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Unformatted text preview: A 2 Circle your ﬁnal answers to each Problem. ’99—— Problem 1 (6 points) X The scores of students on the ACT college entrance exam had mean u = 24 and standard deviation
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 f of these 70 students? Round your answers to four decimal laces if necessary. (3 points) ‘ ’ pf v
w I: W
3 / A1723 0‘ “73967 L} b) What is the approximate probability that the mean score 36 of these students is 23 or higher? Use your
ﬁnal answers from part a. Round your ﬁnal answer to four decimal places. (3 points) 1707 >/ 23) “I A» 23 ~Z‘f
X X, 2, “v 7 7 r 7 Problem 2 (6 points) x A college decides to admit 1500 students. Past experience shows that 70%_of the students admitted will 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) V) :: J§DO)(0,7)= [Ufa
X’VNEQﬂJ “9043] F ( . .
4/1090?) :1/0goa)@,7)(o,z) : l7.7.Lr$23‘a b) What is the probability that more than 1006 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) P (www) Z” 7 “2.81787
a :2 —2.‘51 C? T: 0. "DULY Problem 3 (6 points) A P A SRS odults 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 paramete1= 0.6. I‘— P. a) What are the mean and standard deviation of the sample proportion f) ? Round your answers to four
decimal places if necessary. (3 points) /\ r 90'?) ..
FWE’ME—FLT—S] g“ [o,é)(0.‘9 _,
7” / 0.0I5Ylé((« 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 part a. Use actual values from Table A. Round your
ﬁnal answer to four decimal places. (3 points) (25's 3 $< aeL 0, “0.6 A1 . Z m»
i— 4 r A? < 06 W «Lower < 2' < toms 0.89% "0.1%! =: 0.71% “410$ < a, c” [903 QEOJ‘IOl 5?: 085'” Problem 4 (6 points) Round all of your ﬁnal answers to four decimal laces if necessary. Jen and Elise are playing in the club bowling tournament. Their scores vary as they play repeatedly. I en’s score X has theristribution, and Elise’s score Y has the Ng194, 6) distribution. Their
scores are independent. a) What is the standard deviation of (XY)? (2 points) @2wa 6i+6$ : (831+ (4)1: (00 /66(_ﬂ:qr“,o° : H ' \‘Iym ﬂCKx) ' i b) What is the probability that Jen will score at least two points lower than Elise? (2 points) 506*) ’0 l .0381} 2 < ' ' ' c) What is the probability that Jen will score at leastﬁve points higher than Elise? (2 points) Myawg)
P (X—‘T a?) {Xa‘ri 'A{,(_~9 > gr‘ .
TM to I—— o, 4m : Am,“ F muoo N026) Lia)
Problem 5 (6 points) 8 ’> 7 h :10 0 N 63%} b) 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(134, 3 0) distribution if Method B is used. a) Let us call the mean DRP score for 100 students in the A group 3? . Let us call the mean DRP score for
100 students in the B group y . What is the probability that the mean score for the B group will be at least three points higher than the mean score for the A group? Round your ﬁnal answer to four decimal places.
(2 points) ., __, Lt ’
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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) W
We a. 6>=P(F)+?(é>~?/m a) We! O: 949%)
' 06 +00 — 0,47. :ﬂJb)(o,7) : ‘7: 0H2 0) Suppose thatX and Y are two events with P(X) = 0.4, P(Y) = 0.3, and P(XlY)=0.6. What is P(X and Y)?
(2 points) M P(X M ‘0 : WY)? (WY) :Co.z)(o.e> : 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. y W {Odbl OM? OM \M “WM 0.2% a) What is the probability that a voter is white and voted for candidate Smith? (2 points) P(WW%): I 0.16} b) What is the conditional probability that a voter is Hispanic given that the voter voted for candidate
Smith? (2 points) WNW) 1’ 0/ 0"‘t [0) 'l” “)‘OuD c) What is the probability that a voter is a black voter or a voter who voted for candidate Smith? (2 points)
My 61 7a):?d3)/H’®M «MW?» — OIL+ "l’ ’ —— l J 010833333 =— 1: 0.6 ' RT; Problem 7 (6 points) Round all of your ﬁnal answer to four decimal places if necessary. Consider the following information about random variables X and Y: R: (0.1!) X —) (0,8 ) Y 5 “X: 9 6X
My: Cy=6 R = 0.2X+ 0.8Y. a) If X and Y are independent, what is HR? (2 points) MR = (mm + mm,
an —..—, (0.7,)(4) «révaM/a) = b) If X and Y are independent, what is (5R? (2 points) R:ﬂj.L)X—l (01$)Y 6;+6 +52 : F +6 Consider another random variable: : (mainfbgogﬁgzr 6% :. «imam
":(O.2)L(;)1+/mz(g)’~ : Lr, 703040 c) IfX and Y are correlated andp=—0.5, what is 6R? (2 points) If: : (5/ LI', 5103 j Z _ L 2. '
6 a 4;. + 66 + ZeéFéG
:. '1. Z. 7. 'L 4’ é 6
0.2) “i' ﬂﬁS) Z x@:3) :: /0,Dl(§)l at ﬂawlﬁs); +l/'0,53{5~1) KC) {Om/é) 5% : _—; Q3343‘4‘2H’V ‘3 24.33343 m V] .: 3 I: 012‘
Problem 8 (6 points) 0) ; 0.7‘1 a) The proportion of the American adult population that supports candidate Walker is
An SRS ofiﬂadults asks if they agree with the statement “I support candidate Walker. What is
the probability that at least 2 of those surveyed would agree with that statement? Round your
ﬁnal answer to four ecrmal places if necessary. (2 points) P(MMI 5); PM ;S)4 P018»: 7, s)
[Mom‘er :: onao‘mqf] + >< X; Mam22¢ X: ang'?‘ nu ¢A
P(aaa g m 0: {0,708
4AL 8
WWW n s) = a» (0.74) PKW a, s) :2; [5,7437] 1
1': 8 [(0.0).[bﬁqj7] b) Suppose thatA and B are two events with P(A) = 0.3, P(B) = 0.2, and P(AlB)=O.4. What is P(A or B)?
(2 points) PM a» 3) :JWMWB‘) v 4704M a) P(AM e) : We) 96MB)
: 0.3 a» 0.1  am : [MA/0.9 Tn”; 1" 000% 0) Suppose thatX and Y are two events with P(X) = 0.1, P(Y) = 0.2, and P(YIX)=0.4. What is P(X or Y)?
(2 points) IVX «r» ‘0 :WN WV) «Pam v) P(Xm w) 1: PMQMX)
: o.) + 0.1 —— 0.0% :(m (0M) =  .2c '5’ 0'0“ Problem 9 (6 points) P{$) :: 05+ Round all of your ﬁnal answer to four decimal places if necessary. a) A college administration decides to choose an SRS of ﬁve students to form an advisory board that
represents student opinion. Suppose thatl60% pf all students oppose candidate Smith for college
president and that the opinions of the ﬁve students on the board are independent. Then the probability is hat 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 that has disease Z is p=0.02. If 40 people are randomly
selected from the population, what is the probability that at least 1 of them has disease Z? (2 points) 7%
P/WMAM 2):. ((3.4st / 0.§b"i 2445,66
ML Maﬁa» Me a) :: l—(my {OJWU} c) The proportion of the American population that has disease Wis p=0.03. If 30 people are randomly
selected from the population, What is the probability that at least 1 of them has disease W? (2 points) ,/ A
PM M ﬁfe W) = £3.47)”: /, 0a 5 “'3 7”“ » Lg”. w) 1:, j«L/M7fb Problem 10 (6 points)
Round all of your ﬁnal answer to four decimal places if necessary. The joint probability distribution of X and Y appears below: 11 ON 0 IL
a) What is E(X Y=0)? (2 points) = O Pﬂﬂ’l‘ﬁﬂ—k 3P[X:3lTsaj 4' Q Ya: EIY30)
=0 ﬁ)+ +é(%&j = E b) What is E(YlX=6)? (2 points) ‘ ECHXWL): o P{Y:0IY=€)~+ Wag/m)
 £1 0.;
‘ o 0.») + 3 c) What is E[E(YLX)]? (2 points) E ECG/M] : WV)
5 (Y) = o (OHM 356) m5 H 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 A second random variable is deﬁned as follows: Y = 0.2X— 2. HE A third random variable is deﬁned as follows: Z = 3X2 — 1. a) What is 01%? (2 points) MK: 2(o.lr)'+ “(0‘63 L: a; c) What isag? (2 points) #2 3 {0.q)+ H7{0.6> : K: 2 x?”
Z ‘L
5’“ °"’ 61; : [II32.4) @43 *(“74263 [06)  2:3(z)‘~r=n Bumrum __~ . v
. « 311.0% 1' 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 eamed 70% of the bachelor’s degrees and 50% of the master’s degrees. {97 F W?) = 0,4,7. 8A6, ijsodg
Ob H‘ 0.3 M 0"3 OW: 0.3 p: 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? (70%/rm W F) :: (0,42%; b) You choose an Economics degree at random. What is the probability that it had been awarded to a
woman? Pg?) 5 Oﬁlfaz, 7» (0,622 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? P(AA$T)F") :2 = 0.3225306 .; d) You choose an Economics degree at random and ﬁnd that it had been awarded to a man. What is the
conditional probability that it is a bachelor’s degree? (9J8
P (WCHM) 3 [9,8 +01] 1 O.‘~I73é8 = 1059737, e) You choose an Economics degree at random and ﬁnd that it had been awarded to a man. What is the
conditional probability that it is a master’s degree? [)(ﬂ/i/ST’M) :: [o'mfjil 2 0,5763} :1 0.5265 5 f) You choose an Economics degree at random. What is the probability that it is either a bachelor’s
degree or a degree that had been awarded to a man? mm: a m) = mm” m ~sz My : 0.6 f 0.33  00$ :1 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 0 ulation of homeowners, the mean loss from ﬁre is u=
$100 and the standard deviation of the loss i What is the standard deviation of the total loss for 6 policies? (Note that losses on separate polic1es are 1ndependent.) (2 points) 2; A+e+c:9+5:“ _ 9(9))?“
62’ : 5,94’65’13 ‘97 __ '3 6L—l’67rvl'r/ .
=Cc) 61 ’ b) The insurance company sees that in the ent q ulation of homeowners, the mean loss from ﬁre is n=
$100 and the standard deviation of the loss is What is the standard deviation of the average loss for 6 policies? (Note that losses on separate policies are independent.) (2 points) 15 50
r _ __..v _. I .. Z ILHzLH g2 >4 4!? " c) An estimator, E , is used to estimate the population mean of Y (uy). Suppose uy = 100 and Suppose an SRS of n = 6 is drawn, and )7 is a weighted average in which the observations are
alternatively weighted by 1/4 and 7/4: , r HG}? it}? {its ﬂimiﬂﬁ tits] What is the standard deviation of y ? (2 points) "' é‘ﬁéﬁé’d’f‘éﬁ, +.. 2
6 l 7
[CEMMEW Z 'L
= (when :7 3 iquﬁvfl} £3755 3 (in 7' (509 85/379)" V?
\l : lgﬂgﬂglg “a 15 Each of the 11 multiple choice questions is worth 2 points (22 points total).
Circle your answers. 1. CAD B)
C) A)
B) D)
E) 4.
A)
B) D) 6% > If X and Y have correlation p, anthen 03% + a; > 7. d.
< am L 61‘
l + \ — l 3— I If X and Y have correlation p, an p < 0, then aim, g 0:, + 03.
> i ,3, 2 64 o
< L L 2. l j: l 4’ l "' 
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 or: 1 oz. 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 ? <2: A.+ 9 .+ c + D
0' = 0 5 oz 2 1 2’ 62’: W
Gil—1.002. 62 : 6A46H” / Y‘ ' ' Lg614’62—L. ' 2.2, 07 = 2.0 oz. 6: 2 L 6‘?
O'Y=4.00Z. 62:: “(5):‘M‘HWQ A number other than any of the other choices listed. Let X equal the count of the number of heads in four tosses of a coin. Here, X M . The W 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, union is not a continuous random variable, intersection XMW quwmm You are interested in the political Viewpoints among the 1000 members of a
sorority. You choose 60 members at random to interview. One question is
“Do you support the death penalty?” Suppose that in fact 30% of the 1000 members would say “Yes.” Here, you am‘fi safely use thel B(6’0, 0.3))
distribution for the count X in your sample who say “Yes.” can a m > n 7,
cannot W o
W 60°“) 4 ~—9/M«r 16
6. X and Y are two independent random variables. Consider the following
statements: (1) The mean of X + Y is the sum of their means. (2) The variance of X + Y is the sum of their variances. (3) The correlation betweenX and Y is zero. j
334’) 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. Z. '2
6):;t': 6x+6~r Which of the following statements is correct?
A) Statement (1) is false.
B) Statement (2) is false.
C) Statement (3) is false.
@ Statement (4) is false.
E) Statement (5) is false.
F) Statements (1), (2), (3), (4), and (5) are all true. 7. Call a household prosperous if its income exceeds $15 0,000. 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 prosperous and educated is P(A and B) = 0.21. What is the P(Ac and BC)?
A) 0.1276
7 B) r 0.27 C 0.3976
0.48
0.79 F) A probability other than any of the other choices listed. 8. Toss a balanced coin 20 times and count the number X of heads. The count
X .4144; have a binomial distribution. Deal 25 cards from a shufﬂed deck
(52 cards) and count the number Y of black cards. The count Y M have
a binomial distribution. A) does, does CE) does, does not C) does not, does
D) does not, does not 17
9. 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 My EventsAandB/m'lw . . are disjoint, are independent ’4) i? W are disjoint, are not independent A 8 Mt W 9 ,4 W A, @me
. . . . ) )
C) are not disj omt, are 1ndependent D) are not disjoint, are not independent 8) 41’ PC 3 10. Consider tossing a pair of 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 EventsAandBA/w‘rl. APM (I .. [a Wﬂ7mu+ A) are disjoint, are independent A E‘
B: are disjoint, are not independent M 3‘ M 1"“ '9 W M , (rm are not disjoint, are independent A) e W '9 A W, MW M BMW m
D) are not disjoint, are not independent P (e) : P€67’ k) 11. 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 . a
probability that, in the survey, a higher proportion of female voters than
male voters support candidate Brown (rounded to four decimal places)? é)? 0.4443 B) 0.4483 we 1%
C) 0.4522 W D) ' 0.4562 ’ E) 0.4602 F) 0.4641 G) 0.4681 H) 0.4721 82
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