AP Statistics - Chapter 9 Warm-Ups
Using computer software, I generate 1000 random numbers that are supposed to follow a standard normal distribution. I 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 greater than or equal to 2 (2  value). The results are given in the following table. The expected counts are
computed using the 68–95–99.7 rule.
Between
Between
< –2
–2 and 0
0 and 2
2
Observed
18
492
468
22
Expected
25
475
475
25
1. To test if the distribution of observed counts differ significantly from the expected distribution of counts we use
the X2 statistic. We know that this X2 statistic has approximately a 2 distribution. How many degrees of freedom
does this distribution have?
A) 3
B) 4
C) 999
D) 1000
2. To test if the observed counts differ significantly from the expected distribution of counts we use the X2 statistic.
The component of this X2 statistic corresponding to the “less than 2” category is
A) (O – E)2/E = 0.28
B) (O – E)2/E = 1.96
C) (O – E)2/E = 2.72
D) (O – E)2/E = 49
3. To test if the observed counts differ significantly from the expected distribution of counts we use the X2 statistic.
The value of this X2 statistic is found to be 3.03. The P-value of our test is
A) greater than 0.20
B) between 0.20 and 0.10
C) between 0.10 and 0.05
D) less than 0.05
4. Which of the following is true of chi-square distributions?
A) They take on only positive values
B) They are skewed to the left
C) As the number of degrees of freedom increases, they look less and less like a normal curve
D) all of the above
I teach a large introductory statistics course. In the past, the proportion of students that receive a grade of A is 0.20. The proportion
that receives a B is 0.30. The proportion that receives a C is 0.30. The proportion that receives a D is 0.10. The proportion that
receives an F is 0.10. This year, there were 200 students in the class and I gave the following grades.
A
B
C
D
F
Number
56
74
60
9
1
9. I wish to test whether the distribution of grades this year is the same as in the past. To do so I plan to use the X2
statistic. We know that this X2 statistic has approximately a 2 distribution. How many degrees of freedom does
this distribution have?
A) 200
B) 199
C) 5
D) 4
10. I wish to test whether the distribution of grades this year is the same as in the past. To do so I plan to use the X2
statistic. The component of this X2 statistic corresponding to a grade of C is
A) (O – E)2/E = 0
B) (O – E)2/E = 1
C) (O – E)2/E = 30
D) (O – E)2/E = 11,880.3
11. I wish to test whether the distribution of grades this year is the same as in the past. To do so I use the X2 statistic
and I compute its value to be 33.77. The P-value of our test is
A) greater than 0.10
B) between 0.10 and 0.05
C) between 0.05 and 0.01
D) less than 0.01
12. I wish to test whether the distribution of grades this year is the same as in the past. To do so I use the X2 statistic. I
may assume the X2 statistic has an approximate chi-square distribution because of which of the following?
A) The expected number of people in each grade category is greater than 5
B) The sample size is 200, which is large enough for the chi-square approximation to be valid
C) The number of categories is small relative to the number of observations
D) I may not assume X2 has an approximate chi-square distribution because there is only 1 person in the F category
13. I wish to test whether the distribution of grades this year is the same as in the past. To do so I use the X2 statistic.
The grade category that contributes the largest component to the X2 statistic is
A) A
B) B
C) D
D) F
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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 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.
16. This is an r x c table. The number r has value
A) 2
B) 3
C) 4
D) 6
17. The proportion of men in the table who wore glasses is
A) 0.24
B) 0.37
C) 0.42
D) 0.64
Amount of Reading
Above average
Average
Below average
Wear Glasses
Yes
No
47
26
48
80
31
70
18. The proportion of all above-average readers who wear glasses is
A) 0.24
B) 0.37
C) 0.42
D) 0.64
19. Suppose we wish to test the null hypothesis that there is no association between the amount of reading you do and
whether you wear glasses. Under the null hypothesis, the expected number of above average readers who wear
glasses is
A) 81.1
B) 47
C) 30.7
D) 27.2
20. Suppose we wish to display in a graphic the proportion of all above average readers that wear glasses and do not
wear glasses, respectively. Which of the following graphical displays is best suited to this purpose?
A) a stemplot
B) a scatterplot
C) a boxplot
D) a bar graph
A study was performed to examine the personal goals of children in grades 4, 5, and 6. A
random sample of students was selected for each of the grades 4, 5, and 6 from schools in
Georgia. 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. Results are presented in the table below by the sex of the child.
Make good grades
Be popular
Be good in sports
Boys
96
32
94
31. Which hypotheses are being tested by the chi-square test?
A) The null hypothesis is that personal goals and gender are independent and the alternative is that they are dependent
B) The null hypothesis is that the mean personal goal is the same for boys and girls and the alternative is that the
means differ
C) The distribution of personal goals is different for boys and girls
D) The distribution of gender is different for the three different personal goals
32. The numerical value of the chi-square statistic for this table is
A) 3.84
B) 5.99
C) 16.105
D) 89.966
33. Recent revenue shortfalls in a Midwestern state led to a reduction in the state budget for higher education. To
offset the reduction, the largest state university proposed a 25% tuition increase. It was determined that such an
increase was needed simply to compensate for the lost support from the state. Random samples of 50 freshmen, 50
sophomores, 50 juniors, and 50 seniors from the university were asked whether or not they were strongly opposed
to the increase, given that it was the minimum increase necessary to maintain the university's budget at current
levels. The results are given in the following table. Which hypotheses are being tested by the chi-square test?
Year
Strongly Opposed
Freshman
Sophomore
Junior
Senior
Yes
39
36
29
18
No
11
14
21
32
A) The null hypothesis is that year in school and whether or not you are strongly opposed are independent and the
alternative is that they are dependent
B) The null hypothesis is that the mean number of who are strongly opposed is the same for each of the four years and
the alternative is that these means are different
C) The null hypothesis is that the distributions of the number who are strongly opposed versus not strongly opposed
are the same for the four years. The alternative says that these distributions are different
D) The null hypothesis is that the distributions of the total number of students sampled in each of the four years are
the same. The alternative is that these distributions are different
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Girls
295
45
40
Answer Key
1.
2.
3.
4.
9.
10.
11.
12.
13.
16.
17.
18.
19.
20.
31.
32.
33.
A
B
A
A
D
A
D
A
D
B
C
D
C
D
A
D
C
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