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Four
Section
I
Time-l hour and 30 minutes
Questions l-40
Percent of total grade_S0
Directions: The questions or incomplete statements that
follow are each followed by five
suggested answers or completions. choose the response
that best answers the question or
completes the statement.
l
' The boxplots shown below summarize two
following must be true?
Set
X
Set
Y
50
sets
60 70 80
of data, X and y. Which of
90
the
roo
(A) 50 is an outlier for Set X.
(B) Set Y contains more data than Set X.
(c)
The interquartile range (reR) of Set y is smaller than the IeR
of Set X"
(D) The data in Set Y have a smaller range than the data in set X.
(E) The means of the two data sets are equal.
Answer
81
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g
Sample Examination Four
6Z
2. A regression lir-re includes the point (2, 14) and has the equation
are the sample means of the x and v values. then V =
Section I
y=*r+ 4.If T and )
(A) r+ 4
(B) 5r+4
(c)
t
(D)
r-
\L-'
^
I
+I
L
Ansrver
3. In a recent poll adults r.r'ere asked to estimate the percentage of children rvho live in
poverty in the United States^ The mean response tvas 7o/o and the median was 12Yo.
What does this suggest about the shape of the distribution nf rccnnnccc?
(A) This distribution is most likely symtnetric.
(B) The standard deviation of the distribution is u102.
(C) The distribution is most likely sken'ed to the left.
(D) The distribution is most likely skewed to the right"
G) This is a uniform distribution.
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Answer
4. The scores on the real estate licensing examination given ir-r a particular state are
normally distributed with a standard deviation of 70. What is the mean test score if 25%
ofthe applicants score above 475?
(A)
(B)
(c)
416
428
468
(D) s22
(E) There is not enough information to answer this question
Answer
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Section I
Sarnple Examination Four
83
5. A survey was conducted to determine the percentage of college fieshmen that planned
to take a math colrrse while in college" The resuits *,ere given as 750h with a margin of
error of +l- 4o/a. What does the margin of error of +l- 4o/o mean?
(A)
(B)
(C)
(D)
(E)
The percentage of the population that was surveyed was betu,een 7 lVa and 79ak.
4% of the population n,as not included in the sun/ey.
4% of the students surveyed will most likely change theirminds.
4% of the students surueyed refused to parlicipate in tl-re poll^
The diff'erence between the sample propoition and the population proportion is
likely to be less than 4o/o.
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Answer
6. Which of the follou,ing statements is true?
(A) Values of r near 0 indicate a strong linear relationship.
(B) Changing the measurement units of x and y may affect the conelation between x
y.
(C) Strong correlation means that there is a definite cause-and-effect relationship
between x and y.
(D) Conelation changes when the x and y variables are reversed.
(E) The correlatiot-t can be strongl1, affected by a few outlying observations.
and
Answer
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ll
7. If the standard deviation of a set of data is zero. we can conclude that
(A) an error in computation was made
(B) the obseruations are equally spaced
(C) it was not a simple random sample
(D) the mean must also be zero
(E) all the observations are the same
Answer
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Sample Examination Four
Section I
8. The histogram shows the ages of cars in the parking lot at a local high school.
Which of the following statements are TRUE?
I.
II.
IIL
The mean age is greater than the median age.
The median age is greater than the mean age.
The median is between 6 and 8 vears.
(A) I only
(B) II only
(C) III only
(D) I and II
(E) II and III
Ansrver
9. The 'least-squares' method of determining a regression equation of the form y = a't bx
where the y-values are represented on the vertical axis and the x-values are represented
on the horizontal axis
(A) minimizes the sum of the squares of the horizontal
distances between the
points
regression line and the data
(B) minimizes the absolute value of the correlation coeffrcient of the data set
(C) minimizes the coefficient of determination of the regression model
(D) minimizes the sum of the squares of the vertical distances between the regression
line and the data points
(E) creates an equation whose predictions match the actual data
Answer
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Section
I
Sample Examination Four
85
10. In a study to compare movie preferences among different age groups, a fl2 statistic was
used. If a small value of the test statistic is obtained, it suggests that
(A)
(B)
(C)
(D)
the null hypothesis may not be rejected, since the differences between
observed and expected values are relatively large
the null hypothesis may be rejected, since the differences between the
and expected values are relatively large
the null hypothesis may be rejected, since the differences between the
and expected values are relatively small
the null hypothesis may not be rejected, since the differences between
observed and expected values are relatively small
(E) There is not enough information to
the
observed
observed
the
answer this question
Answer
1
L Billie decides to participate in one of two games of chance involving
dice. In the first
game each player gets a chance to throw the dice 100 times and wins if the dice show
a sum of 7 between l5o/o and 20oh of the time. In the second game, he gets to toss the
dice 125 times and wins if his dice show a sum of 6 or 8 between 30o/o and 35% of the
time. Billie decides to play the first game because he has a better chance of winning in
that game, according to
(A) the Law of Large Numbers
(B) the Law of Averages
(C) the Central Limit Theorem
(D) Chebyshev's Theorem
(E) Bayes'Theorem
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Answer
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86
Sarnp[e Exarnination Four
Section I
12. SonTe supermarkets have been accused of selling ineat that weighs less than is stated on
the package. An inspector randomly selects 5 packages of ground beef and will
consider the store to be in violation if more tlian 2 of the packages are underweight.
Experience has shown him that the packages are either correctly marked or overweight
85% of the tirne. What is tlie piobabiliry that the store i.vil! be considered in violatioir?
(A\ 0.0244
rB) 0.0266
rc)
0.1382
(D) 0.8s00
E)
4.e734
Answer
13.
A controversial radio talk show host proclaims that his listeners are family oriented
people rvho spend 8 hours per week of "quality time" on average with their children.
Members of a rival radio station suspect that his listeners spend less than 8 hours per
rveek of "quality time" on average with their children and they hire a statistician to
perform a study to examine the talk shorv host's claim. The statistician intervier.l,s a
simple random sample of 25 of his listeners, repofts a mean of 7 hours with a standard
deviation of 1.53 hours and conducts a hypothesis test. In which interval is the p-value
for this test found?
(A) p < 0.000s
(B) 0.0005 <p < 0.001
(C) 0.001<p<0.002s
(D) 0.0025 <p < 0.005
(E) p > 0.00s
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Answer
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Section I
14.
Sarnple Examination Four
A study is conducted on a sample of arlhritis sufferers to test a new arthritis
87
clrug
against aspirin. The subjects are randomly assigned
to one of 2 groups: a group that
receives aspirin and another that receives the new drug. The subjects have no
knowledge as to which pills they are receiving, and the pills are identical in appearance.
This is an example of
(A) confounding
(B) Sirnpson's Faradox
(C) an observational study
(D) a census
(E) blinding
r
Ansrver
ffi
15. Which of the following statements best describes the following histograms?
B
(A)
(B)
(C)
(D)
(E)
Histogram
Histogram
Histogram
Histogram
Histogram
A suggests a rectangular distribution.
B suggests a normal distribution.
C suggests a normal distribution.
D suggests a right-skewed distribution.
C suggests a left-skewed distribution.
Answer
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88
Section
Sample Examination Fomr
I
16. A market research firm is hired by a nationally known cosmetics company to test new
formulations of moisturizer. Using their extensive list of possible subjects, the market
researchers first divide the subjects into 5 age groups and then randomly select names
from each age group to pafiicipate in a stud1. This is an example of
(A) a simpie randorn sampie
(B) stratified samplirrg
(C) cluster sampling
(D) convenier-rce sarnpling
(E) systematic sampling
Answer
[]
tt
17. Aperfume manufacturer ships perfume to 3 different department stores. Although the
perfume comes with a suggested retail value, each store decides on an appropriate price
based on the area in which it is located. What is the average price of the perfume sold?
(A)
(B)
(c)
(D)
(E)
Store
Number of bottles sold
Price per bottle (in dollars)
A
150
20
B
300
25
C
300
30
$20.00
$25.00
$26.00
$30.00
It cannot be determined from the information given'
Answer
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Section
I
Sample Examination Four
89
18. A distribution that is symmetric and mound-shaped has a mean of 30 and a standard
deviation of 2. According to the Empirical Rule, approximately 6801o of the data lie
between
(A) 24 and 36
(B) 26 and 34
(C) 28 and 32
(D) 20 and 40
(E) 25 and 35
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Answer
19.
The figure below shows the probability density functions
of two
symmetric
distributions with the same mean.
\
\
-.\
,t
---B
Which of the following statements are true?
(A) median A > median B; O 1= oa
(B) median A < median B;6 e> o a
(C) median A = median B; oe> oa
(D) median A = median B; o t1C a
(E) It cannot be determined from the information siven.
Answer
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Sarnple Exarnination Four
90
Section I
20. A normal population has a mean of 100 and a standard deviation of 16. lf sarnples of
size 64 are drawn from the population. what proporlion of the samples should have
mean values betr,veen 91 and 103?
(A) 0.0668
(B) 0.4987
(c)
(D)
(E)
0.eos0
0.e319
0.e332
Answer
21.
survey was taken to determine the number of hours per week students studied
mathematics. The distribution was approximately normal with a mean of 20 hours and
a standard deviation of four hours. Approximately u,hat percent of the students studied
between 27 and 30 hours?
A
(A)
(B)
(c)
0.00338%
0.004se4
0.4e37%
(D) 33e%
(E\
4.937%
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Answer
22. Size AAA batteries have a mean life of two years with a standard deviation of six
months. The life spans of AAA batteries are normaliy distributed. lf the money-back
guarantee allorvs you to return the batteries if they do not last at least eight months,
rvhat is the probability that a randomly selected battery u'ill qualifu for the guarantee?
(A)
(B)
(c)
(D)
(E)
0.0038
0,004e
0.3800
0.4e00
0.4e62
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Sample Examination Four
Section I
91
23. A.normal distribution has a mean /1and a variance of 16. trf the 35th percentile is equal
to 8, then I is approximately equal to
(A) -2.2e
(B) -1.03
(c) 0.2e
(D)
e.ss
(E)
10.67
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Answer
21. A simple random sample of 150 policemen found that the average time these men had
worked for the police department was f = 18 years with a standard deviatior-l s = 3 years.
Assume the distribution of the time the population of men has worked for the police
department is approxirnately normal with mean 1l.Ten years ago the average amount
of time policeman stayed on the job was lt = 15 years. Are these data evidence that p
has changed? Test the hypotheses
Ho:
lt=
15, Ha:
P+L5
using the one-sample /-test. The appropriate degrees of freedom forthis test are
(A) 14
(B) 1s
(c)
(D)
(E)
18
14e
l s0
Answer
25. The /-distribution critical value for a90o/o confidence level and a sample size of 17 is
(A) 1.341
(B) 1.734
(c)
1^740
(D) 1.746
(E) 2.s83
Answer
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Sample Exarnination Four
92
Section
I
26. A normal distribution
(A) is sometimes symmetrical
(B) is sometimes skewed
(C) alinays has a mean of zerc
(D) is a distribution of a continuous randorn variable
(E) alwavs has a variance of one
Answer
27. A fair die is tossed 3 times. The probabiiity that a
die is
I
occurs on exactly one roil of the
(A) / I \/
1\/ I \
\61\6i \6/
(B)
./t\'/s\t
-\6/
\6/
, .2,.1
(c)
(D)
rp\
^/1\ /5\
'\6i
\6/
..t,
.2
/1\ /5\
\6/ )1\6/
11\ /r"\
\6/ \6/
Answer
28. What sample size should we select if lre wish to develop a90Yo confidence interval fbr
the average diameter of the washers produced by our company, yet we wish to have a
margin of error of no more than +0.02 mm? Assume our manufacturing process results
in a standard deviation of 0.035 mm.
(A)
(B)
J
(c)
9
5
(D) 17
(E) 95
Answer
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Section I
Sarnple Examination Four
93
29. Data are obtained tiom a group of high school seniors comparing age and the number
of hours spent on the telephone. The resulting regression equation is
Predicted number of hours
:
0.123 (age) + 2.57 with
r:
0.866
What percentage of the variation in the number of hours spent on the telephone can be
explained by this least-squares regression model?
(A) 0.7s%
(B) 0.866%
(c)
7s%
(D\ 86.6%
(E) This value cannot be found
rvith the information siven.
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Answer
30. Which of the following statements about correlation r are true?
I.
Il.
III.
when r= 0, there is no linear relationship between the variabres.
When r=0.75,then75oh of the variables are closely related.
When r = 1, there is a perfect cause and effect relationship between the
variables.
(A)
(B)
(C)
(D)
(E)
I only
II only
III only
I and III
I, II, and III
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Answer
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94
Sample Examination Four
Section
X
31. Which of the followine statements are true?
n
(A) The linear conelation coefficient fbr the data in plot A will be near zero.
(B) The slope for the line of best fit for the data in plot B ivill be near zero.
(C) The intercept for the line of best fit for the data in plot C rvill be positive.
(D) Plot B suggests that there is no relationship betu,een the two variables x and y.
(E) All of the above are true.
Ansrver
32. A population has a normal distribution with a mean of 50 and a standard deviation of
10. If a random sample of size 9 is taken from the population, then what is the
probabilitv that this sample mean rvill be between 48 and 54?
(A)
(B)
(c)
(D)
(E)
0.060
0.228
0.38s
0.3ee
0.611
Answer
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Section I
Sarnple Exarnination Four
95
33. The variance of a random variable is 100 and a sample of size 25 is taken. The standard
errol of the nrean is
(A)
(B)
4
(c)
5
L
(D) l0
(E) 100
Answer
mall a survey was taken. It was found that in a random sample of 45 women
over the age of 25, 15 had children. The 900% confidence interr,,al estimate for the true
proporlion of women over the age of 25 in the mall who have children is
34. In
a large
(A) (0.0208, 0.0653)
(B) (0.1374,0.1626)
(c)
(0.2177,0.4489)
(D) (0.3333, o.sss5)
(E) (0.4337, 0.6774)
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Answer
ll
35. Which of the following statements are true if one is constructing ahistogram?
(A) There can be no gaps between bars.
(B) Bars should be square so that the height and the width equal the class count.
(C) The heights of the bars should relate to the frequency of each class.
(D) The scale of the horizontal axis should be the frequency.
(E) The horizontal axis must begin at 0.
Answer
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96
Sarnple Exannination Four
Sectioxr
E
36. Mr. and Mrs. Jones have five children. The five births were independenl with nc
multiple births. Assuming that the chances of having a beiy or a girl are the same, what
is the probabiliry that the farnily has a1 least two girls?
(A) 0.187s
(B) 0.3 l2s
(c)
0.4000
(D) 0.s000
(E) 0.8 i 2s
Answer
37. Assume that
X
and
Y are events in the same sample space. if P(X)=0.30
and
P(Y)= 0.75 then which of the follorving inequalities must be true?
I. P(XandY)t O.Os
II. P(XandY)= O.gs
IIL P(XandY) .0.:O
(A)
(B)
(C)
(D)
(E)
I and II
I and III
II and III
II only
I, II, and III
Ansll,er
38. The probability that a certain virus infects a random person is 0.2. If a random sample
of l2 people is taken, what is the probabilify that exactly 5 people are infected?
(A)
(B)
0.02
(c)
0.0e
0.05
(D) 0.20
(E) 0.e8
Answer
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Section I
Sample Examination Four
97
39. A type II error is
(A) rejecting the null hypothesis when it is false
(B) rejecting the null hypothesis when it is true
(C) failing to reject the null hyporhesis when it is false
(D) failing to reject the null hypothesis when it is true
(E) incorrectly stating the alternative hypothesis
Answer
40. Consider the distribution below,
X
0
I
2
a
J
4
P(n
I
.2
.J
.4
.5
which of the following statements about the distribution is conect?
(A) This distribution is not a valid probability distribution
(B)
(C)
(D)
(E)
because the sum of the X's
is greater than one.
This distribution is not a valid probabilify distribution because the sum of the
probabilities is greater than one.
This distribution is avalid probability'distribution because each of the p(X)
values is between zero and one.
This distribution is a valid probability distribution because the sum of the X's is
exactly 10.
This distribution is a valid probability distribution because the sum of the X's is
exactly 10 and each P(X)is between zero and one.
Answer
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Section IE
E rgA
Time*65 minutes
Questicns l-.5
Fercent of Section II grade-75
Direetiosls: Shcw ail youi' 'work. Indicate elearly the rnethods you use, because you v,,ill
be graded ott the correctness of ,"'our method as iveli as on the accuracv of y6u. results
and explanaticns.
I
" In a nationwide random sample of 550 high school boys and 430 high school girls, 253
bo1'5 und 172 adrc'itled to cheating on an exam. Is this sufrlcient evidence to say that
boys are more likely to admit to cheating than girls? Provide appropriate statistical
evidence.
98
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II
Section
Sarnple Examination Four
99
of the major problems faced by educators today is the continuing increase in
school violence. To help understand this problern a study was conducted by
investigators in a major US city. 750 randomly selected public high sehool male
students were surveyed, and 82 of them admitted that they had carriecl a conceaied
2. One
weapon into their school building in the past 6 months. The investigators intend to use
this infonnation to construct a95o/o confidence interval for the proportion of all publie
school students who carry concealed weapons in their schools.
(a)
What assumptions are necessary in order for this confidence interval to be
appropriate?
(b)
Discuss whether or not each of your assumptions in part (a) appear to be satisfied
by this scenario.
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100
Sample Exarreination Four
Section
[I
3. Consider the sampling distribution obtained by random sampling fiom an infinite
population. A recent study reported that the average difference between the mean test
scores of 17-yearold urban students and 17-year-old suburban students in reading in
1999 was 30 points with a standard deviation of 4 points for the i7-year-old urban
students and a standard deviation of 8 points for the l7-year-old suburban students. A
random sampling of 100 students is selected from the population of each group.
(a) How is the mean of the sampling distribution of the differences between
the
average test scores for the two groups related to the difference in the popuiation
average test scores for the t*'o groups?
(b) How is the standard
deviation of the sampling distribution related to the standard
deviations of the trvo populations?
(c)
The study goes on to say that "This is not an IQ test, but this is a story about
schooling." "IQ" stands for Intelligence Quotient, a generally used method for
measuring relative intelligence. If the lQ's of the two groups are essentially the
same. explain how this disparity in reading test scores might have occurred.
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Section
II
Sarnple Examination
Four l0l
4. The loss of bone mass density (BMD) in men and women can be reduced by drug
treatments, A pharmaceutical company has developed a new calcium supplement,
which claims to reduce the loss of BMD. Volunteers who have been diagnosed with loss
of bone mass density and who are currently not on medication or calcium supplements
will be recruited to parlicipate in a study. It is known that men and women experience
different average losses of BMD.
(a)
Explain how you rvould carry out a completely randomized experiment for the
studv.
(b)
Describe an experimental design that would improve the design in part (a) by
incorporating blocking.
(c)
Can the experimental design in part (b) be carried out
in a double-blind manner?
Exnlain.
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102
Sample Exarnination Four
Section
EI
5. Stuart, a piace kicker" for a professional football team was only hitting 45% of his
attempted field goals from a ceftain range last season. Duling the suminer, he worked
on developing a longer, more accurate kick in the hope of improving his field goal
successes from that range. In the first 8 games of the season, Stuari- rnade 24 out of 40
attempts from this same range. Did Stuart's rvork over the summer help him improve
his field goal kicking success from that range? Provide statistical evidence tc support
your answer.
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Section Itr
Part B
Time-25 minutes
Question 5
Percent of Section [l erade-25
I)irections: Show ali your work. Indicate clearly the methods you use, because you will
be graded on the correctness of your method as well as on the accuracy of your results
and exolanations.
5. In a large university,30% of the students admitted came from private school while
the
remaining 7\Yo came from public schools.
(a)
Use the random digit table belorv to simulate groups of 10 students attending
freshman orientation. Perform the simulation 30 times and develop a frequency
table. Based on your simulation, what is the probability that in a group of l0
students attending orientation, 4 or more came from private schools?
8417706757
5311s84469
1311971020
9426808844
03309726As
t761315582
9486857967
8594091932
2629464759
2935723737
s150681435 410s092031 0644905059 5988431180
0581184s14 7501113006 6339555041 1s86606589
0648874987 5435552704 9035902649 4749671567
08989s7024 9728400637 8928303514 s919s0763s
6788103668 3387635841 s286923114 1s86438942
103
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104
San'nple
Examination Four
Seetiein [X
(b)
Calculate the true theoretical probability that in a group of ten students attending
orientation, four or more students came from private schools (provided that 30%
of the university's students come from private schools andT\Ya come frorn
nrrhlic qchonlc\
(c)
Suppose you believe that the university's 30% figure is too high. You take a
random sample of 960 freshmen and discover that 250 came from private
schools. Is this sufficient evidence to dispute the collese's claim? Justifi, vour
response.
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