Section 14.5 Exercises

CLARIFYING THE CONCEPTS

Question 14.91

1. Explain how the Kruskal-Wallis test is an extension of the Wilcoxon rank sum test from Section 14.4. (p. 14-37

Question 14.92

2. What is the difference between the Kruskal-Wallis test and the analysis of variance (ANOVA) from Chapter 12? (p. 14-36

Question 14.93

3. True or false: To calculate the test statistic for the Kruskal-Wallis test, we temporarily combine all the data values from all the samples and find the ranks of the combined data values, just as we did in Section 14.4 for the Wilcoxon rank sum test. (p. 14-39

Question 14.94

4. What is the meaning of the notation for the Kruskal-Wallis test? (p. 14-38

Question 14.95

5. Explain what the notation means for the Kruskal-Wallis test. (p. 14-39

Question 14.96

6. When the conditions are met, what distribution does the test statistic for the Kruskal-Wallis test follow? (p. 14-37

PRACTICING THE TECHNIQUES

image CHECK IT OUT!

To do Check out Topic
Exercises 7–14 Example 16 Calculating the
Kruskal-Wallis test
statistic
Exercises 15–18 Example 17 Performing the
Kruskal-Wallis test

For Exercises 7–10, calculate , the sum of the ranks for the first sample, , , and, if appropriate, and . Also find the sample sizes and the total sample size. The data represent independent random samples.

Question 14.97

7.

Sample 1 2 3 4 5 5
Sample 2 6 9 10 7 9
Sample 3 5 3 3 1 2

Question 14.98

8.

Sample 1 9 6 8 3 1
Sample 2 6 7 4 10 9
Sample 3 9 3 10 1 2

Question 14.99

9.

Sample 1 184 152 168 164 183 143
Sample 2 193 182 112 155 145
Sample 3 144 149 150 112 127 133
Sample 4 129 172 193 172 162 187
Sample 5 158 152 137 172 114

Question 14.100

10.

Sample 1 113 186 162 122 197 190
Sample 2 127 197 178 102 162 144
Sample 3 120 142 198 167 165 156 178
Sample 4 167 102 122 113 109
Sample 5 124 138 187 109 100 159 142

For Exercises 11–14, calculate .

Question 14.101

11. Use the data and the statistics you calculated in Exercise 7.

Question 14.102

12. Use the data and the statistics you calculated in Exercise 8.

Question 14.103

13. Use the data and the statistics you calculated in Exercise 9.

Question 14.104

14. Use the data and the statistics you calculated in Exercise 10.

For Exercises 15–18, we are interested in whether the population medians differ. Do the following:

  1. State the hypotheses.
  2. Find the critical value and state the rejection rule.
  3. Find the value of the test statistic .
  4. State the conclusion and the interpretation.

Question 14.105

15. Use the data in Exercise 7 and the value you calculated for in Exercise 11. Use level of significance .

Question 14.106

16. Use the data in Exercise 8 and the value you calculated for in Exercise 12. Use level of significance .

Question 14.107

17. Use the data in Exercise 9 and the value you calculated for in Exercise 13. Use level of significance .

Question 14.108

18. Use the data in Exercise 10 and the value you calculated for in Exercise 14. Use level of significance .

APPLYING THE CONCEPTS

Question 14.109

cafeanova

19. Student-Run Café Business. In Chapter 2, Example 8, we looked at data from a student-run café business. The table contains the number of food items sold per day. Test whether the population median number of items sold is the same for wraps, muffins, and chips, using level of significance .

Wraps Muffins Chips
12 6 7
13 3 16
19 10 8
5 1 4
22 8 10

Question 14.110

prosdartsanova

20. The Pros versus the Darts. In the Chapter 3 Case Study, we examined stock market returns for professional financial analysts, compared with random darts and the Dow Jones Industrial Average (DJIA). The table contains daily stock market returns. Test whether the population median stock market returns are the same across all three groups, using level of significance .

14-43

Pros Darts DJIA
10.6 20.6 4.4
27.8 18.5 11.2
29.1 1.8 3.7
2.2 11.7 17.6
14.1 1.8 0.2

Question 14.111

weightages

21. Weight and Age. The Chapter 4 Case Study looked at body measurements for physically fit males and females. Is there a difference in weight among different age groups? The table contains the weights of five randomly chosen females from each of three age groups: younger (18-22), middle (23-30), and older (31+). Test whether the population median weight is the same for younger, middle, and older females, using level of significance .

Younger Middle Older
119.0 142.9 121.3
124.8 104.3 107.4
130.7 115.1 169.3
130.1 98.8 122.8
155.4 110.2 155.4

Question 14.112

fullmoon

22. The Full Moon and Emergency Room Visits. Is there a difference in emergency room visits before, during, and after a full moon? A study looked at the admission rate (number of patients per day) to the emergency room of a Virginia mental health clinic over a series of 12 full moons. The data are provided in the table. Assume the data represent independent random samples. Is there evidence of a difference in emergency room visits before, during, and after the full moon? Test whether the population median number of emergency room visits is the same before, during, and after a full moon, using level of significance .

Before During After
6.4 11.5 5 13 5.8 13.5
7.1 13.8 13 16 9.2 13.1
6.5 15.4 14 25 7.9 15.8
8.6 15.7 12 14 7.7 13.3
8.1 11.7 6 14 11.0 12.8
10.4 15.8 9 20 12.9 14.5

Question 14.113

infantmortality

23. Global Infant Mortality. The following data set represents the infant mortality rate for states or provinces in the United States, Canada, and Mexico. The infant mortality rate is defined as the number of children who die before their first birthday, for every 1000 live births. The data represent independent random samples. Test whether the population median infant mortality rate is the same for the United States, Canada, and Mexico. Use level of significance .

U.S. state Infant mortality rate
California 5.8
Florida 7.2
Georgia 8.5
Illinois 8.4
Pennsylvania 7.1
Texas 6.4
Virginia 7.7
Canadian province Infant mortality rate
Alberta 4.8
Manitoba 7.5
Nova Scotia 4.4
Ontario 5.5
Quebec 5.6
Mexican state Infant mortality rate
Campeche 26.0
Chihuahua 23.4
Sonora 22.6
Tabasco 25.3
Veracruz 28.0
Yucatan 27.0
Table 14.69: Source: The Poverty Mapping Project at the Earth Institute at Columbia University.

Question 14.114

epirating

24. Environmental Performance Index. The Environmental Performance Index (EPI) is a measure of a nation's commitment to environmental protection and global sustainability. Data for 2008 were released at the World Economic Summit's annual meeting in Davos, Switzerland. The following data represent independent random samples of the EPI ratings for nations from four continental regions. Test whether the population median EPI is the same in the four continental regions, using level of significance .

Americas EPI European Union EPI
Canada 88.3 France 87.8
Brazil 82.7 Germany 86.3
USA 81.0 United Kingdom 86.3
Mexico 79.8 Portugal 85.8
Jamaica 79.1 Italy 84.2
Spain 83.1
Ireland 82.7

14-44

Sub-Saharan Africa EPI Asia and Pacific EPI
Kenya 69.0 Japan 84.5
South Africa 69.0 Taiwan 80.8
Ethiopia 58.8 Australia 79.8
Rwanda 54.9 Vietnam 73.9
Chad 45.9 China 65.1
India 60.3
Table 14.71: Source: Yale Center for Environmental Law and Policy.

Question 14.115

cerealnutrition

25. The data in the accompanying table represent the nutritional ratings of breakfast cereals for three manufacturer brands.11 The data were selected independently and randomly. Test whether the population median nutritional rating differs by manufacturer, using level of significance

Table 14.72: Nutritional ratings of breakfast cereals
Cereal Manufacturer Nutritional
rating
Just Right Fruit & Nut Kellogg's 36.4715
Corn Chex Ralston-Purina 41.4450
Count Chocula General Mills 22.3965
Rice Krispies Kellogg's 40.5602
Wheat Chex Ralston-Purina 49.7874
Product 19 Kellogg's 41.5035
Honey Nut Cheerios General Mills 31.0722
Apple Jacks Kellogg's 33.1741
Kix General Mills 39.2411
Double Chex Ralston-Purina 44.3309
Triples General Mills 39.1062
Rice Chex Ralston-Purina 41.9989
Raisin Nut Bran General Mills 39.7034
Just Right Crunchy
Nuggets		 Kellogg's 36.5237
Raisin Bran Kellogg's 39.2592
Frosted Flakes Kellogg's 31.4360
Cinnamon Toast Crunch General Mills 19.8236
Almond Delight Ralston-Purina 34.3848

Question 14.116

antcolony

26. Ant Sizes. A study compared the sizes of ants from different colonies. Researchers measured the masses (in milligrams) of random samples of ants from three different colonies, which were selected independently. The samples are shown here. Test whether the population median sizes differ in the three ant colonies, using level of significance .

Colony Size Colony Size Colony Size Colony Size
3 78 2 59 2 77 1 75
3 89 2 74 3 116 1 87
1 78 1 43 3 29 3 144
2 111 1 130 3 153 1 112
2 147 3 122 3 93 1 65