stat_exam2A_solutions_spring11

stat_exam2A_solutions_spring11 - A 2 Circle your final...

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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 (X-Y)? (2 points) @2wa 6i+6$ : (831+ (4)1: (00 /66(_fl:qr“,o° : H ' \‘Iym flCK-x) ' 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 ’ X”N/’°‘2%) / XmNéewé.) / W/Ndfbfl) were) / w (m. / may, a 6275/) : 6,3462 .: (491+ (3)1 :25’/ 56%;): m :@ MOM) : '36439 :1 [7:] P(S?‘ > >7 +? > We} a 7;) ’& >2 ('75) “tom 5’ , ,, , great 4 >/ 2 (g 1‘: 043%? 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. :flJ-b)(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$)Y 6;+6 +52 : F +6 Consider another random variable: -: (mainfbgogfigzr 6% :. «imam ":(O.2)L(;)1+/mz(g)’~ : Lr, 703040 c) IfX and Y are correlated andp=—0.5, what is 6R? (2 points) If: : (5/ LI', 5103 j Z _ L 2. ' 6 a 4;. + 66 + ZeéFéG :. '1. Z. 7. 'L 4’ é 6 0.2) “i' flfiS) Z x@:3) :: /0,Dl(§)l at flawlfis); +l/'0,53{5~1) KC) {Om/é) 5% : _—; Q3343‘4‘2H’V ‘3 24.33343 m V] -.: 3 I: 012‘ Problem 8 (6 points) 0) ; 0.7‘1 a) The proportion of the American adult population that supports candidate Walker is An SRS ofifladults asks if they agree with the statement “I support candidate Walker. What is the probability that at least 2 of those surveyed would agree with that statement? Round your final answer to four ecrmal places if necessary. (2 points) P(MMI 5); PM ;S)4 P018»: 7, s) [Mom‘er :: onao‘mqf] + >< X; Mam-22¢ X: ang'?‘ nu ¢A P(aaa g m 0: {0,708 4AL 8 WWW n s) = a» (0.74) PKW a, s) :2; [5,7437] 1 1': 8 [(0.0).[bfiqj7] b) Suppose thatA and B are two events with P(A) = 0.3, P(B) = 0.2, and P(AlB)=O.4. What is P(A or B)? (2 points) PM a» 3) :JWMWB‘) v 4704M a) P(AM e) : We) 96MB) : 0.3 a» 0.1 -- am : [MA/0.9 Tn”; 1" 000% 0) Suppose thatX and Y are two events with P(X) = 0.1, P(Y) = 0.2, and P(YIX)=0.4. What is P(X or Y)? (2 points) IVX «r» ‘0 :WN WV) «Pam v) P(Xm w) 1: PMQMX) : o.) + 0.1 —— 0.0% :(m (0M) = - .2c '5’ 0'0“ Problem 9 (6 points) P{$) :: 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 {OJ-WU} 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:3lT-saj 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 35-6) 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; : [II-32.4) @43 *(“74263 [0-6) - 2:3(z)‘~r=n Bum-rum __~ . 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 medium-size 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 M £4 : PU'P) _, (OIs'zXom) 6% n '— N [00 M7? : FF = 0's“! 6%: _: Fflf) __ (050(05-{4} V) [DD . MA , FF‘P‘M 3 —0'0) L 6 M r..- 52” 0’?- 6 é; “am 2 twmmm» ' (06.003344) + (015.1)[o’w) r/l/ loo 3 [Do A 4 ?(a>m> A ’\ P fi'FM >5] ’\ A r (‘0? BA fig???) > 0- (-om) 6&9?” 0,0700%.“ WWW J vmfl 2: 7 (ZN/Wu"; E > any (?'.:. 0,5557 2’ 0.9557 2.: ...
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stat_exam2A_solutions_spring11 - A 2 Circle your final...

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