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Unformatted text preview: Circle your ﬁnal answers to each Problem. Problem 1 (6 points) The scores of students on the ACT college entrance exam had mean u = 24 and standard deviation 6 = 4.
The distribution of scores is roughly normal. a) A SRS of 70 students who took the exam is taken. What are the mean and standard deviation of the
sample mean score 55 of these 70 students? Round your answers to four decimal places if necessary.
(3 points) b) What is the approximate probability that the mean score a? of these students is 23 or higher? Use your
ﬁnal answers from part a. Round your ﬁnal answer to four decimal places. (3 points) » Problem 2 (6 points) A college decides to admit 1500 students. Past experience shows that 70% of the students admitted will I accept. Assuming that students make their decisions independently, the number who accept has the
B(1500, 0.7) distribution. a) What are the mean and standard deviation of the numberX of students who accept? Round your
answers to four decimal places if necessary. (3 points) b) What is the probability that more than 1000 accept? Use your ﬁnal answers from part a. Use an actual
value from Table A. Round your ﬁnal answer to four decimal places. (3 points)  Problem 3 (6 points) A SRS of 700 adults asks if they agree with the statement “I support President Obama.” Suppose that
60% of all US adults would agree if asked this question so the population parameter=vO.6. ’ a) What are the mean and standard deviation of the sample proportion ? Round your answers to four
decimal places if necessary. (3 points) b) What is the probability that the survey result differs from the truth about the population by more than 2
perCentage points? Use your ﬁnal answers from pm“: a. Use actual values from Table A. Round your
v ﬁnal answer to four decimal. places. (3 points) Problem 4 (6 points) . Round all of yourflnal answers to four decimal places if necessary. Jen and Elise are playing in the club bowling tournament. Their scores vary as they play repeatedly.
Jen’s score X has the N(195, 8) distribution, and Elise’s score Y has the N(194, 6) distribution. Their scores are independent. ’ a) What is the standard deviation of (X—Y)? (2 points) b) What is the probability that Jen will score at least’two points lower than Elise? (2 points) 0) What is the probability that Jen will seore at ZeastﬁVe points higher than Elise? (2 points) Problem 5 (6 points) An experiment on the teaching of reading compares two methods, A and B. The response variable is the '
Degree of Reading Power (DRP) score. The experimenter uses Method A in a class of 100 students and
Method B in a comparable class of 100 students. The classes are assigned to the teaching methods at
random. Suppose that in the population of all children of this age the DRP score has the N(.136, 40)
distribution if Method A is used and the N(l34, 30) distribution if Method B is used. a) Let us call the mean DRP score for 100 students in the A group 55 . Let us call the mean DRP score for
100 students in the B group i. What is the probability that the mean score for the B group will be at least three points higher than themean score for the A group? Round your ﬁnal anSwer to four decimal places.
'(2 points) b)‘ Suppose that F and G are two independent events with P(F) '= 0.6 and P(G) = 0.7. What is P(F or G)?
(2 points) " . ‘ ' 0) Suppose thatX and Y are two events with P00: 0.4, P(Y) = 0.3, and P(X] Y)=0.6. What is P(X and Y)?
(2 points) Problem 6 (6 points)
Round all of your ﬁnal answers to four decimal places if necessary. The voters in a large city are 40% white, 40% black, and 20% Hispanic. Candidate Smith wins 40% of
the white vote, 70% of the black vote, and 20% of the Hispanic vote. a) What is the probability that a voter is white and voted for candidate Smith? (2 points) b) What is the conditional probability that a voter is Hispanic given that the voter voted for candidate
Smith? (2 points) . c) What is the probability that a voter is a black voter or a voter who voted for candidate Smith? (2 points) Problem 7 (6 points)
' Round all of your ﬁnal answer to four decimal places if necessary.
Consider the following information about random variables and Y:
“X: 9 I 0X: 5
W: .10 6y = 6
Consider another random variable: R = 0.8Y. a) If X and Y are independent, what is MR? (2 points) b) If X and Y are independent, what is 0R? (2 points) ' c) If X and Y are correlated and p = O.5, what is 6R? (2 points) Problem 8 (6 points) a) The proportion of the American adult population that supports candidate Walker is p=0.21.
An SRS of 8 adults asks if they agree with the statement “I support candidate Walkerf What is
the probability that at least 2 of those surveyed would agree with that statement? Round your
ﬁnal answer to four decimal places if necessary; (2 points)  ' b).Suppose thatA and B are two events With P(A) = 0.3, P(B) =i0.2, and P(AlB)=O.4. What is P(A or B)?
(2 points) i c) Suppose thatX and Y are two events with P(X) = 0.1, P(Y) = 0.2, and P(YIX)=0.4. What is P(X or Y)?
(2 points) ’ Problem 9 (6 points) Round all of your ﬁnal answer to four decimal places if necessaiy. a) A college administration decides to choose an SRS’ of ﬁve students to form an advisory board that
represents student opinion. Suppose that 60% of all students oppose candidateSmith for college
president and that the opinions of the ﬁve students on the board are independent. Then the probability is
0,60'that each opposes candidate Smith. What is the probability that a majority of the advisory board
opposes candidate Smith? (2 points) b) The proportion of the American population thathas disease Z is p=0.02. If 40 people are randomly
selected ﬁ'om the population, what is the probability" that at least 1' of them has disease Z? (2 points) 0) The proportion of the American population that has disease W is p=0.03. If 30 people are randomly I
selected from the population, what is the probability that at least 1 of them has disease W? ~ (2 points) 10. Problem 10 (6 points) Round all of your ﬁnal answer to four deoimal placés if necessary. The joint probability distribution of X and Y appears below: a) What is E(X] Y=0)? (2 points) b) What is E(Y]X=6)? (2 points) 0) What is E[E(Y1)Q]? ' (2 points) 11 12
Problem 11 (6 points) Round all of your ﬁnal answers to four decimal places if necessary. The probability distribution of random Variable X is given below: Valueofx .
Probability. ~ _ V A second random variable is defined as follows: Y = 0.2X— 2. A third random variable is deﬁned as follows: Z = 3X2 '— l. a) What is 0'; ? (2 points) ‘ b) What is 0'; ? (2 points) c) What is 0'; ? (2 points) . 13
Problem 12 (6 points) Round all of your ﬁnal answers to four decimal places if necessary. Of all the degrees in. Economics given by University X last year, 60% were bachelor’s degrees and 40%
were master’s degrees. Women earned 70% of the bachelor’s degrees and 50% of the master’s degrees. a) You choose an Economics degree at random. What is the probability that it was a bachelor’s degree
that had been awarded to a woman? ‘ b) You choose an Economics degree at random. What is the probability that it had been awarded to a
woman? ‘ c) You choose an Economics degree at random and ﬁnd that‘it had been awarded to a woman. What is
the conditional probability that it is a master’s degree? d) You choose an Economics degree at random and ﬁnd that it had been awarded to a man. What is the
conditional probability that it is a bachelor’sdegree? I e) You choose an Economics degree at random and ﬁnd that it had been awarded to a man. What is the
conditional probability that it is a master’s degree? ' D You choose an Economics degree at random. What is the probability that it is either a bachelor’s
degree or a degree that had been awarded to a man? V 14
Problem 13 (6 points) Round all of your ﬁnal answers to four decimal places if necessary. a) The insurance company sees that in the entire population of homeowners, the mean loss from ﬁre is u=
$100 and the standard deviation of the loss is <5 = $50. What is the standard deviation of the total loss for
6 policies? (Note that losses on separate policies are independent.) (2 points) b_) The insurance company sees that in the entire population of homeowners, the mean loss from ﬁre is u= _
$100 and the standard deviation of the loss is <5 = $50. What is the standard deviation of the average loss
for 6 policies? (Note that losses on separate policies are independent.) (2 points) 0) An estimator,_ y , is used to estimate the population mean of Y (m). Suppose} m = 100 and cry = 50.
Suppose an SRS‘ of n = 6 is drawn from a population of 10,000, and i is a weighted average in which the
observations are alternatively weighted by 1/4 and 7/4: Hater+(2Jn+0n+(2in+iaJn+04 What is the standard deviation of ? (2 points) ' Each of the 11 multiple choice questions is worth 2 points. (22 points total). 15 Circle your answers. v 1.
,A)
B)
. C) 2. A)
B)
C) 3. A)”
B)
C)
D)
. E) 4. A)
B)
. C)
D) A)
B) If X and Y have correlation p, and p > 0, then of, + a;
> , O'X_Y_. “A If X and Y have correlation p, and p < 0, then 0;,” a}, + 03. l'/\ V The weight of mediumsize tomatoes Selected at random from a bin at the
local supermarket is a random variable with mean ,u = 10 oz. and standard ‘ deviation 0': 102. The weights of the selected tomatoes are independent. Suppose we pick four tomatoes from the bin at random and put them in a ‘
bag. Deﬁne the random variable Y = the weight of the bag containing the
four tomatoes. What is the. standard deviation of the random variable Y (rounded to one decimal place)? 031‘: 
0;»: 1.0 oz.
O'y = 2.0 oz. '
UY = 02. A number other than any of the other choices listed. LetX equal the count of the number of heads in four tosses Of a coin. Here, X . The of any collection of events is the event that at least one of the collection occurs. is a continuous random variable, union ' ' is a continuous random variable, intersection is not a continuous random variable,vunion is not a' continuOus random variable, intersection You are interested in the political viewpoints among the 1000 members of a
sorority. You choose 60 members at random to interview. Onequestion is
“Do you support the death penalty?” Suppose that in fact 30% of the 1000
members would say “Yes.” Here, you safely use the B(60, 0.3)
distribution for the count X in your sample who say “Yes.” can ' cannot 6. A)
B)
C)
D)
F) A) B) .'
C) D)
E)
F) 8., A) B)’ C)
D) . 16
X and Y are two independent random variables. Consider the following
statements: ' ' (l) The mean of X + Y is‘the sum of their means. (2) The variance of X + Y is the sum oftheir variances.‘
(3) The Correlation between X and Y is zero. ' (4) The variance'of the difference X — Y is the difference of their variances. '  (5) The mean of the difference X — Y is the difference of their means. Which of the following statements is correct?
Statement (1) is false. ‘ Statement (2) 'is false. Statement (3) is false. Statement (4) is false. Statement (5) is false. _ I ’
Statements ( 1), (2), (3*), (4), and (5) are all true. Call a household prOsperousrif its income exceeds $150,000. Call the ‘ household educated if the householder Completed college. Select an _ American household at random, and let A be the event that the selected
household is prosperous and B'the event that it is educated. According to a .
government survey, P(A) = 0.29 and,P(B) = 0.44, and the probability that a
household is both prosperousand educated is P(A and B) = 0.21. What is '
the P(Ac and BC)? , ' '  0.1276  . 0.27 0.3976 0.48 0.79 .
A probability other than any 'of the other choices listed. Toss a balanced coin 20 times and count the numberX of heads. The count 
X have a binomial distribution. Deal 25 cards from a shufﬂed deck
(52 cards) and count the number Y of black cards. The count Y v have
a binomial distribution. does, does does, ,dOes ,nOt does not, does does not, does not 9. A)
B)
C) : D) 10. .A)
B)
C) D) , 11. A)
B)
_ C)
D)
E)
F)
G)
H) . .17 Consider tossing a pair of fair dice two times. Consider the following
events: A: 5 on the first roll, B = 4 or less on the ﬁrst roll. Events A and B . ' Events A and B
are disjoint, are independent
are disjoint, are not independent
are not disjoint, are independent
are not disjoint, are not independent Consider tossing a pair 0f fair dice two times. . Consider the following
events: A= 7 on the ﬁrst roll, B = 8 or less on the second roll. Events A'and
B . Events A and B . ' , , p '
are disjoint, are independent  are disj Oint, are not independent  are not disjoint, are independent are not disjoint, are not independent Suppose that the proportion of male voters who support candidate Brown is
0.52 (pM = 0.52). Suppose that the proportion of female voters who support
candidate Brown is 0.51 (pp = 0.51). A sample survey interviews/SRSS of
100 male voters (from a population of 100,000) and 100 female voters
(from a population of 100,000). The samples are independent. What is the " probability that, in the surVey, a higher proportion of female voters than male voters support candidate Brown (rounded to four decimal places)? _
0.4443 ' ' ' ’ 0.4483 0.4522 0.4562 0.4602 0.4641 " 0.4681 ' 5 0.4721 ...
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