STAT 401, Section 1 Questions from first Mid-term Spring, 2004Question 1. Suppose that Y is a normal random variable with mean 12 and variance 25. Whichof the following is closest to P (7 < Y < 17)?(A) 95%(B) 50.0%(C) 99.7%(D) 68%Question 2. Suppose the joint pmf of X and Y is given byx1 2y 1 k 3k2 3k kWhat is k?(A) 1(B)14(C)18(D)13(E) None of the aboveQuestion 3. Which one of the following random variables is discrete?(A) B is a binomial(5, .2) random variable.(B) Z is a standard normal random variable.(C) E is an exponential random variable with parameter λ = 2.(D) X is a random variable with density function f(x) = k/(1 + x2) for some value of k.(E) None of the aboveQuestion 4. Suppose that we perform an experiment consisting of flipping a dime, a nickel, apenny, and a quarter. Let S denote the sample space for this expe riment. (In other words, Sconsists of the 16 outcomes {HHHH, HHHT, . . . , T T T T }.)Suppose that A is the event that the quarter comes up heads. How many outcomes arecontained in A?(A) 2(B) 8(C) 4(D) 16(E) None of the above1Question 5. A personnel director will interview 14 job candidates in random order, of whom 9have master’s degrees. If she interviews 7 of the candidates on the first day, what is theprobability that she will interview 2 people with master’s degrees on the first day?(A)29×5514= .0159(B) 72!91425145= .0504(C)9255147= .0105(D) 149!279575= .0047(E) None of the aboveQuestion 6. A probability experiment consists of simultaneously rolling a six-sided die, flipping acoin, and selecting one card at random from a standard 52-card deck. How many outcomesare contained in the sample space for this experiment?(A) 526!+ 522!= 20, 359, 846(B) 6 + 2 + 52 = 60(C) 6 × 2 × 52 = 624(D) (6 + 2 + 52)3= 216, 000(E) None of the aboveQuestion 7. Suppose that X is a random variable with probability density function given byf(x) =kx20 < x < 10 otherwise.Let µ = E (X). What is the value of µ + k?(A) 3.25(B) 3.5(C) 4(D) 3.75(E) None of the aboveQuestion 8. Suppose that P (A) and P (B) both equal13. Furthermore, suppose that A and B areindependent events. What is P (A ∪ B)?(A)79(B)69(C)59(D)39(E) None of the above2Question 9. Suppose that 25% of the people who make purchases in a particular electronics storebuy an expensive item costing more than $100. Of those who buy these expensive items, 80%use credit cards to make their purchases. Of the customers who do not buy an item costingmore than $100, only 40% use c redit cards.Suppose we are told that a shopper used a credit card to make a purchase. What is theprobability that that shopper bought an item costing more than $100?(A)23= .667(B)14= .25(C)15= .20(D)25= .40(E) None of the aboveQuestion 10.Assume that you are given the following information about events A1, A2, and A3:P (A1∩ A2) = .25 P (A1∩ A3) = .20 P (A2∩ A3) = .35P (A1∩ A2∩ A3) = .15What is the probability of the shaded event depicted in the Venn diagram accompanying thisquestion?(A) .60(B) .80(C) .50(D) .70(E) None of the aboveQuestion 11. Suppose X1and X2are independent random variables with E (X1) = 3 and E (X2) =1. What is E (3X1− 4X2+ 10)?(A) 9(B) −1(C) 15(D) 5(E) None of the above3Question 12. Suppose that X1, . . . , X20are independent random variables, each with mean 120and variance 100. Let X denote the sample mean120P20i=1Xi. What isV(X)?(A) 10(B) 6(C) 5(D) 120(E) None of the aboveQuestion 13. For events A and B, suppose P (A) = .4 and P (B) = .5 and P (A ∪ B) = .8. Whichof the following is equal to P (A|B)?(A) .4(B) .3(C) .2(D) .5(E) None of the aboveQuestion 14. Suppose that 25% of all drivers are older than 50, 10% of all drivers have gottena speeding ticket in their lives, and 5% of all drivers are older than 50 and have gottena speeding ticket in their lives. What percent of drivers older than 50 have ever gotten aspeeding ticket?(A) 40%(B) 10%(C) 30%(D) 20%(E) None of the aboveQuestion 15. Suppose that the random variable Y can only take the values 1, 2, and 3. Itsprobability mass function satisfies P (Y = y) = ky for y = 1, 2, 3. What is the value of k?(A) 1(B) k cannot be determined without knowing the expectation of Y .(C)12(D)13(E) None of the aboveQuestion 16. Suppose that X is a continuous random variable with cdfF (x) =0 x < 3(x − 3)/2 3 ≤ x ≤ 51 x > 5.What is the median (also known as the 50th percentile) of X?(A) 3.5(B) 3.75(C) 4(D) 4.25(E) None of the above4Question 17. Suppose I have a fair 10-sided coin (i.e., a coin with 10 faces for which each face comesup with probability110). I roll this coin 400 times. Let N denote the numbe r of times thenumber 9 comes up. Which of the following is the bes t approximation for P (39 ≤ N ≤ 50)?(A) Φ9.56− Φ−0.56= .477(B) 40039!.139.9361− 40050!.150.9350= .050(C) 40040!.140.9360= .066(D) Φ10.56− Φ−1.56= .559Question 18. Suppose that X and Y are jointly distributed with joint densityf(x, y) =kxy 0 < x < 1, 0 < y < 10 otherwise.What is k?(A) 2(B) 3(C) 1(D) 4(E) None of the aboveQuestion 19. Suppose that X has cumulative distribution function (cdf) given byF (x) =0 x < 0.2 0 ≤ x < 1.3 1 ≤ x < 2.5 2 ≤ x < 3.6 3 ≤ x < 4.8 4 ≤ x < 51 5 ≤ x.What is P (2 ≤ X < 4)?(A) .3(B) .4(C) .6(D) .5(E) None of the aboveQuestion 20. Suppose that B is a binomial(10, .3) random variable. If Y = 7B − 22, what is theexpected value of Y ?(A) 3(B) 2.1(C) 0(D) −1(E) None of the above5Question 21. Suppose that a fast food restaurant runs a promotion in which 10% of all its softdrink cups contain a coupon for free food. If a pers on purchases 20 soft drink cups duringthis promotion, what is the probability that exactly two of these cups will contain a couponfor free fo od?(A) 202!.12.918= .285(B)102901810020= .318(C) 2010!1020101 −102010= .176(D)220= .1(E) None of the aboveQuestion 22. A man plays Bingo every day. On the weekend, a lot of people play and theprobability that the man loses is .95. On the weekdays, when fewer people play, the probabilityof losing drops to .80. The daily results (win or loss) are independent of one another.Suppose the man plays every day for a week (i.e., two weekend days and five weekdays).What is the probability that he will lose on all seven days?(A)5 × .95 + 2 × .807= .907(B).955+.802= .590(C) .952× .805= .296(D).955×.802= .076(E) None of the