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Chapter 16 Review Problems

Chapter 16 Review Problems - Chapter 16 Random...

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Unformatted text preview: Chapter 16 Random Variables (Review) Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the expected vaiue of the random variable. Round to three decimal places. P(X=x) 0.5 0.2 0.2 A) 1 B) 1.1 C) 0 D) 0.4 E) 0.6 2) x 30 60 90 P(X=x) 0.5 0.4 0.6 A) 90 B) 75 ' C) 60 D) 111 E) 93 3) The number of golf balls ordered by customers of a pro shop has the following probability distribution. ‘ x 3 6 9 12 15 p(x) 0.14 0.26 0.36 0.14 0.10 A) 8.4 B) 9 C) 7.98 D) 5.73 - s) 9.12 4) x 100 200 300 400 P(X=x) 0.1 0.7 0.1 0.1 A) 230 s) 220 C) 210 D) 150 E) 250 5) Sue Anne owns a mediumwsized business. The probability model below describes the number of emyloyees that may call in sick on any given day. Number of Employees Sick 0 1 2 3 4 P(X = x) 0.1 0.45 0.25 0.15 0.05 What is the expected value of the number of employees calling in sick each day? A) 1.70 B) 1.65 C) 2.00 D) 1.00 E) 1.60 6) In a certain game, a fair die is rolled and a player gains 20 points if the die shows a "6." If the die does not Show a "6," the piayer loses 3 points. If the die were to be rolled 100 times, what would be the expected total gain or £055 for the player? A) A gain of about 83 points B) A loss of about 300 points C) A gain of about 1,700 points D) A loss of about 250 points E) A gain of about 583 points 1) 2) 3) 4) 5) 6) Create a probability model for the random variable. 7) You pick a card from a deck. If you get a face card, you win $15. If you get an ace, you win $25 7) plus an extra $40 for the ace of hearts. For any other card you win nothing. Create a probability model for the amount you win at this game Amount won $0 $15 $25 $40 31?. £21.. .5. __1__ 52 52 52 52 $0 $15 $25 $65 n. 22; a ._.1__ 52 52 52 52 $0 $15 $25 $40 3e 12. e _1_ 52 52 52 52 $0 $15 $25 $65 3_6 12. i .1. 52 52 52 52 $0 $15 $25 $65 32 16 3 1 52533535 A) P(Amount won) Amount won 3) P(Amount won) Amount won C) P(Arnount won) Amount won D) P(Amount won) Amount won E) P(Amount won) 8) You roll a fair die. If you get a number greater than 4, you Win $80, If not, you get to roil again. if 8) you get a number greater then 4 the second time, you win $20. Otherwise you win nothing. Create a probability model for the amount you win at this game. Amount won $80 $20 $0 2 8 16 A .. _ ___. } P(Amount won) 6 3 6 36 Amount won $80 $20 2 4 6 6 $80 $20 $0 2 2 2 6 6 6 $100 $80 $20 $0 4: 2 2 16 "a? 56' 315' E $100 $80 $.20 $0 4 8 8 16 36 553523" B) P(Amount won) Amount won C) P(Arnount Won) Amount won D) P(Arnount won) Amount won E) P(Amount won) 9) A couple plans to have children until they get a boy, but they agree that they will not have more than four chiidren even it ail are girls. Create a probability model for the number of children they will have. Assume that boys and girEs are equaliy Eikely. A) Children I 1 2 3 I’(Chilclren) 0.5 0.25 0.25 B) Children 1 2 3 4 P(Children) 0.5 0.25 0.125 0.0625 C) Children 1 2 3 4 P(Children) 0.25 0.25 0.25 0.25 D) Children 1 2 3 4 5 P(Chfldren) 0.5 0.25 0.125 0.0625 0.0625 E) Children 1 2 3 4 P(Children) 0.5 0.25 0.125 0.125 10) A company bids on two contracts. It anticipates a profit of $90,000 if it gets the larger contract and a profit of $10,000 if it gets the smaller contract. It estimates that there's a 10% chance of winning the larger contract and a 50% chance of winning the smaller contract. Create a probability model for the company's profit. Assume that the contracts will be awarded independently. A) Profit $0 $10,000 $90,000 I’(Profit) 0.4 0.5 0.1 B) Profit $0 $10,000 $90,000 $100,000 P(Profit) 0.35 0.5 0.1 0.05 C) Profit $0 $10,000 $90,000 $100,000 P(Profit) 0.45 0.45 0.05 0.05 D) Profit $0 $10,000 $90,000 P(Profit) 0.45 0.45 0.05 13.) Profit $0 $10,000 $90,000 $100,000 P(Profit) 0.45 0.45 0.05 0.6 11) A carnivai game offers a $120 cash prize for anyone who can break a balioon by throwing a dart at it. It costs $10 to piay and you're Willing to spend up to $40 trying to win. You estimate that you have a 10% chance of hitting the balloon on any throw. Create a probability model for the amount you will win. Assume that throws are independent of each other. Round to four decimal places if necessary. A) Amount won $120 $110 $100 $90 —$40 P{Amount won) 0.1 0.09 0.0810 0.0729 0.0656 8) Amount won $110 $100 $90 $80 “$40 P(Amount won) 0.1 0.09 0.0810 0.0729 0.0656 C) Amount won $110 $100 $90 $80 —$40 P(Amount won) 0.1 0.09 0.0810 0.0729 0.6561 {3) Amount won $120 $110 $100 $90 “$40 P(Amount won) 0.1 0.09 0.0810 0.0729 0.6561 E) Amount won $110 $100 $90 $80 P(Amount won) 0.1 0.09 0.0810 0.7290 9) 10) n) Find the expected value of the random variable. Round to three decimal places. 12) Suppose you pay $1.00 to roll a fair die with the understanding that you will get back $3.00 for 12) rolling a 4 or a 2, nothing otherwise. What is the expected amount you win? A) $1.00 B) $3.00 C) —$1.00 D) $0.33 E) $0.00 13) You pick a card from a deck. If you get a face card, you win $5. if you get an ace, you win $30 plus 13) an extra $60 for the ace of hearts. For any other card you win nothing. find the expected amount you will win. A) 05.00 B) $3.85 C) 04.04 D) 05.19 E) $4.62 14) You pick a card from a deck. if you get a club, you win $90. If not, you get to draw again (after 14) replacing the first card). if you get a club the second time, you win $30. If not, you lose. Find the expected amount you will win. A) $30.00 B) $36.56 C) $32.34 D) $28.13 B) $45.00 15) The probabilities that a batch of 4 computers will contain 0, 1, 2, 3, and 4 defective computers are 15) 0.6274, 0.3102, 0.0575, 0.0047, and 0.0001, respectively. Find the expected number of defective computers in a batch of 4. A) 0.34 B) 0.53 C) 2.00 D) 1.07 E) 0.44 16) A carnival game offers a $100 cash prize for anyone who can break a baiioon by throwing a dart at 16) it. It costs $8 to play and you're willing to spend up to $32 trying to win. You estimate that you have a 10% chance of hitting the balloon on any throw. Find the expected amount you will win. Assume that throws are independent of each other. A) $638.73 8) $9.27 C) 43511.52 D) $14.88 E) $6.88 17) An insurance policy costs $140 per year, and wiil pay policyholders $12,000 if they suffer a major 1'7) injury (resulting in hospitalization) or $7000 if they suffer a minor injury {resulting in lost time from work). The company estimates that each year 1 in every 1800 policyholders will have a major injury and 1 in every 500 a minor injury. What is the company‘s expected profit on this policy? A) $119.33 B) 02509999040 C) —$238.44 D) —$20.67 ‘0) $131.33 Find the standard deviation of the random variable. Round to two decimal places if necessary. 18) P(X:x) 0.7 0.2 0.1 A) 0.68 B) 0.66 C) 0.44 D) 0.72 E) 0.69 19) x 30 60 90 P(X:x) 0.4 0.1 0.5 A) 27.74 B) 25.75 C) 24.06 D) 26.89 E) 28.30 20) Find the standard deviation for the given probability distribution. B) 2.19 C) 2.11 D) 1.22 E) 1.51 21) Sue Anne owns a medium-sized business. The probability model below describes the number of employees that may call in sick on any given day. Number of Employees Sick 0 1 2 3 4 P(X:x) 0.05 0.4 0.25 0.2 0.1 What is the standard deviation of the number of employees calling in sick each day? A) 1.09 B) 1.31 C) 1.19 D) 0.98 E) 1.20 22) The random variable x is the number of houses sold by a realtor in a single month at the Sendsom'e Rea} Estate Office. Its probability distribution is as follows. Find the standard deviation of the number of houses sold. Houses Sold (x) Probability P(x) 0.24 0.01 0.12 0.16 0.01 0.14 0.11 0.21 A) 1.62 B) 6.86 C) 2.62 D) 4.45 s) 2.25 Find the standard deviation of the random variable. 23) You pick a card from a deck. If you get a face card, you win $10. If you get an ace, you win $20 plus an extra $40 for the ace of hearts. For any other card you Win nothing. Find the standard deviation of the amount you will win. A) $94.08 B) $9.70 C) $8.92 D) $10.67 B) $103.51 19) 20) 21) 22) 2s) 24) You pick a card from a deck. If you get a club, you win $90. If not, you get to draw again (after repiacing the first card). It you get a ciub the second time, you win $20. if not, you lose. bind the standard deviation of the amount you wiil win. A) $39.51 B) $1410.94 C) $34.56 D) $31.93 E) $37.56 25) A teacher grading statistics homeworks finds that none of the students has made more than three errors. 13% have made three errors, 27% have made two errors, and 40% have made one error. Find the standard deviation of the number of errors in students‘ statistics horneworks. A) 0.86 B) 1.08 C) 0.94 D) 0.88 E) 0.80 26) A poiice department reports that the probabilities that 0, 1, 2, and 3 burglaries wiil be reported in a given day are 0.46, 0.44:, 0.07, and 0.03, respectively. Find the standard deviation of the number of burglaries in a day. A) 1.00 a) 0.86 (1)0574 13) 0.99 s) 0.54. 27) A carnival game offers a $80 cash prize for anyone who can break a baiioon by throwing a dart at it. It costs $5 to play and you're willing to spend up to $20 trying to win. You estimate that you have a 8% chance of hitting the baiioon on any throw. Find the standard deviation of the number of darts you throw. Assume that throws are independent of each other. A) 0.79 B) 0.81 C) 1.01 D) 0.88 E) 0.94 Create a probability modei for the random variabie. 28) Your soccer team, Mill Valley, plays two games against Fairfield soccer team . The probability that your team wins the first game is 0.6. If your team wins the first game, the probability that they also Win the second game is 0.8. If your team ioses the first game, the probability that they win the second game is 0.5. Let the random variable X be the number of games won by your team, Mill Valley. Find the probability model for X. A) Games won 0 1 2 P(Games won) 0.2 0.2 0.48 B) Games won 0 1 2 P(Games won) 0.2 0.12 0.48 C) Games won 0 1 2 P(Games won) 0.08 0.44 0.48 D) Games won 0 1 2 1°(Garnes won) 0.2 0.32 0.48 E) Games won 0 1 2 P(Garnes won) 0.2 0.5 0.3 24) 25) 26) 27) 28) 29) In a box of 9 batteries, 3 are dead. You choose two batteries at random from the box. Let the random variable X be the number of good batteries you get. Find the probabiiity model for X. A) Number good 0 1 2 P{Number good) 0.417 0.500 0.250 B) Number good 0 1 2 P(Number good) 0.083 0.500 0.417 C) Number good ‘ 0 1 2 P(Number geod) 0.111 0.444 0.444 D) Number good 0 1 2 P(Number good) 0.083 0.250 0.417 E) Number good 0 1 2 P(Nurnber good) 0.045 0.409 0.545 Find the expected value of the random variable. Round to three decimal places. 30) Your soccer team, Mill Valley, plays two games against Fairfield soccer team . The probability that your team Wins the first game is 0.3. If your team Wins the first game, the probability that they also win the second game is 0.7. If your team loses the first game, the probability that they Win the second game is 0.3. Let the random variable X be the number of games won by your team, Mill Valley. Find the expected value of X. A) p. a 1.00 B) a a 0.72 C) p a 0.51 D) p. = 0.63 a) a e 0.60 31) in a box of 8 batteries, 6 are dead. You choose two batteries at random from the box. Let the random variable X be the number of good batteries you get. Find the expected value of X. A) 0:086 B) 020.29 C) u=0.63 D) a: 1.50 s) 030-50 Find the standard deviation of the random variable. 32) Your soccer team, Mill Valley, plays two games against Fairfield soccer team . The probability that your team Wins the first game is 0.4. If your team Wins the first game, the probability that they also win the second game is 0.4. if your team loses the first game, the probability that they win the second game is 0.2. Let the random variable X be the number of games won by your team, Mill Valley. Find the Standard deviation of X. A) o a 0.84 B) o =: 0.63 C) O m 0.73 D) or m 0.69 E) o = 0.54 33) in a box of 11 batteries, 7 are dead. You choose tw0 batteries at random from the box. Let the random variable X be the number of good batteries you get. Find the standard deviation of of X. A) 030.65 B) o: 0.79 C) are 0.42 D) o a 0.62 E) 0u0.46 29) 30) 31) 32) 33) Solve. 34) Given independent random variabies with means and standard deviations as shown, find the mean 34.) and standard deviation of the variable 4X. A) p:280,o:24 B) n=280,o=6 C)n=74,o=24 D) n=280,o=96 E) y=74eo=6 35) Given independent random variables with means and standard deviations as shown, find the mean 35) and standard deviation of the variable X+ Y. Round to two decimal piaces if necessary. A)ti=90,o=74 3)}.12 1800,62 12 C)p.=90,0'212 D) a: 90, cm 8.60 E) a a 1800, o a 8.60 36) Given independent random variables with means and standard deviations as shown, find the mean 36) and standard deviation of the variable Y — 6. Round. to two decimai places if necessary. A) a=264,6~=27 B) n=264,o“=21 C) p. = 264, 6 = 27.66 D) p=270,o‘=27 E) a = 264, o = 26.32 37) Given independent random variables with means and standard deviations as showzi, find the mean 37) and standard deviation of the variabte BY - 9. Round to two decimal places if necessary. A) a = 480, o a51.79 B) a z 471, o a 50.20 C) 0:471, 0 W42 D) p2480,0':51 E) nz471,o==51 38) Given indeyendent random variables with means and standard deviations as shown, find the mean 38) and standard deviation of the variabie X+ 4Y . Round to two decimal places if necessary. A) p. a 150, o a 30.46 B) n m 390, o a 30.46 C) nw390,c7:40 D)nm150,o: 40 E) n a 320, U x 30.46 39) An insurance company estimates that it should make an annual profit of $140 on each 39) homeowners policy written, with a standard deviation of $5700. If it writes 3 of these policies, what are the mean and standard deviation of the annual profit? Assume that poiicies are independent of each other. A) a a $420, (I = $9872.69 B) a = $242.49, 0 = $9872.69 C) a : $242.49, 0 = $17,100 D) a = $420, 0 = $17,100 E) a = $420, 0 = $51,300 40) A slot machine at a casino pays out an average of $0.92, With a standard deviation of $125. It costs a dollar to play. If a person plays ‘7 times, what are the mean and standard deviation of the casino's profit? A) p = $0.56, (I 7» $6125 B) p. 3 $6.44, 0' a $875 C) a = $6.44, 0' = $330.72 D) a = $0.56, 0 ._.. $330.72 E) p = $0.56, (7 = $875 41) A company selling vegetable seeds in packets of 4-D estimates that the mean number of seeds that Wiil actuaiiy grow is 35.9 with a standard deviation of 1.6 seeds. If a customer buys 5 different seed packets, what are the expected value and standard deviation of the number of bad seeds? Assume that packets are independent of each other. A) a = 80.27, (I = 3.58 13m = 179.5, 0 = 40 C) a 2 179.5, 0" = 3.58 D) a: 179.5,o = 8 E) we 80.210 = 8 42) Suppose that in one town adult men have a mean weight of166 1b with a standard deviation of1‘7 lb. Adult women have a mean weight of 143 lb with a standard deviation of 11 1b. 10 year old children have a mean weight of 83 lb with a standard deviation of 6 lb. Suppose that a man, a woman, and a 10-year old child get into an elevator. What are the mean and standard deviation of their total weight? A) 5:392:19, 0244611) B) a a 234.29 ib, Gm 21.1215 C) 5239215, 3234133 D) p a 39215, om 21.1215 3) a a 234.29 lb, or w: 34 lb 10 4.0) 41) 42) 43) At a furniture factory, tables must be assembled, finished, and packaged before they can be shipped to stores. Based on past expefience, the manager finds that the means and standard deviations (in minutes) of the times for each phase are as shown in the tabie: Phase Assembly 26.8 Finishing 35.7 Packaging 15.1 SD 3.1 2.4 What are the mean and standard deviation of the totai time to prepare a table for shipping? Assume that the times for each phase are independent. A) pt: 77.6 Inin,0’= 4.70 min B) nm77.61m'n,o=8.1min C) M47112 mime": 8.1 min D) am 47.12 min, 0*: 4.70 min E) [.1 m 77.6 min, 0' = 22.13 min 44) Sue buys a large packet of rice. The amount of rice that the manufacturer puts in the packet is a random variable with a mean of 1018 g and a standard deviation of 9 g. The amount of rice that Sue uses in a week has a mean of 210 g and a standard deviation of 21 g. Find the mean and standard deviation of the amount of rice remaining in the packet after a week. A) pmSOSg,Om12g B) pw1228 g,cs:-22.85g C)pm1228g,cr-=30g D) p2808 g, 0:22.85g E) pm808 g,0“218.97g 45) The amount of money that Maria earns in a week is a random variable with a mean of $1000 and a standard deviation of $35. The amount of money that Elena earns in a week is a random variable with a mean of $810 and a standard deviation of$15. How much more do you expect Maria to earn in a week than Elena? What is the standard deviation of this difference? Assume that Maria's earnings are independent of Elena's earnings. A) $190, $31.62 B) $1810, $38.08 C) $1810, 850 D) $190, $38.08 E) $190, $20 11 43) 44) 45) 46) Janet is planning to rent a booth at a festival for a day to sell clothes that she has made. She sells jackets for $180 and skirts for $115. Her past experiences suggests that sales of jackets will have a mean of 7.3 with a standard deviation of1.8,‘ and sales of skirts will have a mean of14.00 with a standard deviation of 3.0. The cost of renting the booth for the clay is $190. What are the mean and standard deviation of her net income? {Hint you should first define random variables and use them to express her net income} A) {a = $2734.00, 0 = $479.00 B) )1 = $2924.00, 0 = $40.23 C) in : $2734.00, 0 = $669.00 D) i1 2 $2734.00, 0' = $473.29 E) y. 2 $2924.00, 0 = $473.29 Find the indicated probability. 47) Miguel buys a large bottle and a small bottle of juice. The amount of juice that the manufacturer puts in the large bottle is a random variable with a mean of1016 ml and a standard deviation of 8 ml. The amount of juice that the manufacturer puts in the small bottle is a random variable with a mean of 510 ml and a standard deviation of 5 ml. If the total amount of juice in the two bottles can be described by a normal model, what's the probability that the total amount of juice in the two bottles is more than 1540.2 ml? A) 0.067 3) 0.933 C.) 0.055 D) 0.081 E) 0.919 48) A slot machine at a casino pays out an average of $0.90, with a standard deviation of$110. It costs a dollar to play. If gamblers play this machine 6,969,600 times in a month, what is the probability that the casino will come out ahead? Assume that the casino's total profit follows a Normal model. A) 0.986 a) 0.971 C) 0.992 D) 0014 E) 0.008 49) The amount of money that Maria earns in a week is a random variable with a mean of $960 and a standard deviation of $35. The amount of money that Elena earns in a week is a random variable with a mean of $830 and a standard deviation of $15. if the difference between Maria's weekly income and Elena‘s weekly income can be described be a Normal model, what is the probability that Maria's weekly income is at least $149.04 more than Elena's weekly income? (In other words, what is the probability that the differenceM — E is at least $149.04?) Assume that Maria's earnings are independent of Elena's earnings. A) 0.691 B) 0.655 C) 0.345 D) 0.309 E) 0.274 12 46) 47) .48) 49) 50) At a furniture factory, tables must be assembled, finished, and packaged before they can be 50) shipped to stores. Based on past experience, the manager finds that the means and standard deviations (in minutes) of the times for each phase are as shown in the table: Phase Assembly 24.3 Finishing 35.2 Packaging 14.6 3.2 1.8 Find the probability that a table can be prepared for shipping in less than 64.0? minutes. Assume that the times for each phase are independent and that the times for each phase follow a Normal model. ' A) 0.023 B) 0.045 C) 0.955 D) 0.014 E) 0.986 Provide an appropriate response. 51) Some marathons allow two runners to "sph‘t" the marathon by each running a half marathon. Alice 51) and Sharon plan to spiit a marathon. Alice’s half—marathon times average 92 minutes with a standard deviat...
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