There are 3 – signs and 12 + signs
n = = 15
Larson/Farber 4th ed

13. Solution: Using the Sign Test
α =
Critical Value:
0.05
Use Table 8 (n ≤ 25)
Critical value is 3
Larson/Farber 4th ed

14. Solution: Using the Sign Test
Test Statistic:
x = 3 (n ≤ 25; use smaller number of + or – signs)
Decision:
Reject H0
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 4th ed

15. 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 4th ed

16. Solution: Using the Sign Test
Ha:
α =
Critical value:
median ≥ 6000
Test Statistic:
Decision:
There are 60 – signs and 40 + signs.
n = = 100
x = 40
median < 6000
0.01
n > 25 z
-2.33
0.01
-2.33
Fail to Reject H0
At the 1% level of significance you cannot reject the dealership’s claim.
-1.9
Larson/Farber 4th ed

17. The 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.
A sample must be randomly selected from each population.
The samples must be dependent (paired).
The difference between corresponding data entries is found and the sign of the difference is recorded.
Larson/Farber 4th ed

18. Performing The Paired-Sample Sign Test
In Words In Symbols
State the claim. Identify the null and alternative hypotheses.
Specify the level of significance.
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.
State H0 and Ha.
Identify .
n = total number of + and – signs
Larson/Farber 4th ed

19. Performing The Paired-Sample Sign Test
In Words In Symbols
Determine the critical value.
Find the test statistic.
Use Table 8 in Appendix B.
x = lesser number of + and – signs
Make a decision to reject or fail to reject the null hypothesis.
Interpret the decision in the context of the original claim.
If the test statistic is less than or equal to the critical value, reject H0. Otherwise, fail to reject H0.
Larson/Farber 4th ed

20. 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 4th ed

21. Example: Paired-Sample Sign Test
Prison 1 2 3 4 5 6 7 8 9 10
Before 21 34 45 30 54 37 36 33 40
After 19 22 16 31 18 17
Sign + –
Solution:
H0:
Ha:
The number of repeat offenders will not decrease.
The number of repeat offenders will decrease.
Determine the sign of the difference between the “before” and “after” data.
Larson/Farber 4th ed

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

23. Solution: Paired-Sample Sign Test+ –
Test Statistic:
Decision:
x = 1 (the smaller number of + or – signs)
Reject H0
At the 2.5% level of significance, you can support the psychologist’s claim that the number of repeat offenders will decrease.
Larson/Farber 4th ed

24. 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 4th ed

25. Section 11.2
The Wilcoxon Tests
Larson/Farber 4th ed

26. 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 4th ed

27. 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 4th ed

28. Performing The Wilcoxon Signed-Rank Test
In Words In Symbols
State the claim. Identify the null and alternative hypotheses.
Specify the level of significance.
Determine the sample size n, which is the number of pairs of data for which the difference is not 0.
Determine the critical value.
State H0 and Ha.
Identify .
Use Table 9 in Appendix B.
Larson/Farber 4th ed

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

30. Performing The Wilcoxon Signed-Rank Test
In Words In Symbols
Make a decision to reject or fail to reject the null hypothesis.
Interpret the decision in the context of the original claim.
If ws is less than or equal to the critical value, reject H0. Otherwise, fail to reject H0.
Larson/Farber 4th ed

31. 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 music 40 36 42 35
Larson/Farber 4th ed

32. Solution: Wilcoxon Signed-Rank Test
H0:
Ha:
α =
n =
There is no difference in the length of the athletes’ workout sessions.
There is a difference in the length of the athletes’ workout sessions.
0.05 (two-tailed test)
10 (the difference between each data pair is not 0)
Larson/Farber 4th ed

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

34. Solution: Wilcoxon Signed-Rank Test
Test Statistic:
With music
Without music
Difference
Absolute value
Rank
Signed rank 45 38 40 28 33 39 36 41 42 47 62 54 44 35 7 1 2 3 5 6 7 8 9
7.5 -1 -2 3
-4.5
4.5 6 7 9
7.5 10 -2 2 -5
4.5 3 3 -1 1 6 6 8 9 7
7.5 5
4.5 9 10
Larson/Farber 4th ed

35. 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) + (+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 4th ed

36. Solution: Wilcoxon Signed-Rank Test
Decision:
Reject H0
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 4th ed

37. 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.
n1 represents the size of the smaller sample and n2 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 4th ed

38. 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 4th ed

39. Performing The Wilcoxon Rank Sum Test
In Words In Symbols
State the claim. Identify the null and alternative hypotheses.
Specify the level of significance.
Determine the critical value(s).
Determine the sample sizes.
State H0 and Ha.
Identify .
Use Table 4 in Appendix B.
n1 ≤ n2
Larson/Farber 4th ed

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

41. Performing The Wilcoxon Rank Sum Test
In Words In Symbols
Calculate the test statistic.
Make a decision to reject or fail to reject the null hypothesis.
Interpret the decision in the context of the original claim.
If z is in the rejection region, reject H0. Otherwise, fail to reject H0.
Larson/Farber 4th ed

42. 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 58 73 94 81 78 74 66 75 97 79
Female 57 65 71 67 64 77 80 70
Larson/Farber 4th ed

43. Solution: Wilcoxon Rank Sum Test
H0:
Ha:
There is no difference between the males’ and the females’ earnings.
There is a difference between the males’ and the females’ earnings.
α =
Rejection Region:
0.10 (two-tailed test) Z
0.05
1.645
Larson/Farber 4th ed

44. 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
Sample
Rank 57 F 1 58 M 2 64 3 65 4 66
5.5 67 7 70 8 71 9 73
10.5
Ordered data
Sample
Rank 74 M 12 75 13 77 F 14 78
15.5 79 17 80 18 81
19.5 94 21 97 22
Larson/Farber 4th ed

45. Solution: Wilcoxon Rank Sum Test
Because the smaller sample is the sample of males, R is the sum of the male rankings.
R =
= 138
Using n1 = 10 and n2 = 12, we can find μR, andR.
Larson/Farber 4th ed

46. Solution: Wilcoxon Rank Sum Test
H0:
Ha:
no difference in earnings.
Test Statistic
difference in earnings.
α =
Rejection Region:
0.10
Decision:
Fail to reject H0
At the 10% level of significance, you cannot conclude that there is a difference between the males’ and females’ earnings. Z
0.05
1.645
1.52
Larson/Farber 4th ed

47. 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 4th ed

48. The Kruskal-Wallis Test
Section 11.3
The Kruskal-Wallis Test
Larson/Farber 4th ed

49. 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 4th ed

50. The 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.
H0: There is no difference in the distribution of the populations.
Ha: There is a difference in the distribution of the populations.
Larson/Farber 4th ed

51. 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 4th ed

52. The Kruskal-Wallis Test
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 4th ed

53. Performing a Kruskal-Wallis Test
In Words In Symbols
State the claim. Identify the null and alternative hypotheses.
Specify the level of significance.
Identify the degrees of freedom
Determine the critical value and the rejection region.
State H0 and Ha.
Identify .
d.f. = k – 1
Use Table 6 in Appendix B.
Larson/Farber 4th ed

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

55. Performing a Kruskal-Wallis Test
In Words In Symbols
Make a decision to reject or fail to reject the null hypothesis.
Interpret the decision in the context of the original claim.
If H is in the rejection region, reject H0. Otherwise, fail to reject H0.
Larson/Farber 4th ed

56. 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 4th ed

57. 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
45.35
41.70
38.25
53.25
43.85
43.57
Larson/Farber 4th ed

58. Solution: Kruskal-Wallis Test
H0:
Ha:
There is no difference in the hourly pay rates in the three states.
There is a difference in the hourly pay rates in the three states.
α =
d.f. =
Rejection Region:
0.01
k – 1 = 3 – 1 = 2 χ2
9.210
0.01
Larson/Farber 4th ed

59. 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
35.97
2.5
35.98 4
37.10 CA 5
38.25 6
38.65 7
40.12 8
40.37 MD 9
40.50 10
Ordered Data
Sample
Rank
41.70 CA 11
42.05 IN
12.5
43.20 14
43.57 MD 15
43,85 16
44.94 17
44.98 18
45.35 19
45.70 20
Ordered Data
Sample
Rank
47.78 CA 21
48.79 MD 22
48.80 23
49.20 24
49.68 25
49.88 26
52.94 27
53.25 28
53.82 29
Larson/Farber 4th ed

60. Solution: Kruskal-Wallis Test
The sum of the ranks for each sample is as follows.
R1 = = 160.5
R2 =
= 63.5
R3 =
= 211
Larson/Farber 4th ed

61. Solution: Kruskal-Wallis Test
H0:
Ha:
no difference in hourly pay rates
Test Statistic
H ≈ 13.12
difference in hourly pay rates.
Decision:
Reject H0
α =
d.f. =
Rejection Region:
0.01
At the 1% level of significance, you can conclude that there is a difference in actuaries’ hourly pay rates in California, Indiana, and Maryland.
3 – 1 = 2 χ2
9.210
0.01
13.12
Larson/Farber 4th ed

62. 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 4th ed

63. Section 11.4
Rank Correlation
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64. Section 11.4 Objectives
Use the Spearman rank correlation coefficient to determine whether the correlation between two variables is significant.
Larson/Farber 4th ed

65. 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
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66. 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 4th ed

67. 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 4th ed

68. The Spearman Rank Correlation Coefficient
The null and alternative hypotheses for this test are as follows.
H0: ρs = 0 (There is no correlation between the variables.)
Ha: ρs  0 (There is a significant correlation between the variables.)
Larson/Farber 4th ed

69. Testing the Significance of the Spearman Rank Correlation Coefficient
In Words In Symbols
State the null and alternative hypotheses.
Specify the level of significance.
Determine the critical value.
Find the test statistic.
State H0 and Ha.
Identify .
Use Table 10 in Appendix B.
Larson/Farber 4th ed

70. Testing the Significance of the Spearman Rank Correlation Coefficient
In Words In Symbols
Make a decision to reject or fail to reject the null hypothesis.
Interpret the decision in the context of the original claim.
If |rs| is greater than the critical value, reject H0. Otherwise, fail to reject H0.
Larson/Farber 4th ed

71. 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 At α = 0.05, can you conclude that there is a correlation between the beef and lamb prices? (Source: U.S. Department of Agriculture)
Year
Beef
Lamb
1999
63.4
74.5
2000
68.6
79.8
2001
71.3
66.9
2002
66.5
74.1
2003
79.7
94.4
2004
85.8
101.0
2005
89.7
110.0
Larson/Farber 4th ed

72. Solution: The Spearman Rank Correlation Coefficient
Ha:
ρs = 0 (no correlation between beef and lamb prices)
ρs ≠ 0 (correlation between beef and lamb prices)
α =
n =
Critical value:
0.05 7
Table 10
The critical value is 0.786
Larson/Farber 4th ed

73. Solution: The Spearman Rank Correlation Coefficient
Beef
Rank
Lamb d d2
63.4
74.5
68.6
79.8
71.3
66.9
66.5
74.1
79.7
94.4
85.8
101.0
89.7
110.0 1 3 -2 4 3 4 -1 1 4 1 3 9 2 2 5 5 6 6 7 7
Σd2 = 14
Larson/Farber 4th ed

74. Solution: The Spearman Rank Correlation Coefficient
Ha:
ρs = 0
Test Statistic
ρs ≠ 0
α =
n =
Critical value:
0.05 7
Decision:
Fail to Reject H0
The critical value is 0.786
At the 5% level of significance, you cannot conclude that there is a significant correlation between beef and lamb prices between 1999 and 2005.
Larson/Farber 4th ed

75. Section 11.4 Summary
Used the Spearman rank correlation coefficient to determine whether the correlation between two variables is significant.
Larson/Farber 4th ed

76. Section 11.5
The Runs Test
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77. Section 11.5 Objectives
Use the runs test to determine whether a data set is random.
Larson/Farber 4th ed

78. 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 4th ed

79. 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. P P F F F F P F F F P P P P P P
Solution:
There are 5 runs.
P P F F F F P F F F P P P P P P
Length of run: 2 4 1 3 6
Larson/Farber 4th ed

80. 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.
H0: The sequence of data is random.
Ha: The sequence of data is not random.
Larson/Farber 4th ed

81. Test Statistic for the Runs Test
When n1  20 and n2  20, the test statistic for the runs test is G, the number of runs.
When n1 > 20 or n2 > 20, the test statistic for the runs test is
where
Larson/Farber 4th ed

82. Performing a Runs Test for Randomness
In Words In Symbols
State the claim. Identify null and alternative hypotheses.
Specify the level of significance. (Use α = for the runs test.)
Determine the number of data that have each characteristic and the number of runs.
State H0 and Ha.
Identify .
Determine n1, n2, and G.
Larson/Farber 4th ed

83. Performing a Runs Test for Randomness
In Words In Symbols
Determine the critical values.
Calculate the test statistic.
If n1  20 and n2  20, use Table 12. If n1 > 20 or n2 > 20, use Table 4.
If n1  20 and n2  20, use G.
If n1 > 20 or n2 > 20, use
Larson/Farber 4th ed

84. Performing a Runs Test for Randomness
In Words In Symbols
Make a decision to reject or fail to reject the null hypothesis.
Interpret the decision in the context of the original claim.
If G  the lower critical value, or if G  the upper critical value, reject H0. Otherwise, fail to reject H0.
Larson/Farber 4th ed

85. 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 I N I I N N N
Larson/Farber 4th ed

86. Solution: Using the Runs Test
Ha:
The occurrence of injuries is random.
The occurrence of injuries is not random.
I I N N N I N I I N N N
n1 = number of Is =
n2 = number of Ns =
G = number of runs =
α =
Critical value: 5 7 6
0.05
Because n1 ≤ 20, n2 ≤ 20, use Table 12
Larson/Farber 4th ed

87. Solution: Using the Runs Test
Critical value:
n1 = 5 n2 = G = 6
Value of n1
The lower critical value is 3 and the upper critical value is 11.
Larson/Farber 4th ed

88. Solution: Using the Runs Test
Ha:
random
Decision:
Fail to Reject H0
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.
not random
n1 = n2 =
G =
α =
Critical value: 5 7 6
0.05
lower critical value = 3 upper critical value =11
Test Statistic:
G = 6
Larson/Farber 4th ed

89. 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? M M F F F F M M M M M M F F F F F M M M M M M M F F F M M M M F M M F M
Larson/Farber 4th ed

90. Solution: Using the Runs Test
Ha:
The selection of employees is random.
The selection of employees is not random.
M M F F F F M M M M M M F F F F F M M M M M M M F F F M M M M F M M F M
n1 = number of Fs =
n2 = number of Ms =
G = number of runs = 14 22 11
Larson/Farber 4th ed

91. Solution: Using the Runs Test
Ha:
random
Test Statistic:
not random
n1 = n2 =
G =
α =
Critical value: 14 22 11
0.05
Decision:
n2 > 20, use Table 4 Z
-1.96
0.025
1.96
Larson/Farber 4th ed

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

93. Solution: Using the Runs Test
Ha:
random
Test Statistic:
not random
z = -2.53
n1 = n2 =
G =
α =
Critical value: 14 22
Decision:
Reject H0 11
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.
0.05 Z
-1.96
0.025
1.96
-2.53
Larson/Farber 4th ed

94. Section 11.5 Summary
Used the runs test to determine whether a data set is random.
Larson/Farber 4th ed