Download AP STATISTICS EXAM REVIEW - Glen Ridge Public Schools

AP STATISTICS
EXAM REVIEW
(click on topic of choice)
TOPIC I:
Describing Data
(15 questions)
TOPIC II:
Experimental Design
(15 questions)
TOPIC III:
Probability
(18 questions)
TOPIC IV:
Inference
(15 questions)
TOPIC I: Describing Data
• Univariate Data
• Normal Distributions
• Bivariate Data
Topic I
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I.1. Which observation has the higher z-score?
I. x=25.4; μ =12.9; σ=3.7
II. x=25.4; μ=15.3; σ=2.7
A.
B.
C.
D.
I
II
z-scores are equal
cannot be determined since we don’t
know the standard deviations of the
populations
E. cannot be determined since we don’t
know if the populations are normal
Topic I
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SOLUTION
I.1. Which observation has the higher z-score?
I. x=25.4; μ =12.9; σ=3.7
II. x=25.4; μ=15.3; σ=2.7
The z-score for I is
I
3.38; the z-score for II
II
z-scores are equal
is 3.74
cannot be determined since we don’t
know the standard deviations of the
populations
E. cannot be determined since we don’t
Topic I
know if the populations are normal
Menu
A.
B.
C.
D.
I.2. The Quartile and Percentile positions of
the value 10 in the set
{10, 6, 8, 9, 12, 17, 32, 16}
are:
A.
B.
C.
D.
E.
Q3; 38
Q2; 38
Q2; 26
Q3; 26
none of these
Topic I
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SOLUTION
I.2. The Quartile and Percentile positions of
the value 10 in the set
A.
B.
C.
D.
E.
{10, 6, 8, 9, 12, 17, 32, 16}
are:
When the values are in
Q3; 38
order, there are 3 values
Q2; 38
below
10.
This
puts
10
in
Q2; 26
the 37.5 percentile and
Q3; 26
nd quartile.
the
2
none of these
Topic I
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I.3.
A.
B.
C.
D.
E.
The Standard Deviation of the set
{5, 7, 7, 8, 10, 11} is 2. Which of the
following sets also has a Standard
Deviation of 2?
{4, 5, 8, 12, 14}
{2, 4, 6, 8, 10, 12}
{3, 5, 5, 6, 8, 9}
{10, 14, 14, 16, 20, 22}
none of the above
Topic I
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SOLUTION
I.3.
A.
B.
C.
D.
E.
The Standard Deviation of the set
{5, 7, 7, 8, 10, 11} is 2. Which of the
following sets also has a Standard
Deviation of 2?
{4, 5, 8, 12, 14}
{2, 4, 6, 8, 10, 12}
{3, 5, 5, 6, 8, 9}
{10, 14, 14, 16, 20, 22}
none of the above
There is a uniform
decrease of 2 units in
this set. The st.
deviation must be
the same.
Topic I
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I.4.
A.
B.
C.
D.
E.
If 12% of the values of a data set lie
between a and b and d is added to each
value, then which of the following is true?
12% still lies between a and b
12% lies between a + d and b + d
(12+d)% lies between a and b
(12+d)% lies between a + d and b + d
there is no way to tell how much data
is between a and b
Topic I
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SOLUTION
I.4.
A.
B.
C.
D.
E.
If 12% of the values of a data set lie
between a and b and d is added to each
value, then which of the following is true?
A uniform
shift of all the
12% still lies between a and b
data
12% lies between a + d and b + d
maintains the
(12+d)% lies between a and b
(12+d)% lies between a + d and b + dpercentages
there is no way to tell how much dataof
is data in
between a and b
shifted
intervals. Topic I
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I.5.
A.
B.
C.
D.
E.
If a distribution is relatively symmetric
and mount-shaped, order the following
(from least to greatest)
1. a z-score of 1
2. the value of Q3
3. a value in the 70th percentile
1, 2, 3
1, 3, 2
3, 2, 1
3, 1, 2
2, 3, 1
Topic I
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SOLUTION
I.5. If a distribution is relatively symmetric
and mount-shaped, order the following
(from least to greatest)
1. a z-score of 1
2. the value of Q3
3. a value in the 70th percentile
A.
B.
C.
D.
E.
1, 2, 3
1, 3, 2
3, 2, 1
3, 1, 2
2, 3, 1
The percentile of a z-score of 1
is about 84%, and the
percentile of Q3 is 75%
Topic I
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I.6. Which of the following would NOT be a
correct interpretation of a correlation
coefficient of r = -.30
A. The variables are inversely related
B. The coefficient of determination is 0.09
C. 30% of the variation between the
variables is linear
D. There exists a weak relationship
between the variables
E. All are correct
Topic I
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SOLUTION
I.6. Which of the following would NOT be a
correct interpretation of a correlation
coefficient of r = -.30
The value of
A. The variables are inversely related
r2 explains
B. The coefficient of determination is 0.09
C. 30% of the variation between the
the variation
variables is linear
between the
D. There exists a weak relationship
variables.
between the variables
Not r.
E. All are correct
Topic I
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I.7. Which of the following displays is best
suited for categorical data?
A.
B.
C.
D.
E.
Box Plot
Bar Graph
Stem and Leaf Plot
Dot Plot
Scatterplot
Topic I
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SOLUTION
I.7. Which of the following displays is best
suited for categorical data?
A.
B.
C.
D.
E.
Box Plot
Bar Graph
Stem and Leaf Plot
Dot Plot
Scatterplot
In a bar graph, each
column is separate,
allowing for categorical
separation.
Topic I
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I.8.
A.
B.
C.
D.
E.
Linear regression usually employs the
method of least squares. Which of the
following is the quantity that is
minimized by the least squares process?
ŷi
xi  xi
2
ˆ
(
y

y
)
 i i
( xi , yi )
(x  x )
i
2
i
Topic I
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SOLUTION
I.8. Linear regression usually employs the
method of least squares. Which of the
following is the quantity that is
minimized by the least squares process?
A.
B.
C.
D.
E.
ŷi
xi  xi
2
ˆ
(
y

y
)
 i i
Least Squares Regression
minimizes the residuals
in the y-direction.
( xi , yi )
(x  x )
i
2
i
Topic I
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I.9.
Which of the following is NOT true?
A. Two sets of data can have the same
means but different variances
B. Two sets of data can have the same
variance but different means
C. Two different values in a data set can
have the same z-score
D. All the absolute values of z-scores for a
data set can be equal
E. All of the above are true
Topic I
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SOLUTION
I.9.
A.
B.
C.
D.
E.
Which of the following is NOT true?
Since each
value is a
Two sets of data can have the same
distinct
means but different variances
distance
Two sets of data can have the same
from the
variance but different means
Two different values in a data set can
mean, the
have the same z-score
z-scores
All the absolute values of z-scores for a
must all be
data set can be equal
different
All of the above are true
Topic I
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I.10.
In a symmetric, mount-shaped
distribution, what percentile has a z-score
of -2?
Topic I
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SOLUTION
I.10.
In a symmetric, mount-shaped
distribution, what percentile has a z-score
of -2?
2.5th
percentile
We should be able to
approximate this with the
normal distribution. Area to
the left of -2?
On the TI-83:
Topic I
Normalcdf(-1E99, -2) = .02275
Menu
I.11.
A.
B.
C.
D.
E.
Lee’s z-score on his math test was 1.5.
The class average was a 62.1 and the
variance was 6.76. What was Lee’s actual
grade on the test?
60
62
64
66
68
Topic I
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SOLUTION
I.11.
A.
B.
C.
D.
E.
Lee’s z-score on his math test was 1.5.
The class average was a 62.1 and the
variance was 6.76. What was Lee’s actual
grade on the test?
60
62
64
66
68
We need to solve the
following equation for x:
z
x

x  62.1
1.5 
6.76 so x=66
Topic I
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I.12. Does the following problem have a unique
solution? If so, find it. If not, show at
least two answers:
5 numbers have Q1=12, Median=15,
Q3=18. Find the mean.
Topic I
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SOLUTION
I.12. Does the following problem have a unique
solution? If so, find it. If not, show at
least two answers:
5 numbers have Q1=12, Median=15,
Q3=18.
the17,mean.
{11,Find
13, 15,
19} μ=15 { 9,
15, 15, 17, 19} μ=15
YES!!!
surprisingly!
the data sets are not
unique, but the mean is
always 15!
Topic I
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I.13. The average grade on a math test given to
two sections is 60.98. Section I has 27
students with a mean grade of 57.30. If
the mean grade of Section 2 is 65.30, how
many students are in section 2?
Topic I
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SOLUTION
I.13. The average grade on a math test given to
two sections is 60.98. Section I has 27
students with a mean grade of 57.30. If
the mean grade of Section 2 is 65.30, how
many students are in section 2?
23 students.
57.30 * 27  65.30n
 60.98
27  n
Topic I
we arrive at n=23
Menu
I.14. If the mean of 75 values is 52.6 and the
mean of 25 values is 48.4; find the mean
of all 100 values.
A.
B.
C.
D.
E.
51.55
52.76
56.55
56.88
59.12
Topic I
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SOLUTION
I.14. If the mean of 75 values is 52.6 and the
mean of 25 values is 48.4; find the mean
of all 100 values.
A.
B.
C.
D.
E.
51.55
52.76
56.55
56.88
59.12
75(52.6) + 25(48.4)
100
Topic I
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I.15.
A.
B.
C.
D.
E.
In skewed-right distributions, what is
most frequently the relationship of the
mean, median, and mode?
mean > median > mode
median > mean > mode
mode > median > mean
mode > mean > median
mean > mode > median
Topic I
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SOLUTION
I.15.
A.
B.
C.
D.
E.
In skewed-right distributions, what is
most frequently the relationship of the
mean, median, and mode?
The median is
resistant, the mean,
mean > median > mode
not at all. So a right
median > mean > mode
skewed distribution
mode > median > mean
will have a mean
mode > mean > median
much higher than
mean > mode > median
median, much
higher than mode.Topic I
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TOPIC II: Experimental Design
• Sampling
• Designing Experiments
• Observational Studies
Topic II
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II.1. A personnel director studied the eating
habits of employees by watching a group
of employees at lunch. He wishes to see
who buys in the cafeteria, who brings a
home lunch, and who goes out.
The study is categorized as:
A.
B.
C.
D.
E.
a census
a survey sample
an observational study
a designed experiment
none of these
Topic II
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SOLUTION
II.1. A personnel director studied the eating
habits of employees by watching a group
of employees at lunch. He wishes to see
who buys in the cafeteria, who brings a
home lunch, and who goes out.
The study is categorized as:
A.
B.
C.
D.
E.
a census
a survey sample
an observational study
a designed experiment
none of these
The director is observing
behavior, not
implementing
treatments on the
Topic II
group
Menu
A personnel director studied the eating
II.2.
habits of employees by watching a group
of employees at lunch. He wishes to see
who buys in the cafeteria, who brings a
home lunch, and who goes out.
If the director only looks at those in one
department, she is performing:
A.
B.
C.
D.
E.
a simple random sample
a quota sample
a convenience sample
a multi-stage cluster sample
a census
Topic II
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SOLUTION
A personnel director studied the eating
II.2.
habits of employees by watching a group
of employees at lunch. He wishes to see
who buys in the cafeteria, who brings a
home lunch, and who goes out.
If the director only looks at those in one
department, she is performing:
A.
B.
C.
D.
E.
a simple random sample
a quota sample
a convenience sample
a multi-stage cluster sample
a census
Without a properly
randomized
selection, she is
introducing bias Topic
to II
Menu
the study.
A personnel director studied the eating
II.3.
habits of employees by watching a group
of employees at lunch. He wishes to see
who buys in the cafeteria, who brings a
home lunch, and who goes out.
If the director selects 50 employees at
random and categorizes by gender, she is:
A.
B.
C.
D.
E.
blocking for gender
testing for a lurking variable
promoting sexual harassment
testing for bias
none of these
Topic II
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SOLUTION
A personnel director studied the eating
II.3.
habits of employees by watching a group
of employees at lunch. He wishes to see
who buys in the cafeteria, who brings a
home lunch, and who goes out.
If the director selects 50 employees at
random and categorizes by gender, she is:
A.
B.
C.
D.
E.
blocking for gender
She is investigating
testing for a lurking variable
whether gender
promoting sexual harassment
affects
lunchtime
testing for bias
behavior
Topic II
none of these
Menu
II.4. Which of the following is NOT a concern
in data collection?
A.
B.
C.
D.
E.
lurking variables
blocking
bias
non-response
all of the above are concerns
Topic II
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SOLUTION
II.4. Which of the following is NOT a concern
in data collection?
A.
B.
C.
D.
E.
Even blocking methods
need to be analyzed
lurking variables
blocking
bias
non-response
all of the above are concerns
Topic II
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II. 5. Which of the following is NOT a valid
sample design?
A. Code every member of a population and select 100
randomly chosen members
B. Divide a population by gender and select 50 individuals
randomly from each group
C. Select individuals randomly and place into gender
groups until you have the same proportion as in the
population
D. Select five homerooms at random from all the
homerooms in a large high school.
E. All of these are valid
Topic II
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SOLUTION
II. 5. Which of the following is NOT a valid
sample design?
A. Code every member ofBut
a population
andbe
select
100to
you may
able
randomly chosen members
question the validity of
B. Divide a population by gender and select 50 individuals
answer (C)
randomly from each group
C. Select individuals randomly and place into gender
groups until you have the same proportion as in the
population
D. Select five homerooms at random from all the
homerooms in a large high school.
E. All of these are valid
Topic II
Menu
An insurance company conducted a study
II.6.
to determine the percent of cardiologists
who had been sued over the last 5 yrs.
The variable of interest is:
A. the doctor’s specialty, e.g. cardiology,
obstetrics, etc.
B. the number of doctors who are
cardiologists
C. all cardiologists in the American Medical
Association directory
D. a random sample of 100 cardiologists
E. none of these
Topic II
Menu
SOLUTION
An insurance company conducted a study
II.6.
to determine the percent of cardiologists
who had been sued over the last 5 yrs.
The variable of interest is:
A. the doctor’s specialty, e.g. cardiology,
obstetrics, etc.
we are
are interested in the
B. the number of doctors who
cardiologists
percentage of doctors
C. all cardiologists in the American
Medical
who have
been sued
Association directory
D. a random sample of 100 cardiologists
Topic II
E. none of these
Menu
An insurance company conducted a study
II.7.
to determine the percent of cardiologists
who had been sued over the last 5 yrs.
The population of interest is:
A. the set of all doctors who were sued for
malpractice
B. the set of cardiologists who were sued
for malpractice
C. all doctors
D. all cardiologists
E. all doctors who have malpractice
insurance
Topic II
Menu
SOLUTION
An insurance company conducted a study
II.7.
to determine the percent of cardiologists
who had been sued over the last 5 yrs.
The population of interest is:
A. the set of all doctors who were sued for
malpractice
B. the set of cardiologists who were sued
Just cardiologists, not
for malpractice
all doctors.
C. all doctors
D. all cardiologists
E. all doctors who have malpractice
Topic II
insurance
Menu
An insurance company conducted a study
II.8.
to determine the percent of cardiologists
who had been sued over the last 5 yrs.
Which could be used to gather the data?
A. a designed experiment
B. a census of all cardiologists
C. an observational study of randomly
selected cardiologists
D. a survey sent to randomly selected
cardiologists
E. any answer except (A)
Topic II
Menu
SOLUTION
An insurance company conducted a study
II.8.
to determine the percent of cardiologists
who had been sued over the last 5 yrs.
This
population
Which could be used to gather the data?
of this
A. a designed experiment
observation
B. a census of all cardiologists
al study is
C. an observational study of randomly
too large to
selected cardiologists
D. a survey sent to randomly selected
track
cardiologists
everyone
E. any answer except (A)
down.Topic II
Menu
II.9.
Which of the following is NOT a source
of bias in sample surveys?
A.
B.
C.
D.
E.
non-response
wording of questions
voluntary response
use of a telephone survey
all are sources of bias
Topic II
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SOLUTION
II.9.
A.
B.
C.
D.
E.
Which of the following is NOT a source
of bias in sample surveys?
non-response
wording of questions
voluntary response
use of a telephone survey
all are sources of bias
and don’t forget…
even a huge
sample size can’t
correct a poorly
selected sample.
Remember the
Literary Digest
Topic II
Poll!
Menu
II.10.
Which of the following is NOT a
requirement of a controlled experiment?
A.
B.
C.
D.
E.
control
comparison
replication
randomization
all of these are required
Topic II
Menu
SOLUTION
II.10.
Which of the following is NOT a
requirement of a controlled experiment?
A.
B.
C.
D.
E.
control
comparison
replication
randomization
all of these are required
Topic II
Menu
II.11.
A randomized block design is NOT:
A. similar to a stratified random sample for
surveys
B. a strategy to control for an influence that
would affect the outcome of the experiment
C. a strategy that depends on randomization
D. only used for gender comparisons
E. all of these describe a randomized block
design.
Topic II
Menu
SOLUTION
II.11.
A randomized block design is NOT:
You can
block with
A. similar to a stratified random sampleany
for
surveys
categorical
B. a strategy to control for an influencevariables!
that
would affect the outcome of the experiment
C. a strategy that depends on randomization
D. only used for gender comparisons
E. all of these describe a randomized block
design.
Topic II
Menu
A research team is comparing
II.12. performance in AP Statistics based on
whether traditional or activity-based
instruction methods were used. The final
grades of 500 students will be collected.
The population of interest is:
A. the 500 students chosen
B. the students taught by activity-based
statistics
C. the students taught by traditional
methods
D. all students in high school.
E. none of these
Topic II
Menu
A researchSOLUTION
team is comparing
II.12. performance in AP Statistics based on
whether traditional or activity-based
instruction methods were used. The final
grades of 500 students will be collected.
The population of interest is:
We are
A. the 500 students chosen
B. the students taught by activity-based interested in
knowing
statistics
C. the students taught by traditional about ALL AP
methods
STATISTICS
D. all students in high school.
STUDENTS Topic II
E. none of these
Menu
A research team is comparing
II.13. performance in AP Statistics based on
whether traditional or activity-based
instruction methods were used. The final
grades of 500 students will be collected.
An appropriate design for the study is:
A.
B.
C.
D.
E.
a blocked design experiment
a stratified random sample
a completely randomized design
a simple random sample
none of these
Topic II
Menu
A researchSOLUTION
team is comparing
II.13. performance in AP Statistics based on
whether traditional or activity-based
instruction methods were used. The final
grades of 500 students will be collected.
An appropriate design for the study is:
A.
B.
C.
D.
E.
a blocked design experiment
a stratified random sample
a completely randomized design
a simple random sample
none of these
B or C.
Either one
is fine.
Topic II
Menu
II.14.
A survey is to be conducted in your
school. There is to be a total of 40
students in the sample. Describe how you
would choose the participants if
there are to be the same number of
freshmen, sophomores, juniors, and
seniors in the sample.
Topic II
Menu
SOLUTION
II.14.
A survey is to be conducted in your
school. There is to be a total of 40
students in the sample. Describe how you
would choose the participants if
there are to be the same number of
freshmen, sophomores, juniors, and
seniors in the sample.
Select a simple random sample
of 10 from each class.
Topic II
Menu
II.15.
A survey is to be conducted in your
school. There is to be a total of 40
students in the sample. Describe how you
would choose the participants if
there are to be the same number of males
and females in the sample
Topic II
Menu
SOLUTION
II.15.
A survey is to be conducted in your
school. There is to be a total of 40
students in the sample. Describe how you
would choose the participants if
there are to be the same number of males
and females in the sample
Select a simple random sample of 20
males and 20 females.
Topic II
Menu
TOPIC III: Probability
•
•
•
•
Random Variables
Binomial Distributions
Geometric Distributions
Sampling Distributions
Topic III
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5
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7
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16
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18
back to main
If 3 people, Joe, Betsy, and Sue, play a
III.1.
game in which Joe has a 25% chance of
winning and Betsy has a 40% chance of
winning, what is the probability that Sue
will win?
A.
B.
C.
D.
E.
25%
35%
40%
65%
cannot be determined
Topic III
Menu
SOLUTION
III.1. If 3 people, Joe, Betsy, and Sue, play a
game in which Joe has a 25% chance of
winning and Betsy has a 40% chance of
winning, what is the probability that Sue
will win?
A.
B.
C.
D.
E.
25%
35%
40%
65%
cannot be determined
Assuming one
winner, the
probabilities must
add up to 100%
Topic III
Menu
III.2.
A local law enforcement agency
published the following chart.
The percentage of altercations
involving at least one teenager is:
A.
B.
C.
D.
E.
8%
37%
45%
55%
82%
Altercations Between
Two teens
A teen and an adult
Two adults
Percent
45%
37%
18%
Topic III
Menu
SOLUTION
III.2.
A local law enforcement agency
published the following chart.
The percentage of altercations
involving at least one teenager is:
p(two teens) + p(teen and adult) =
A.
B.
C.
D.
E.
8%
37%
45%
55%
82%
Altercations Between
Two teens
A teen and an adult
Two adults
Percent
45%
37%
18%
Topic III
Menu
III.3.
18 – 30
31 – 40
41 – 50
over 50
Democrat Republican
25
18
32
21
17
25
14
32
Independ
12
10
17
15
What proportion of Republicans are over 50?
A.
B.
C.
D.
E.
61/238
32/96
96/238
32/61
cannot be determined
Topic III
Menu
III.3.
18 – 30
31 – 40
41 – 50
over 50
SOLUTIONRepublican
Democrat
25
18
32
21
17
25
14
32
Independ
12
10
17
15
What proportion of Republicans are over 50?
A.
B.
C.
D.
E.
61/238
32/96
96/238
32/61
cannot be determined
There are 96
Republicans of whom
32 are over age 50
Topic III
Menu
III.4.
18 – 30
31 – 40
41 – 50
over 50
Democrat Republican
25
18
32
21
17
25
14
32
Independ
12
10
17
15
If one person is chosen at random, what is the
probability he is a Democrat between 41 and
50 years old?
A.
B.
C.
D.
E.
17/238
17/88
61/238
17/61
88/238
Topic III
Menu
III.4.
18 – 30
31 – 40
41 – 50
over 50
SOLUTIONRepublican
Democrat
25
18
32
21
17
25
14
32
Independ
12
10
17
15
If one person is chosen at random, what is the
probability he is a Democrat between 41 and
50 years old?
A. 17/238
Total # of adults is 238.
B. 17/88
17 are democrats
C. 61/238
between 41 and 50
D. 17/61
E.
88/238
Topic III
Menu
III.5.
18 – 30
31 – 40
41 – 50
over 50
Democrat Republican
25
18
32
21
17
25
14
32
Independ
12
10
17
15
Given that a person chosen is between 31 and
40, what is the probability the person is an
Independent?
A.
B.
C.
D.
E.
10/238
10/63
10/54
54/238
63/238
Topic III
Menu
III.5.
18 – 30
31 – 40
41 – 50
over 50
SOLUTIONRepublican
Democrat
25
18
32
21
17
25
14
32
Independ
12
10
17
15
Given that a person chosen is between 31 and
40, what is the probability the person is an
Independent?
A.
B.
C.
D.
E.
10/238
10/63
10/54
54/238
63/238
There are 63 people
between 31 and 40 of
whom 10 are Independent.
Topic III
Menu
III.6.
18 – 30
31 – 40
41 – 50
over 50
Democrat Republican
25
18
32
21
17
25
14
32
Independ
12
10
17
15
What proportion of the citizens sampled are
over 50 OR Independent?
A.
B.
C.
D.
E.
54/238
61/238
100/238
115/238
cannot be determined
Topic III
Menu
III.6.
A.
B.
C.
D.
E.
18 – 30
31 – 40
41 – 50
over 50
SOLUTIONRepublican
Democrat
25
18
32
21
17
25
14
32
Independ
12
10
17
15
What proportion of the citizens sampled are
over 50 OR Independent?
61/238 + 54/238 – 15/238
54/238
61/238
100/238
115/238
cannot be determined
15 adults are in both
categories
Topic III
Menu
III.7.
If P(A)=0.4, P(B)=0.2, and P(A and B)= 0.08
Which is true?
A. Events A and B are independent and
mutually exclusive
B. Events A and B are independent but not
mutually exclusive
C. Events A and B are mutually exclusive but
not independent
D. Events A and B are neither independent nor
mutually exclusive
E. Events A and B are independent but
whether they are mutually exclusive cannot
be determined.
Topic III
Menu
SOLUTION
III.7.
If P(A)=0.4, P(B)=0.2, and P(A and B)= 0.08
Which is true?
A. Events A and B are independent and
mutually exclusive
B. Events A and B are independent but not
mutually exclusive
C.
EventsP(A
A and
B are
exclusivethey
but are not
--Since
and
B)mutually
is non-zero,
not independent
mutually
D.
Events Aexclusive.
and B are neither independent nor
--Since
P(A
and B) does equal P(A)P(B), the
mutually
exclusive
E.
Events
A and
B are independent butjust happen in
events
are
independent—they
Topic III
whether they are mutually exclusive cannot
sequence
Menu
be determined.
III.8.
What is the probability that a family with
6 children will have 3 boys and 3 girls?
POSSIBLE NAME OF DISTRIBUTION?
1. Binomial Model
2. Geometric Model
3. Uniform Model
4. Normal Model
Topic III
Menu
SOLUTION
III.8.
What is the probability that a family with
6 children will have 3 boys and 3 girls?
This is a binomial distribution:
n=6
p=0.5
x=3
POSSIBLE NAME OF DISTRIBUTION?
3 (0.5)3 = 0.3125
C
(0.5)
1.
Binomial Model
6 3
2. Geometric Model
3. Uniform Model
4. Normal Model
Topic III
Menu
III.9. What is the probability that a person is
over 6 feet tall if the mean height of her
age group is 5’6” and a standard deviation
of 10”?
POSSIBLE NAME OF DISTRIBUTION?
1. Binomial Model
2. Geometric Model
3. Uniform Model
4. Normal Model
Topic III
Menu
SOLUTION
III.9. What is the probability that a person is
over 6 feet tall if the mean height of her
age group is 5’6” and a standard deviation
of 10”?
This is a normal distribution:
POSSIBLE NAME OF DISTRIBUTION?
1. 1E99,
Binomial
Model 10/12)
normalcdf(6,
5+6/12,
2. Geometric Model
3. =Uniform
0.2743Model
4. Normal Model
Topic III
Menu
III.10.
What is the probability that a shipment of
100 fruit will have no more than 6 rotten
fruits if the probability that any one fruit is
rotten is 0.04?
POSSIBLE NAME OF DISTRIBUTION?
1. Binomial Model
2. Geometric Model
3. Uniform Model
4. Normal Model
Topic III
Menu
SOLUTION
III.10.
What is the probability that a shipment of
This is 100
a binomial
fruit will distribution:
have no more than 6 rotten
6) =one
0.8936
fruits binomcdf(100,
if the probability0.04,
that any
fruit is
rotten is 0.04?
Did you think to use the normal
POSSIBLE NAME OF DISTRIBUTION?
approximation? 1. Binomial Model
μ = np = 4 2. Geometric Model
σ=√p(1–p)/n
.0196 Model
3. = Uniform
Model
normalcdf(-1E99,4.6, Normal
4, .0196)
= 1.00
Topic III
Menu
How come that didn’t work?
III.11.What is the probability that the first base
hit will occur during the fourth at-bat if
the probability that the hitter gets a base
hit is 0.27 for any at-bat?
POSSIBLE NAME OF DISTRIBUTION?
1. Binomial Model
2. Geometric Model
3. Uniform Model
4. Normal Model
Topic III
Menu
SOLUTION
III.11.What is the probability that the first base
hit will occur during the fourth at-bat if
the probability that the hitter gets a base
hit is 0.27 for any at-bat?
This is a geometric (waiting time) distribution:
n=??? (there
isn’t one!
That’s
why
it’s Geometric!)
POSSIBLE
NAME
OF
DISTRIBUTION?
p=0.27; x=4
1. Binomial Model
2. Geometric Model
3. 4)
Uniform
Model
geometpdf(0.27,
= (.73)(.73)(.73)(.27)
4. Normal
Model
= 0.7160
Topic III
Menu
III.12.
Suppose a basketball player scores 70% of
her free throws. Assume each shot is
independent and the probability is the
same on each trial.
Find the probability she scores on 3 of her
next 5 attempts
Topic III
Menu
SOLUTION
III.12.
Suppose a basketball player scores 70% of
This is a binomial distribution:
her free throws. Assume each shot is
independent and the probability is the
n=5
same3 on each
trial.
2
p=0.70 5C3 (0.7) (0.3) = 0.3087
x=3
Find the probability she scores on 3 of her
next 5 attempts
Topic III
Menu
III.13.
Suppose a basketball player scores 70% of
her free throws. Assume each shot is
independent and the probability is the
same on each trial.
Find the probability that the first time she
scores is on her 3rd attempt
Topic III
Menu
SOLUTION
III.13.
ThisSuppose
is a geometric
distribution:
a basketball player scores 70% of
her free throws. Assume each shot is
n=???independent
(we wait ‘tiland
a success—there’s
no
n!)
the probability is the
p=0.70; x=3 same on each trial.
geometpdf(0.7, 3) = (.3)(.3)(.7)
= 0.063
Find the probability that the first time she
scores is on her 3rd attempt
Topic III
Menu
III.14. The Central Limit Theorem for sample
means is critical because…
A. It states that for large sample sizes, the population
distribution is approximately normal
B. It states that for large sample sizes, the sample is
approximately normal
C. It states that for any population, the sampling
distribution is normal regardless of sample size
D. It states that for large sample sizes, the sampling
distribution is approximately normal regardless of the
population distribution
E. It states that for any sample size, the sampling
Topic III
distribution is normal
Menu
SOLUTION
III.14. The Central Limit Theorem for sample
means is critical because…
A. It states that for large sample sizes, the population
distribution is approximately normal
B. It states that for large sample sizes, the sample is
approximately normal
C. It states that for any population, the sampling
distribution is normal regardless of sample size
D. It states that for large sample sizes, the sampling
distribution is approximately normal regardless of the
population distribution
E. It states that for any sample size, the sampling
Topic III
distribution is normal
Menu
III.15.
The amount of time it takes a high school
class of 1000 freshmen to swim 10 lengths
of the school pool has a distribution that
is skewed left due to some excellent
swimmers. The mean amount of time
needed is 9.2 minutes and the standard
deviation is 5.3 min. If 64 students are
chosen at random, then what is the
probability their mean time will exceed 10
minutes?
Topic III
Menu
SOLUTION
III.15.
The amount of time it takes a high school
class of 1000 freshmen to swim 10 lengths
of the school pool has a distribution that
normalcdf(10,
9.2,to5.3/√64)
= 0.113
is skewed1E99,
left due
some excellent
swimmers. The mean amount of time
needed is 9.2 minutes and the standard
deviation is 5.3 min. If 64 students are
chosen at random, then what is the
probability their mean time will exceed 10
minutes?
Topic III
Menu
III.16.
It is assumed that 50% of all people catch
one or more colds each year. What is the
probability that out of 400 randomly
selected people, 216 or more will catch
one or more colds this year?
A.
B.
C.
D.
E.
0.0055
0.0121
0.055
0.11
0.55
Topic III
Menu
SOLUTION
III.16.
It is assumed that 50% of all people catch
one or more colds each year. What is the
probability
out
of >400
randomly
We that
want
p(p
216/400
= .054)
selected people,
or n(1-p)
more will
catch10,
Since np216
and
exceed
one we
or more
colds this
year?
can apply
the normal
A.
B.
C.
D.
E.
0.0055
0.0121
0.055
0.11
0.55
approximation with µ=.5 and
σ=.025
Normalcdf(.54, 1E99, .5, .025)
= .055
Topic III
Menu
III.17.
The decision whether the distribution ofo
a sample mean follows a normal or a tdistribution depends on:
A. Sample size
B. Whether you have the actual data or
only statistics of the data
C. Whether you know the population
standard deviation
D. Whether np>10 and n(1-p)>10
E. None of the above
Topic III
Menu
SOLUTION
III.17.
The decision whether the distribution ofo
a sample mean follows a normal or a tdistribution depends on:
A. Sample size
B. Whether you have the actual data or
only statistics of the data
C. Whether you know the population
standard deviation
D. Whether np>10 and n(1-p)>10
E. None of the above
Topic III
Menu
National studies show that 14% of male
teenagers and 12% of female teenagers
III.18.
will be involved in a major traffic accident
while driving. What’s the probability that
independent samples of 100 female teens
and 75 male teens will have results that
differ by more than 3% in either direction?
A.
B.
C.
D.
E.
.042
.085
.42
.85
Cannot be determined
Topic III
Menu
National studies
show that 14% of male
SOLUTION
teenagers and 12% of female teenagers
III.18.
will
beboth
involved
in a major
traffic
accident
For
genders,
np and
n(1-p)
exceed
while
driving.
What’s
the
probability
that
5 so we can use the CLT.
independent samples of 100 female teens
and
will have results =that
z = 75 male teens
(.03-.02)
.1938
differ √(.14*.86/75
by more than+3%
in
either
direction?
.12*.88/100)
A.
B.
C.
D.
E.
.042
.085 Using normalcdf(.1938, 1E99) = .423
.42 But we want both directions so add in
.85 the other tail to get an answer of .846
Cannot be determined
TOPIC IV: Inference
• Inference on Means
• Inference on Proportions
• Inference on 2-way
Tables
• Inference on Regression
Topic IV
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
back to main
IV.1.
A.
B.
C.
D.
E.
Given H0: μ=30, HA: μ<30, if you
conclude that the mean is less than 30
when it is actually 27…
you have made a type II error
you have made a type I error
the result of your test was not significant
you have drawn a correct conclusion
all of the above are true
Topic IV
Menu
SOLUTION
IV.1.
A.
B.
C.
D.
E.
Given H0: μ=30, HA: μ<30, if you
conclude that the mean is less than 30
when it is actually 27…
you have made a type II error
you have made a type I error
the result of your test was not significant
you have drawn a correct conclusion
all of the above are true
You are rejecting a false null
hypothesis. No problems.
Topic IV
Correct decision.
Menu
IV.2. The local news station reports that the
97% confidence interval for a candidate’s
support was (43%, 48%). What does the
phrase “97% confidence” mean?
A. 97% of the voters support the candidate
B. 97% of the time, this candidate’s level of support will
be between 43% and 48%
C. There is a 97% probability that the true level of support
is between 43% and 48%
D. There is a 97% probability that any other sample
percentage is in the interval (43%, 48%)
Topic IV
E. none of these is true
Menu
SOLUTION
not (D) because you’re
IV.2. The local news station reports that the
predicting the range of the
97% confidence interval for a candidate’s
TRUE proportion; you’re not
support was (43%, 48%). What does the
interested in predicting the
phrase “97% confidence” mean?
range of other samples!!!
A. 97% of the voters support
the interpretation:
candidate
Correct
B. 97% of the time, this candidate’s level of support will
be between 43%“If
andthis
48%process were repeated
C. There is a 97%over
probability
that the
true level
of support
and over
again,
about
97%
is between 43% and 48%
of our INTERVALS would
D. There is a 97% probability that any other sample
true
proportion.”
percentage is in capture
the intervalthe
(43%,
48%)
Topic IV
E. none of these is true
Menu
A study of 20 teachers in a school district
IV.3.
indicated that the 95% confidence interval
for the mean salary of all teachers is
($38,945, $41,245). What assumptions
must be true for this interval to be valid?
A. no assumptions are necessary. The CLT applies
B. The sample is randomly selected from a population of
salaries that is a t-distribution
C. the distribution of the sample means is approximately
normal
D. the distribution of all teachers’ salaries is approximately
normal
Topic IV
E. None of the above.
Menu
SOLUTION
A study of 20 teachers in a school district
IV.3.
indicated
the 95%size
confidence
interval
Because that
our sample
is only 20,
you
for
the
mean
salary
of
all
teachers
is
have to know the population was
($38,945,
What
assumptions
normal
to$41,245).
begin with.
(D)
is the best
must
be true for this interval to be valid?
response.
A. no assumptions are necessary. The CLT applies
B. The…I
sample
is randomly
a population
of
might
accept selected
choice from
(C) too
since the
salaries that is a t-distribution
CLT is all about when x-bar is normal,
C. the distribution of the sample means is approximately
which is the goal I suppose.
normal
D. the distribution of all teachers’ salaries is approximately
normal
Topic IV
E. None of the above.
Menu
IV. 4.
In order to reduce the width of a
confidence interval, we can:
A. increase sample size only
B. increase confidence level only
C. increase sample size and increase confidence
level
D. increase sample size and decrease
confidence level
E. none of these would reduce the width of the
interval
Topic IV
Menu
SOLUTION
IV. 4.
In order to reduce the width of a
Less confidence
affords
you to
confidence
interval,
we can:
predict a much tighter range of
values.
A. increase sample size only
B. increase confidence level only
C. increase sample size and increase confidence
level
D. increase sample size and decrease
confidence level
E. none of these would reduce the width of the
interval
Topic IV
Menu
IV.5.
A.
B.
C.
D.
E.
If the 95% confidence interval for μ is
(6,9), what conclusion can we draw if we
test Ho: μ=10 vs. Ha: μ≠10 at α=.05?
reject Ho
fail to reject Ho
accept Ho
accept Ha
There is insufficient information given
to draw a conclusion
Topic IV
Menu
SOLUTION
IV.5. If the 95% confidence interval for μ is
(6,9), what conclusion can we draw if
we test Ho: μ=10 vs. Ha: μ≠10 at
α=.05?
A.
B.
C.
D.
E.
A two tailed test of
reject Ho
fail to reject Ho
α=.05 is equivalent
accept Ho
to a 95% confidence
accept Ha
interval. 10 is not
There is insufficient information given
within the interval.
to draw a conclusion
Topic IV
Menu
A bakery determines that it will be
profitable if the time it takes to decorate a
IV.6.
cake does not exceed 45 minutes. The
owner documents the time spent on 20
cakes and performs a test. If the P-value
of the test is 0.032, then he should
conclude:
A.
B.
C.
D.
E.
at α=.05, fail to reject Ho
at α=.05, reject Ho
at α=.03, reject Ho
at α=.025, reject Ho
We cannot draw a conclusion from
this information
Topic IV
Menu
A bakery determines
SOLUTIONthat it will be
profitable if the time it takes to decorate a
IV.6.
cake does not exceed 45 minutes. The
owner documents the time spent on 20
cakes and performs a test. If the P-value
of the test is 0.032, then he should
conclude:
A.
B.
C.
D.
E.
at α=.05, fail to reject Ho
at α=.05, reject Ho
a p-value of
at α=.03, reject Ho
.032 is
at α=.025, reject Ho
significant at
We cannot draw a conclusion from
the 5% level Topic IV
this information
Menu
IV.7.
A.
B.
C.
D.
E.
Do boys perform better in math than
girls? A randomly selected group of
each gender were given the same math
assessment.
What design seems to be employed?
Matched Pairs Design
Simple Random Design
Multi-State Cluster Design
Independent Samples
Design
Randomized Block Design
boys girls
n
110
135
Mean
71.6
68.3
St.Deviation 10.4
11.2
Topic IV
Menu
SOLUTION
IV.7.
A.
B.
C.
D.
E.
Do boys perform better in math than
girls? A randomly selected group of
each gender were given the same math
assessment.
What design seems to be employed?
We have 2 independent
Matched Pairs Design
boys girls.
girls
samples here: boys and
Simple Random Design
Multi-State Cluster Design
Independent Samples
Design
Randomized Block Design
n
110
135
Mean
71.6
68.3
St.Deviation 10.4
11.2
Topic IV
Menu
IV.8.
A.
B.
C.
D.
E.
Do boys perform better in math than
girls? A randomly selected group of
each gender were given the same math
assessment.
What are Ho and Ha to determine if
boys’ scores are higher than girls?
Ho: μb – μg = 0; Ha: μb – μg < 0
Ho: μb – μg = 0; Ha: μb – μg ≠ 0
Ho: μb – μg = 0; Ha: μb – μg > 0
Ho: μb – μg < 0; Ha: μb – μg = 0
Ho: μb = μg; Ha: μb ≠ μg
boys girls
n
110
135
Mean
71.6 68.3
St.Dev
10.4 11.2
Topic IV
Menu
SOLUTION
IV.8.
A.
B.
C.
D.
E.
Do boys perform better in math than
girls? A randomly selected group of
each gender were given the same math
assessment.
way to write
What are Ho andAnother
Ha to determine
if
thisthan
would
be
boys’ scores are higher
girls?
Ho: μb = μg; Ha: μb > μg
Ho: μb – μg = 0; Ha: μb – μg < 0
Ho: μb – μg = 0; Ha: μb – μg ≠ 0
Ho: μb – μg = 0; Ha: μb – μg > 0
Ho: μb – μg < 0; Ha: μb – μg = 0
Ho: μb = μg; Ha: μb ≠ μg
boys girls
n
110
135
Mean
71.6 68.3
St.Dev
10.4 11.2
Topic IV
Menu
IV.9.
A.
B.
C.
D.
E.
Do boys perform better in math than
girls? A randomly selected group of
each gender were given the same math
assessment.
Suppose the p-value is .0344. We can
then conclude:
at α=.025, reject Ho
at α=.02, reject Ho
at α=.01, reject Ho
at α=.025, fail to reject Ho
We cannot draw a conclusion
from this information
boys girls
n
110
135
Mean
71.6 68.3
St.Dev
10.4 11.2
Topic IV
Menu
SOLUTION
IV.9.
A.
B.
C.
D.
E.
Do boys perform better in math than
girls? A randomly selected group of
each gender were given the same math
assessment.
Suppose the p-value is .0344. We can
then conclude:
.0344 is not
significant boys
at .025,
at α=.025, reject Ho
girls
.02, or .01. We must
at α=.02, reject Ho
n
110 135
fail to reject it at
at α=.01, reject Ho
Mean
71.6 68.3
these
levels.
at α=.025, fail to reject Ho
St.Dev
10.4 11.2
We cannot draw a conclusion
Topic IV
from this information
Menu
IV.10.
A.
B.
C.
D.
E.
The rejection region for a test Ho: p=.4
vs. Ha: p<.4, with n=50 and α=.05 is
given by:
Reject Ho if z > 1.96 or z < -1.96
Reject Ho if z > 1.645 or z < -1.645
Reject Ho if z < -1.96
Reject Ho if z < -1.645
Reject Ho if t > 2.59
Topic IV
Menu
SOLUTION
IV.10.
The rejection region for a test Ho: p=.4
vs. Ha: p<.4, with n=50 and α=.05 is
given by:
invnorm(.05)
A.
B.
C.
D.
E.
Reject Ho if z > 1.96 or z < -1.96
because the entire
Reject Ho if z > 1.645 or z < -1.645
.05 rejection
Reject Ho if z < -1.96
region is in the left
Reject Ho if z < -1.645
tail.
Reject Ho if t > 2.59
Topic IV
Menu
IV.11. The power of a test is described by all of
the following EXCEPT
A.
B.
C.
D.
Power = p( rejecting Ho when Ha is true)
Power = 1 – β
Power = α + β
The calculation of power requires knowing
the values of μ0, μa, σ, and α
E. All of these are correct descriptions of the
concept of power.
Topic IV
Menu
SOLUTION
IV.11. The power of a test is described by all of
the following EXCEPT
A.
B.
C.
D.
E.
But it IS true that as one
increases, the other has to
decrease.
Power = p( rejecting Ho when Ha is true)
Power = 1 – β
…just not always by the
Power = α + β
same amount
The calculation of power requires knowing
the values of μ0, μa, σ, and α
All of these are correct descriptions of the
Topic IV
concept of power.
Menu
IV.12. In a matched pairs test of 75 pairs, which
of the following assumptions is necessary?
A. The distribution of the paired differences
should be approximately normal
B. The population variances should be equal
C. The samples are randomly and independently
selected
D. The sets of values for each variable are
approximately normal
Topic IV
E. None of these assumptions is necessary.
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SOLUTION
IV.12. In a matched pairs test of 75 pairs, which
of the following assumptions is necessary?
A. The distribution of the paired differences
should be approximately normal
B. The population variances should be equal
And being good stats students, you will either
C. The samples are randomly and independently
make
sure n>25 or do a quick little graph to
selected
show
there
aren’t
any
outliers.
Right?
D. The sets of values for each variable are
approximately normal
Topic IV
E. None of these assumptions is necessary.
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IV.13. A pharmaceutical company claims that
50% of adult males living in a city in the
Midwest get at least two colds per year. A
random sample of 100 adult males living
in the city reported that only 42% got two
or more colds.
Do these data provide evidence (at the 5%
significance level) that the true proportion
of people is less than 50%
Topic IV
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SOLUTION
IV.13. A pharmaceutical company claims that
50% of
adult
males
living
a city
in the
Of course
you
would
want
to in
run
through
getproperly,
at least two
colds per year. A
the Midwest
entire test
but…
random sample of 100 adult males living
in the city reported that only 42% got two
or more colds.
Do these data provide evidence (at the 5%
significance
truethere
proportion
we fail
to reject level)
at thethat
5%the
level.
is
people istoless
50%
not enoughof
evidence
saythan
the proportion
Topic IV
is less than 50%
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IV.14. If a 90% confidence interval for the slope
of a regression line does not contain 0,
then which of the following is a valid
conclusion?
A. The confidence interval is not valid
B. A significance test will not be significant at the
10% level
C. There is sufficient evidence to conclude that the
slope of the true regression line is 0
D. There is sufficient evidence to conclude that the
slope of the true regression line is not 0.
Topic IV
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E. None of these is valid.
SOLUTION
IV.14. If a 90% confidence interval for the slope
of a regression
line does
not contain
If 0 isn’t
in the interval,
we are
90% 0,
then which
of the
following
is a valid
confident
that the
slope
is NOT zero.
conclusion?
This is exactly what a regression test would
A. The confidence interval is not valid
conclude: the slope is NOT zero.
B. A significance test will not be significant at the
10% level
C. There is sufficient evidence to conclude that the
slope of the true regression line is 0
D. There is sufficient evidence to conclude that the
slope of the true regression line is not 0.
Topic IV
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E. None of these is valid.
What is the expected number of males who
prefer chocolate?
IV.15.
A.
B.
C.
D.
E.
27.8
29.2
31.3
36.3
None of these
male female
chocolate
32
16
vanilla
14
4
strawberry
3
10
Topic IV
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48 of 79SOLUTION
people prefer chocolate
What is(60.76%),
the expected
number
of 60.76%
males who
so we
expect
of the
prefer
chocolate?
IV.15.
49 males
to like chocolate.
.6076(49) = 29.77
A.
B.
C.
D.
E.
or if you prefer the book formula:
(row total x column total)
total number male female
= 29.77
27.8
29.2
31.3
36.3
None of these
chocolate
32
16
vanilla
14
4
strawberry
3
10 IV
Topic
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