Chapter 10
Quiz 1
Name ________________________________________________________
Date ___________________
1. Decide whether each of these statements is true or false. a. When the number of categories is small, the shape of the c 2 distribution is skewed right. b. You can use a chi-square goodness-of-fit test to assess how well a given probability model fits your data. c. In a chi-square goodness-of-fit test, the expected frequencies are always positive integers. d. As the number of categories increases, the shape of the c 2 distribution gradually becomes more skewed and less mound-shaped. Questions 2 and 3 refer to these data: The middle column of this table lists the actual percent of traffic stops, categorized by the age of the driver, for the Las Vegas Metropolitan Police Department in 2002. Suppose that in a random sample of 1000 traffic stops for the following year, the distribution was as shown in the column on the right. Age Younger than 18
Percent Stopped in 2002
Number Stopped in 2003
3.0
40
18–29
40.5
435
30–39
27.4
280
40–49
16.5
150
50 and older
12.6
95
2. If you test the goodness of fit of the 2003 sample to the 2002 distribution, the expected frequency for drivers aged 18–29 in 2003 is A. 40.5 B. 81 C. 200 D. 405 E. 435 3. Suppose you use a chi-square goodness-of-fit test to test the null hypothesis that the age distribution of drivers stopped in 2003 for traffic violations is the same as the age distribution in 2002. You compute c 2 = 14.7. At the 5% significance level, which of these is an appropriate conclusion? A. Because the P-value is less than 0.05, you reject the null hypothesis. B. Because the P-value is less than 0.05, you fail to reject the null hypothesis. C. Because the P-value is greater than 0.05, you reject the null hypothesis. D. Because the P-value is greater than 0.05, you fail to reject the null hypothesis. E. Because the conditions for a chi-square goodness-of-fit test are not satisfied, you can’t draw an appropriate conclusion.
Statistics in Action Instructor’s Resource Book © 2008 Key Curriculum Press
Chapter 10 Quiz 1
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Chapter 10
Quiz 1 (continued)
4. Which of these statements is not true for a chi-square goodness-of-fit test? A. The test may be used to check whether a die is fair. B. The test may be used to check whether observed frequencies come from a population that is the same as a given theoretical distribution. C. When there are two categories, the test is equivalent to a z-test of a proportion. D. You can reject the null hypothesis when the P-value is smaller than a , the level of significance. E. The alternative hypothesis states that all the proportions in your population are different from the proportions in your model. 5. This table gives the final grade distribution in one statistics class of 23 students at a large university. At this university, instructors are encouraged to give roughly equal percentages of grades A, B, C, D, and F. The dean claims that the instructor of this class has departed from this practice. Check the conditions for a chi-square goodness-of-fit test to see whether the dean can use this test to test her claim.
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Chapter 10 Quiz 1
Grade
Count
A
3
B
8
C
6
D
3
F
3
Statistics in Action Instructor’s Resource Book © 2008 Key Curriculum Press
Chapter 10
Quiz 2
Name ________________________________________________________
Date ___________________
1. Decide whether each statement is true or false. a. To determine whether there is an association between voting and subscribing to a newspaper, an investigator takes independent random samples of those who voted and of those who didn’t vote in the last election and then asks each person whether he or she subscribes to a newspaper. The appropriate chi-square test has 4 degrees of freedom. b. In a chi-square test of independence, independent random samples are taken from two or more populations. c. Chi-square tests of independence and homogeneity both are usually right-tailed tests. d. The null hypothesis for a chi-square test of homogeneity is that the proportion of the population that falls into any given category is the same for every population. Questions 2–4 refer to these data: The two-way table shown here presents data about the race/ethnicity and gender, per thousand, of a random sample of drivers stopped for traffic violations by the Las Vegas Metropolitan Police Department. (Note: The “Other” column for race/ethnicity was not included because of its very small numbers.) Race/Ethnicity (per thousand)
Gender (per thousand)
White
Black
Male
372
107
Female
186
Total
558
Hispanic
Asian
Total
184
33
696
50
49
19
304
157
233
52
1000
2. What is the expected number of Hispanic males per thousand persons stopped for traffic violations, assuming race/ethnicity and gender are independent? A.
233 1000
D. 233 ×
1000 696
B.
696 1000
E. 233 ×
C. 696 ×
1000 233
696 1000
3. Suppose you use a chi-square test to test the claim that there is an association between race/ethnicity and gender among those stopped for traffic violations. You compute c 2 = 13.02 . Which of these is an appropriate conclusion at the 5% significance level? A. Use a chi-square test of homogeneity. The P-value is less than 0.05. This is sufficient evidence to reject the null hypothesis that race/ethnicity and gender are not associated.
Statistics in Action Instructor’s Resource Book © 2008 Key Curriculum Press
Chapter 10 Quiz 1
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Chapter 10
Quiz 2 (continued)
B. Use a chi-square test of homogeneity. The P-value is greater than 0.05. This is insufficient evidence to reject the null hypothesis that race/ethnicity and gender are not associated. C. Use a chi-square test of independence. The P-value is less than 0.05. This is sufficient evidence to reject the null hypothesis that race/ethnicity and gender are not associated. D. Use a chi-square test of independence. The P-value is greater than 0.05. This is insufficient evidence to reject the null hypothesis that race/ethnicity and gender are not associated. E. A chi-square test cannot be used because at least one of the expected values is too small. 4. Construct a segmented bar chart for the sample in the preceding two-way table. Does it appear from your plot that the proportion of females among those who have been stopped for traffic violations is the same for each race? Explain using only your plot.
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Chapter 10 Quiz 1
Statistics in Action Instructor’s Resource Book © 2008 Key Curriculum Press
Chapter 10
Test A
Name ________________________________________________________
Date ___________________
1. A fair die is rolled 60 times, and the value of c 2 is computed using expected counts of 10 for each face. If this is repeated many times, the shape of the distribution of the values of c 2 should be A. approximately normal. B. skewed right. C. skewed left. D. uniform. E. bimodal. 2. The major difference between the chi-square test of homogeneity and the chi-square test of independence is the A. number of categories. B. sample size. C. method of sampling. D. size of the c 2 statistic. E. number of degrees of freedom. 3. Which of these statements is not true? A. A chi-square test of independence that is statistically significant shows a cause-and-effect relationship. B. A segmented bar chart is useful in observing when two variables might be associated. C. As the number of categories increases, the c 2 distribution approaches the normal distribution. D. The chi-square tests involve categorical variables. E. The chi-square goodness-of-fit test is an extension of the z-test to more than two categories. 4. The null hypothesis is rejected in a chi-square test of significance when A. the test conditions are satisfied. B. the P-value is larger than a , the level of significance. C. the P-value is larger than 1 - a . D. the c 2 statistic is smaller than the critical value for the given level of significance. E. the c 2 statistic is larger than the critical value for the given level of significance.
Statistics in Action Instructor’s Resource Book © 2008 Key Curriculum Press
Chapter 10 Quiz 1
5
Chapter 10
Test A (continued)
5. Which test is appropriate for determining whether a random-digit generator is truly random in terms of the proportions of each digit it produces? A. the chi-square goodness-of-fit test B. the chi-square test of homogeneity C. the chi-square test of independence D. Either B or C is appropriate. E. None of these tests is appropriate. 6. Suppose that 85% of all Americans are right-handed, 10% are left-handed, and 5% are ambidextrous. A random sample of 114 Mississippians includes 80 right-handers, 31 lefthanders, and 3 ambidextrous. What is the value of the c 2 statistic for the goodness-of-fit test that the distribution of handedness for all Mississippians is the same as the distribution for all Americans, and what is the correct conclusion? A. c 2 » 37.92; not significant at the 0.05 level B. c 2 » 37.92; significant at the 0.05 level C. c 2 » 2.37; not significant at the 0.05 level D. c 2 » 2.37; significant at the 0.05 level E. Because the conditions are not satisfied, significance can’t be determined. 7. An educator randomly selects 300 statistics students to check whether there is a relationship between a student passing the course and his or her number of absences. These results were reported. Number of Classes Missed 0–1
2–3
4 or More
Total
Passed Statistics Course
30
50
100
180
Didn’t Pass Statistics Course
10
30
80
120
Total
40
80
180
300
For a chi-square test of independence, what type of error may have been made? A. Type I: the P-value was greater than 0.05. B. Type I: the P-value was less than 0.05. C. Type II: the P-value was greater than 0.05. D. Type II: the P-value was less than 0.05. E. Either type of error is possible. 8. In Question 7, what is the expected count for the number of students who were absent 0 or 1 time and who also passed the statistics course? Show your computation.
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Chapter 10 Quiz 1
Statistics in Action Instructor’s Resource Book © 2008 Key Curriculum Press
Chapter 10
Test A (continued)
9. Match the survey designs in parts a–c with the most appropriate chi-square test: goodness of fit, homogeneity, or independence. a. You are told that the number of cracked M&M’s depends on color. To check this claim, you randomly select 100 M&M’s and sort by color and whether the M&M is cracked or uncracked. b. You are told that the distribution of M&M’s colors is as follows: 13% red, 16% green, 14% yellow, 20% orange, 24% blue and 13% brown. To check this claim, you randomly select a sample of M&M’s and count the number of M&M’s of each color. c. You are told that the number of cracked M&M’s depends on color. To check this claim, you randomly select 100 M&M’s of each color and count the number of cracked and uncracked M&M’s of each color. 10. A sociology class is interested in knowing whether there is a relationship between personality style and type of extracurricular activity. The class took a random sample of 200 students from a large high school and recorded these data. Personality Style Extrovert Type of Extracurricular Activity
Normal
Introvert
Sports
21
44
25
Club
16
33
20
None
3
16
22
Using the standard four-step process (name test and check conditions; state hypotheses; compute test statistic and P-value and draw a sketch; and write a conclusion in context), test the null hypothesis that personality style and type of extracurricular activity are independent. 11. A pollster was hired by a television studio to take a random sample of 400 high school students, 100 from each of grades 9–12, and ask the students whether they watch at least one reality television show on a regular basis. The percentages that replied yes for grades 9, 10, 11, and 12 were 28%, 18%, 30%, 33%, respectively. The studio wishes to know whether there is evidence that the percentage varies among the grade levels. a. What significance test would you use for this design—a test of goodness of fit, homogeneity, or independence? Explain your choice. b. Perform the test you selected, showing all four steps.
Statistics in Action Instructor’s Resource Book © 2008 Key Curriculum Press
Chapter 10 Quiz 1
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Chapter 10
Test B
Name ________________________________________________________
Date ___________________
1. A fair 20-sided die is rolled 120 times, and the value of c 2 is computed using expected counts of 6 for each face. If this process is repeated many times, the shape of the distribution of the values of c 2 should be A. slightly skewed right. B. extremely skewed right. C. slightly skewed left. D. extremely skewed left. E. bimodal. 2. In which of these three chi-square tests do you take just one random sample from one large population? I. chi-square goodness-of-fit test II. chi-square test of homogeneity III. chi-square test of independence A. I only
B. II only
D. I and III
E. II and III
C. III only
3. Which of these statements is not true? A. In general, you can reject the null hypothesis when c 2 is very large. B. A segmented bar chart is useful in observing when two variables may be associated. C. For a chi-square test of independence, the larger the c 2 statistic, the weaker the association between the two variables. D. In general, a very large c 2 value corresponds to a very small, statistically significant, P-value. E. The chi-square goodness-of-fit test is an extension of the z-test to more than two categories. 4. The middle column of this table lists all alcohol-related motor vehicle fatalities in the United States in 2002, categorized by age. Suppose that in a random sample of 1000 alcohol-related motor vehicle fatalities for the following year, the distribution was as shown in the column on the right. Alcohol-Related Motor Vehicle Fatalities in 2002
Alcohol-Related Motor Vehicle Fatalities in 2003
Percent
Number
Younger than 20
14.5
156
20–29
24.5
223
30–39
23.3
246
40–49
19.1
171
50 and older
18.6
204
Age
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Chapter 10 Quiz 1
Statistics in Action Instructor’s Resource Book © 2008 Key Curriculum Press
Chapter 10
Test B (continued)
Suppose you use a chi-square goodness-of-fit test to test the null hypothesis that the age distribution of alcohol-related motor vehicle fatalities in 2003 is the same as the age distribution in 2002. You compute c 2 = 7.37. At the 5% significance level, which of these is an appropriate conclusion? A. Because the P-value is less than 0.05, you reject the null hypothesis. B. Because the P-value is less than 0.05, you fail to reject the null hypothesis. C. Because the P-value is greater than 0.05, you reject the null hypothesis. D. Because the P-value is greater than 0.05, you fail to reject the null hypothesis. E. Because the conditions for a chi-square goodness-of-fit test are not satisfied, an appropriate conclusion cannot be drawn. 5. You take a random sample from each of the four grade levels in your high school in order to test whether the proportion of students that favor changing the start time of the school day is the same for each grade level. Which test is appropriate for testing your hypothesis (assuming the test conditions are satisfied)? A. the chi-square goodness-of-fit test B. the chi-square test of homogeneity C. the chi-square test of independence D. the z-test E. None of these tests is appropriate. 6. An educator randomly selects 300 AP Statistics students who took this year’s AP Statistics Exam and wishes to check whether there is an association between whether a student passed the AP Statistics Exam with a score of 3 or higher and the number of AP Exams that student took this year. The results are given in the table below. Number of AP Exams Taken
Passed AP Statistics Exam with Score of 3 or Higher Got Score Less Than 3 Total
1–2
3–4
5 or More
Total
100
50
30
180
80
30
10
120
180
80
40
300
For a chi-square test of independence, what type of error may have been made? A. Type I: the P-value was greater than 0.05. B. Type I: the P-value was less than 0.05. C. Type II: the P-value was greater than 0.05. D. Type II: the P-value was less than 0.05. E. Either type of error is possible.
Statistics in Action Instructor’s Resource Book © 2008 Key Curriculum Press
Chapter 10 Quiz 1
9
Chapter 10
Test B (continued)
7. A researcher randomly selects 100 people in order to test whether there is a relationship between the size of a person’s handwriting and his or her gender. These results are given in the table below. Writing Size Large
Medium
Small
Total
Male
18
21
14
53
Female
13
15
19
47
Total
31
36
33
100
The c 2 statistic for these data is about 2.21, with 2 degrees of freedom and the P-value is about 0.33. The best conclusion for these results is that A. because the c 2 statistic is relatively large, the null hypothesis is rejected. B. because the c 2 statistic is relatively small, the null hypothesis is rejected. C. because the P-value is less than 0.5, the null hypothesis is rejected. D. there is evidence of a relationship between gender and writing size. E. there is no evidence of a relationship between gender and writing size. 8. What is the expected count for the number of females with a small writing size in Question 7? Show your computation. 9. Match the survey designs in parts a–c with the most appropriate chi-square test: goodness of fit, homogeneity, or independence. a. You are told that the proportion of arrested shoplifters that are male is the same for each of four age groups. You randomly select 100 arrested shoplifters and categorize each person by sex and by age group. b. You are told that the proportion of arrested shoplifters that are male is the same for each of four age groups. You randomly select 100 arrested shoplifters from each of the four age groups and count the number of males in each age group. c. You are told that males arrested for shoplifting have this age distribution: ages 18–24, 40%; ages 25–35, 35%; ages 36–49, 20%; and age 50 and older, 5%. You randomly select 100 males arrested for shoplifting and count the number in each age group to check the claim. 10. A sociology class is interested in knowing whether there is an association between a teacher’s disciplinary style and his or her level of education. The class took a sample of 200 teachers and recorded these data: Disciplinary Style
Level of Education
Weak
Normal
Strict
Bachelor’s Degree
16
39
16
Master’s Degree
11
35
21
9
19
34
Master’s Degree + 30 Credits
10
Chapter 10 Quiz 1
Statistics in Action Instructor’s Resource Book © 2008 Key Curriculum Press
Chapter 10
Test B (continued)
Using the standard four-step process (name test and check conditions; state hypotheses; compute test statistic and P-value and draw sketch; and write a conclusion in context), test the null hypothesis that disciplinary style and level of education are independent. 11. A high school principal took independent random samples of 50 students from each grade, 9 through 12, and recorded the number who were absent on at least one Friday during the school year. The number of the 50 students with at least one absence on a Friday in each of grades 9 through 12 were 4, 4, 6, and 11, respectively. The principal wishes to test the null hypothesis that the absence rate is the same for each grade level. a. What significance test would you use for this design—a test of goodness of fit, homogeneity, or independence? Explain your choice. b. Perform the test you selected, showing all four steps.
Statistics in Action Instructor’s Resource Book © 2008 Key Curriculum Press
Chapter 10 Quiz 1
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