1. 11 Chapter Nonparametric Tests © 2012 Pearson Education, Inc.
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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
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3. Section 11.1
The Sign Test
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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)
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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).
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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
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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.
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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).
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9. Performing a Sign Test for a Population Median
In Words In Symbols
Identify the claim. State 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 α.
n = total number of + and – signs
If n ≤ 25, use Table 8 in Appendix B. If n > 25, use Table 4 in Appendix B.
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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.
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11. Example: Using the Sign Test
A website administrator for a company claims that the median number of visitors per day to the company’s website is no more than An employee doubts the accuracy of this claim. The number of visitors per day for 20 randomly selected days are listed below. At α = 0.05, can the employee reject the administrator’s claim?
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12. Solution: Using the Sign Test
Ha:
median ≤ 1500 (Claim)
median > 1500
Compare each data entry with the hypothesized median 1500
– – + + 0
– – + + – –
– +
There are 7 – signs and 12 + signs
n = = 19
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13. Solution: Using the Sign Test
α =
Critical Value:
0.05
Use Table 8 (n ≤ 25)
Critical value is 5
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14. Solution: Using the Sign Test
Test Statistic:
x = 7 (n ≤ 25; use smaller number of + or – signs)
Decision:
Fail to reject H0
At the 5% level of significance, there is not enough evidence for the employee to reject the website administrator’s claim that the median number of visitors per day to company’s website is no more than 1500.
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15. Example: Using the Sign Test
An organization claims that the median annual attendance for museums in U.S. is at least 39,000. A random sample of 125 museums reveals that the annual attendances for 79 museums were less than 39,000, the annual attendances for 42 museums were more than 39,000, and and the annual attendances for 4 museums were 39,000. At α = 0.01, is there enough evidence to reject the organization’s claim? (Adapted from American Association of Museums)
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16. Solution: Using the Sign Test
Ha:
α =
Critical value:
median ≥ 39,000 (Claim)
Test Statistic:
Decision:
There are 79 – signs and 42 + signs.
n = = 121
x = 42
median < 39,000
0.01
n > 25
Reject H0
At the 1% level of significance there is enough evidence to reject the claim that the median annual attendance for museums in the U.S. is at least 39,000.
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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.
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18. Performing The Paired-Sample Sign Test
In Words In Symbols
Identify the claim. State 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 α.
n = total number of + and – signs
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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 = smaller 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.
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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?
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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. (Claim)
Determine the sign of the difference between the “before” and “after” data.
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22. Solution: Paired-Sample Sign Test+ –
α =
n =
Critical value:
0.025 (one-tailed)
1 + 9 = 10
Critical value is 1
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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, there is enough evidence to support the psychologist’s claim that the number of repeat offenders will decrease.
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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)
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25. Section 11.2 The Wilcoxon Tests
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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
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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.
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28. Performing The Wilcoxon Signed-Rank Test
In Words In Symbols
Identify the claim. State 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 α.
Use Table 9 in Appendix B.
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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.
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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.
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31. Example: Wilcoxon Signed-Rank Test
A golf club manufacturer believes that golfers can lower their score by using the manufacturer’s newly designed golf clubs. The scores of 10 golfers while using the old design and while using the new design are shown in the table below. At α = 0.05, can you support the manufacturer’s claim?
Golfer 1 2 3 4 5 6 7 8 9 10
Score (old design) 89 84 96 74 91 85 95 82 92 81
Score (new design) 83 76 80 87 90 77
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32. Solution: Wilcoxon Signed-Rank Test
H0:
Ha:
α =
n =
The new design does not lower scores.
The new design lower scores. (Claim)
0.05 (one-tailed test)
9 (the difference between one data pair is 0)
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33. Solution: Wilcoxon Signed-Rank Test
Critical Value
Table 9
Critical value is 8
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34. Solution: Wilcoxon Signed-Rank Test
Test Statistic:
Score
(old design)
(new design)
Difference
Absolute value
Rank
Signed rank 89 83 6 8 84 1 96 92 4
5.5 74 76 –2 2
2.5
–2.5 91 _ 85 80 5 7 95 87 9 82 –3 3 –4 90 81 77
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35. Solution: Wilcoxon Signed-Rank Test
Test Statistic:
The sum of the negative ranks is:
–2.5 + (–4) = –6.5
The sum of the positive ranks is:
= 38.5
ws = 6.5 (the smaller of the absolute value of these two sums: |–6.5| < |38.5|)
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36. Solution: Wilcoxon Signed-Rank Test
Decision:
Reject H0
At the 5% level of significance, there is enough evidence to support the claim that golfers can lower their scores by using the newly designed clubs.
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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, combine both samples and rank the combined data, then sum the ranks for the smaller of the two samples.
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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
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39. Performing The Wilcoxon Rank Sum Test
In Words In Symbols
Identify the claim. State the null and alternative hypotheses.
Specify the level of significance.
Determine the critical value(s) and the rejection region(s).
Determine the sample sizes.
State H0 and Ha.
Identify α.
Use Table 4 in Appendix B.
n1 ≤ n2
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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
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41. Performing The Wilcoxon Rank Sum Test
In Words In Symbols
Calculate the test statistic and sketch the sampling distribution.
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.
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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 earnings 78 93
114
101 98 94 86 95
117 99
Female earnings 77 85 91 87 84 97
100 90
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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. (Claim)
α =
Rejection Region:
0.10 (two-tailed test) z
–1. 645
0.05
1.645
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44. Solution: Wilcoxon Rank Sum Test
To find the values of R, μR, and σR, construct a table that shows the combined data in ascending order and the corresponding ranks.
Ordered data
Sample
Rank 77 F 1 78 M 2 84 3 85 4 86
5.5 87 7 90 8 91 9 93
10.5
Ordered data
Sample
Rank 94 M 12 95 13 97 F 14 98
15.5 99 17
100 18
101
19.5
114 21
117 22
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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 σR.
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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, there is not enough evidence to conclude that there is a difference between the males’ and females’ earnings. z
–1.645
0.05
1.645
1.52
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47. © 2012 Pearson Education, Inc. All rights reserved.
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
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48. The Kruskal-Wallis Test
Section 11.3
The Kruskal-Wallis Test
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49. © 2012 Pearson Education, Inc. All rights reserved.
Section 11.3 Objectives
Use the Kruskal-Wallis test to determine whether three or more samples were selected from populations having the same distribution
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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.
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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.
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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 represents 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.
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53. Performing a Kruskal-Wallis Test
In Words In Symbols
Identify the claim. State 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 α.
d.f. = k – 1
Use Table 6 in Appendix B.
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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.
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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.
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56. Example: Kruskal-Wallis Test
You want to compare the number of crimes reported in three police precincts in a city. To do so, you randomly select 10 weeks for each precinct and record the number of crimes reported. The results are shown in the table in next slide. At α = 0.01, can you conclude that the distributions of crimes reported in the three police precincts are different?
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57. Example: Kruskal-Wallis Test
Number of Crimes Reported for the week
101st Precinct (Sample 1)
106th Precinct (Sample 2)
113th Precinct (Sample 3) 60 65 69 52 55 51 49 64 70 66 61 50 53 67 48 58 57 62 45 54 59 44 56 63
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58. Solution: Kruskal-Wallis Test
H0:
Ha:
There is no difference in the number of crimes reported in the three precincts.
There is a difference in the number of crimes reported in the three precincts. (Claim)
α =
d.f. =
Rejection Region:
0.01
k – 1 = 3 – 1 = 2
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59. Solution: Kruskal-Wallis Test
The table shows the combined data listed in ascending order and the corresponding ranks.
Ordered Data
Sample
Rank 44
101st 1 45 2 48 3 49 4 50
5.5
106th 51
113th 7 52
8.5 53 10
Ordered Data
Sample
Rank 54
106th 11 55 12 56
101st 13 57 14 58
106st 15 59
113th 16 60
17.5 61 19 62
20.5
Ordered Data
Sample
Rank 62
113th
20.5 63 22 64
106th 23 65
24.5 66 26 67 27 69 28 70
29.5
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60. Solution: Kruskal-Wallis Test
The sum of the ranks for each sample is as follows.
R1 = = 77
R2 =
= 177
R3 =
= 211
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61. Solution: Kruskal-Wallis Test
H0:
Ha:
no difference in number of crimes
Test Statistic
H ≈
difference in number of crimes
Decision:
Reject H0
α =
d.f. =
Rejection Region:
0.01
At the 1% level of significance, there is enough evidence to support the claim that there is a difference in the number of crimes reported in the three police precincts.
3 – 1 = 2
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62. © 2012 Pearson Education, Inc. All rights reserved.
Section 11.3 Summary
Used the Kruskal-Wallis test to determine whether three or more samples were selected from populations having the same distribution
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63. © 2012 Pearson Education, Inc. All rights reserved.
Section 11.4
Rank Correlation
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64. © 2012 Pearson Education, Inc. All rights reserved.
Section 11.4 Objectives
Use the Spearman rank correlation coefficient to determine whether the correlation between two variables is significant
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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.
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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.
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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.)
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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 α.
Use Table 10 in Appendix B.
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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.
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71. Example: The Spearman Rank Correlation Coefficient
The table shows the school enrollments (in millions) at all levels of education for males and females from 2000 to At α = 0.05, can you conclude that there is a correlation between the number of males and females enrolled in school? (Source: U.S. Census Bureau)
Year
Male
Female
2000
35.8
36.4
2001
36.3
36.9
2002
36.8
37.3
2003
37.6
2004
37.4
38.0
2005
38.4
2006
37.2
2007
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72. Solution: The Spearman Rank Correlation Coefficient
ρs = 0 (no correlation between the number of males and females enrolled in school)
H0:
Ha:
ρs ≠ 0 (correlation between the number of males and females enrolled in school) (Claim)
α =
n =
Critical value:
0.05 8
Table 10
The critical value is 0.738
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73. Solution: The Spearman Rank Correlation Coefficient
Male
Rank
Female d d2
35.8 1
36.4
36.3 2
36.9
36.8 3
37.3 5
37.6 4
37.4
6.5
38.0
5.5
38.4
7.5 –1
37.2
–1.5
2.25 8
0.5
0.25
Σd2 = 5.5
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74. Solution: The Spearman Rank Correlation Coefficient
Ha:
ρs = 0
Test Statistic
ρs ≠ 0
α =
n =
Critical value:
0.05 8
Decision:
Reject H0
The critical value is 0.738
At the 5% level of significance, there is enough evidence to conclude that there is a correlation between the number of males and females enrolled in school.
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75. © 2012 Pearson Education, Inc. All rights reserved.
Section 11.4 Summary
Used the Spearman rank correlation coefficient to determine whether the correlation between two variables is significant
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76. © 2012 Pearson Education, Inc. All rights reserved.
Section 11.5
The Runs Test
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77. © 2012 Pearson Education, Inc. All rights reserved.
Section 11.5 Objectives
Use the runs test to determine whether a data set is random
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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.
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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
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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.
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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
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82. Performing a Runs Test for Randomness
In Words In Symbols
Identify the claim. State the null and alternative hypotheses.
Specify the level of significance. (Use α = 0.05 for the runs test.)
Determine the number of data that have each characteristic and the number of runs.
State H0 and Ha.
Identify α.
Determine n1, n2, and G.
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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 in Appendix B. If n1 > 20 or n2 > 20, use Table 4 in Appendix B.
If n1 ≤ 20 and n2 ≤ 20, use G.
If n1 > 20 or n2 > 20, use
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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.
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85. Example: Using the Runs Test
As people enter a concert, an usher records where they are sitting. The results for 13 people are shown, where L represents a lawn seat and P represents a pavilion seat. At α = 0.05, can you conclude that the sequence of seat locations is not random? L L L P P L P P P L L P L
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86. Solution: Using the Runs Test
Ha:
The sequence of seat locations is random.
The sequences of seat locations is not random. (Claim)
L L L P P L P P P L L P L
n1 = number of L’s =
n2 = number of P’s =
G = number of runs =
α =
Critical value: 7 6 7
0.05
Because n1 ≤ 20, n2 ≤ 20, use Table 12
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87. Solution: Using the Runs Test
Critical value:
n1 = 7 n2 = G = 7
Value of n1
The lower critical value is 3 and the upper critical value is 12.
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88. Solution: Using the Runs Test
Ha:
random
Decision:
Fail to Reject H0
At the 5% level of significance, there is not enough evidence to support the claim that the sequence of seat locations is not random. So, it appears that the sequence of seat locations is random.
not random
n1 = n2 =
G =
α =
Critical value: 7 6 7
0.05
lower critical value = 3 upper critical value =12
Test Statistic:
G = 7
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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
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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 F’s =
n2 = number of M’s =
G = number of runs = 14 22 11
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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
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92. Solution: Using the Runs Test
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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
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94. © 2012 Pearson Education, Inc. All rights reserved.
Section 11.5 Summary
Used the runs test to determine whether a data set is random
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