Ch. 11 Review
Algebra III/Stats
Use the following to answer questions 1 through 5:
Using his TI-83/84, Zach generates 1000 random numbers that are supposed to follow a standard
normal distribution. He classify these 1000 numbers according to whether their values are less
than –2 (value < –2), between –2 and 0 (–2  value < 0), between 0 and 2 (0  value < 2), or at
least 2 (value ≥). The results are given in the following table. The expected counts are
computed using the 68–95–99.7 rule.
Observed count
Less than
–2
14
Between
–2 and 0
489
Between
0 and 2
476
At least
2
21
Expected count
25
475
475
25
To test to see if the distribution of observed counts differs significantly from the distribution of
expected counts, we use the χ2 statistic.
1. In this case, the χ2 statistic has approximately a chi-square (χ2) distribution. How many
degrees of freedom does this distribution have?
A) 1000 B) 999 C) 4 D) 3
2. The component (O – E)2/E of the χ2 statistic corresponding to the category “at least 2” is
A) 64
B) 16 C) 0.64
D) 0.16
3. The component (O – E)2/E of the χ2 statistic corresponding to the category “less than –2” is
A) 4.84
B) 2.42 C) 1.96
D) 121
4. The value of the χ2 statistic is about
A) 0.00211
B) 0.64
C) 4.84
D) 5.89
5. The P-value of the test is
A) greater than 0.20
B) between 0.10 and 0.20
C) between 0.05 and 0.10
D) between 0.01 and 0.05
6. Which of the following statements is true of chi-square distributions?
A) As the degrees of freedom increases, their density curves look less and less like a
normal curve.
B) Their density curves are skewed to the left.
C) As the degrees of freedom decreases, their density curves look more and more like a
skewed distribution.
D) They take on only positive values.
Use the following to answer questions 7 through 10:
A random sample of 100 traffic tickets given to motorists in a large city is examined. The tickets
are classified according to the race of the driver. The results are summarized in the following
table.
Number of tickets
White
46
Black
37
Hispanic
11
Other
6
The proportion of this city’s population in each of the racial categories listed above is as follows.
White
0.65
Proportion
Black
0.30
Hispanic
0.03
Other
0.02
We wish to test whether the racial distribution of traffic tickets in the city is the same as the
racial distribution of the city’s population. To do so, we use the χ2 statistic.
7. The component (O – E)2/E of the χ2 statistic corresponding to the category “Hispanic” is
A) 2.67
B) 5.82 C) 21.33
D) 36.51
E) 4011.36
8. We compute the value of the χ2 statistic to be 36.52. Assuming that this statistic has
approximately a χ2 distribution, the P-value of our test is
A) greater than 0.20
C) between 0.05 and 0.10
B) between 0.10 and 0.20
D) less than 0.01
9. The category that contributes the largest component to the χ2 statistic is
A) Other B) Hispanic C) Black D) White
10. What percent of the city’s population is not black?
A) less than 1%
B) 30%
C) 70%
D) 97%
Use the following to answer questions 11 through 17:
You are teaching a large introductory statistics course. In the past, the proportions of students
that received grades of A, B, C, D, or F have been, respectively, 0.20, 0.30, 0.30, 0.10, and 0.10.
This year, there were 200 students in the class, and the students earned the following grades.
Grade
Number
A
56
B
74
C
60
D
9
F
1
You wish to test to see whether the distribution of grades this year was different from the
distribution in the past. To do so, you plan to use the χ2 statistic.
11. Assuming that the χ2 statistic has approximately a χ2 distribution, how many degrees of
freedom does the distribution have?
A) 199 B) 5 C) 4 D) 1
12. The component (O – E)2/E of the χ2 statistic corresponding to the grade of A is
A) 0
B) 1 C) 3.2
D) 6.4
13. The χ2 statistic would be about
A) 33.77 B) 77.59 C) 96.1 D) 18.05
14. The P-value of the test is
A) greater than 0.20
B) between 0.10 and 0.20
C) between 0.01 and 0.05
D) less than 0.01
15. The grade category that contributes the smallest component to the χ2 statistic is
A) A B) B C) C D) D E) F
16. The grade category that contributes the largest component to the χ2 statistic is
A) A B) B C) C D) D E) F
17. What percent of the 200 students earned a grade of “C” or lower?
A) 99%
B) 70%
C) 60%
D) 35%
Use the following to answer questions 18 and 19:
Using a TI-83/84, Amber generates 1000 random numbers that are supposed to follow a standard
normal distribution. She classifies these 1000 numbers according to whether their values are less
than 0 or greater than or equal to 0. The results are given in the table below.
Number
Less Than 0
508
Greater Than or Equal to 0
492
Because the standard normal distribution is symmetric about 0, one would expect half of the
random numbers generated to be less than 0 and half to be greater than or equal to 0. To test to
see if the distribution of the observed number in each category differs significantly from the
expected distribution of counts, she uses the χ2 statistic.
18. The value of the χ2 statistic is
A) 0.256 B) 0.128 C) 0.288
D) 0.576
19. In this case, the χ2 statistic has approximately a χ2 distribution. How many degrees of
freedom does this distribution have?
A) 0 B) 1 C) 2 D) 999
Use the following to answer questions 20 through 25:
Are avid readers more likely to wear glasses than those who read less frequently? Three hundred
men in the Korean army were selected at random and classified according to whether or not they
wore glasses and whether the amount of reading they did was above average, average, or below
average. The results are presented in the following table.
Amount of Reading
Above average
Average
Below average
Wear Glasses?
Yes
No
47
26
48
78
31
70
20. This is an r × c table. The number c has value
A) 2 B) 3 C) 4 D) 6
21. The proportion of men in the table who do not wear glasses is
A) 0.58 B) 0.49 C) 0.42 D) 0.356
22. The proportion of all below-average readers who wear glasses is about
A) 0.10 B) 0.27 C) 0.31 D) 0.42
23. Suppose we wish to test the null hypothesis that there is no association between the
amount of reading you do and whether or not you wear glasses. Under the null
hypothesis, the expected number of average readers who do not wear glasses is
approximately
A) 58.58 B) 73.08 C) 25.7 D) 21.4
24. Suppose we wished to display in a graph the proportion of all above-average readers
who wear glasses and do not wear glasses, respectively. Which of the following
graphical displays is best suited to this purpose?
A) bar graph B) scatterplot C) box-and-whisker plot D) stemplot
25. Suppose we wish to test the null hypothesis that there is no association between the
amount of reading you do and whether or not you wear glasses. Under the null
hypothesis, the expected number of below-average readers who wear glasses is
approximately
A) 42.42 B) 58.58 C) 25.7 D) 25.5
Use the following to answer questions 26 through 28:
When a police officer responds to a call for help in a case of spousal abuse, what should the
officer do? A randomized controlled experiment in Dallas, Texas, studied three police responses
to spousal abuse: advise and possibly separate the couple, issue a citation to the offender, and
arrest the offender. The effectiveness of the three responses was determined by re-arrest rates.
The table below shows these rates.
# of Re-arrests
0
1
2
3
4
Arrest
175
36
2
1
0
Assigned Treatment
Citation
Advise/Separate
181
187
33
24
7
1
1
0
2
0
26. This is an r × c table. The number r has value
A) 2 B) 3 C) 4 D) none of these
27. Suppose the χ2 statistic has approximately a χ2 distribution for the above data. How
many degrees of freedom does this distribution have?
A) 0 B) 1 C) 2 D) none of the above
28. Suppose we wish to test the null hypothesis that the distribution of the number of
subsequent arrests is the same regardless of the treatment assigned. When this null
hypothesis is true, the expected number of times that no subsequent arrest would occur
for the treatment “Citation” is about
A) 177 B) 179 C) 181 D) 187
29. Suppose we wish to test some null hypothesis, and the P-value was calculated under the
χ2 distribution curve. Under what circumstances could we reject the null hypothesis?
A) We should always reject the null hypothesis, because P-values are irrelevant.
B) If the P-value is very large.
C) If the P-value is very small.
D) We cannot reject the null hypothesis under any circumstances.
Use the following to answer questions 30 through 32:
Even though Puerto Rico is a territory of the United States, there are many cultural differences
between the states on the continent of North America and the Caribbean island of Puerto Rico.
These differences include the way consumers handle problems with purchases. Two researchers
surveyed owners of i-phones in the Northeastern United States and in Puerto Rico. They asked
those who had experienced problems with their i-phones whether they complained. The results
are given in the table below.
Complained?
No
Yes
N.E. United States
94
330
Puerto Rico
33
64
30. This is an r × c table. The number c has value
A) 1 B) 2 C) 3 D) 4
31. The cell that contributes most to the χ2 statistic is
A) Puerto Ricans who did not complain.
B) Puerto Ricans who complained.
C) Americans in the Northeastern United States who did not complain.
D) Americans in the Northeastern United States who complained.
32. The P-value for testing the null hypothesis that the probability of complaining is the
same for the Northeastern United States and Puerto Rico, against the alternative that the
probability of complaining is different for the Northeastern United States and Puerto
Rico,
A) is between 0.05 and 0.10
B) is between 0.025 and 0.05
C) is between 0.01 and 0.025
D) is less than 0.01
33. The cell that contributes the smallest amount to the χ2 statistic is
A) Puerto Ricans who did not complain.
B) Puerto Ricans who complained.
C) Americans in the Northeastern United States who did not complain.
D) Americans in the Northeastern United States who complained.
Use the following to answer questions 34 through 38:
All current-carrying wires produce electromagnetic (EM) radiation, including the electrical
wiring running into, through, and out of our homes. High-frequency EM radiation is thought to
be a cause of cancer; the lower frequencies associated with household current are generally
assumed to be harmless. To investigate this, researchers visited the addresses of children in the
Denver area who had died of some form of cancer (leukemia, lymphoma, or some other type)
and classified the wiring configuration outside the building as either a high-current configuration
(HCC) or as a low-current configuration (LCC). Here are some of the results of the study.
HCC
LCC
Leukemia
52
84
Lymphoma
10
21
Other Cancers
17
31
Computer software was used to analyze the data. The output is given below. It includes the cell
counts, the expected cell counts, and the value of the χ2 statistic. In the table, expected counts
are printed below observed counts and enclosed within parentheses.
HCC
LCC
Total
Leukemia
52
(49.97)
84
(86.03)
136
Lymphoma
10
(11.39)
21
(19.61)
31
Other Cancers
17
(17.64)
31
(30.36)
48
Total
79
136
215
χ2 = 0.082 + 0.170 + 0.023 + 0.048 + 0.099 + 0.013 = 0.435
34. The appropriate degrees of freedom for the χ2 statistic is
A) 0 B) 1 C) 2 D) 3
35. The P-value of the test is
A) larger than 0.20
B) between 0.10 and 0.20
C) between 0.05 and 0.10
D) between 0.01 and 0.05
36. The cell that contributes the most to the χ2 statistic is
A) the cases of leukemia that occurred in homes with an HCC.
B) the cases of leukemia that occurred in homes with an LCC.
C) the cases of other cancers that occurred in homes with an LCC.
D) the cases of lymphoma that occurred in homes with an HCC.
37. The cell that contributes the smallest amount to the χ2 statistic is
A) the cases of leukemia that occurred in homes with an HCC.
B) the cases of leukemia that occurred in homes with an LCC.
C) the cases of other cancers that occurred in homes with an LCC.
D) the cases of lymphoma that occurred in homes with an LCC.
38. Which of the following may we conclude based on the test results?
A) There is weak evidence that LCC causes cancer in children.
B) There is not much evidence of an association between wiring configuration and the
type of cancer that caused the deaths of children in the study.
C) Cancer is definitely caused by the HCC or LCC wiring of a building.
D) There is weak evidence that HCC causes cancer in children.
Use the following to answer questions 39 through 41:
A study was performed to examine the personal goals of middle school kids in grades 6, 7, and 8.
A random sample of students was selected from each of grades 6, 7, and 8 from middle schools
in Alabama. The students received a questionnaire regarding achieving personal goals. They
were asked what they would most like to do at school: make good grades, be good at sports, or
be popular. The results are presented in the table below, grouped by the sex of the student.
Top Personal Goal
Make good grades
Be popular
Be good in sports
Boys
71
29
95
Girls
301
49
20
39. What percent of all girls chose “make good grades” as most important?
A) about 81.35 %
B) about 53.27 %
C) about 19.19 %
D) about 13.24 %
40. The numerical value of the χ2 statistic for this table is closest to
A) 7.63 B) 58 C) 119 D) 157
41. The P-value of the test is
A) less than 0.001 B) 2.911
C) 2.0
D) 16.105
42. Are avid readers more likely to wear glasses than those who read less frequently? Five
hundred men in the Korean army were selected at random and characterized as to
whether they wore glasses and whether the amount of reading they did was above
average, average, or below average. The results are presented in the following table.
Wear Glasses?
Yes
No
Amount of Reading
Above average
111
77
Average
84
55
Below average
95
78
The numerical value of the χ2 statistic for testing the null hypothesis that the amount of
reading you do and whether or not you wear glasses are independent is about
A) 0 B) 1 C) 2 D) 3
43. The appropriate degrees of freedom for the above problem is
A) 0 B) 1 C) 2 D) 3
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Algebra III/Statistics