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PROBABILITY DISTRIBUTIONS -- DISCRETE: CHAPTER 7

1. Number of accidents that occur annually on a busy intersection is an example of: a. continuous random variable b. discrete random variable c. discrete probability distribution d. continuous probability distribution

2. A random variable is a variable whose value is determined by:

a. the outcome of an experiment and associated with probability b. the random population c. the random space d. all of the above

3. A discrete random variable is a random variable:

a. that can assume any value in one or more intervals b. whose values are countable c. that is derived from a random population d. that is determined by random probability

4. A table, formula, or graph that shows all possible values a random variable can assume,

together with their associated probabilities is called a: a. probability distribution b. random variable c. bivariate distribution d. probability tree

5. A continuous random variable is a random variable:

a. that can assume any value in one or more intervals b. whose values are countable c. that is derived from a random population d. that is determined by random probability

6. Which of the following is not an example of a discrete random variable?

a. The number of days it rains in a month in New York b. The number of stocks a person owns c. The number of persons allergic to penicillin d. The time spent by a physician with a patient

7. Which of the following is an example of a discrete random variable?

a. The weight of a box of cookies b. The length of a window frame c. The number of horses owned by a farmer d. The distance from home to work for a worker

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8. The probability distribution table of a discrete random variable lists: a. some of the values that the random variable can assume and their corresponding

probabilities b. all the values that the random variable can assume and their corresponding

probabilities c. all the values that the random variable can assume and their corresponding

frequencies d. all of these

9. If X and Y are random variables, the sum of all the conditional probabilities of X given

a specific value of Y will always be: a. 0.0 b. 1.0 c. the average of the possible values of X d. a value larger than zero but smaller than 1.0

10. For a discrete random variable x, the probability of any value of x is:

a. always greater than 1 b. always less than zero c. always in the range zero to 1 d. never greater than zero

11. Which of the following is true for the probability of a discrete random variable x?

a. p(x) < 0 b. p(x) > 1 c. p(x) = 2

d. 0 ≤ P(x) ≤ 11

12. For the probability distribution of a discrete random variable x, the sum of the probabilities of all values of x must be:

a. equal to zero b. in the range zero to 1 c. equal to .5 d. equal to 1

13. Which of the following is true for the probability distribution of a discrete random

variable x? a. ΣP(x) < 0 b. ΣP (x) = 1 c. ΣP (x) = 2 d. ΣP (x) > 1

Following table lists the probability distribution of a discrete random variable x (number of cars) for families in Bloomington.

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x P(x) 0 .04 1 .11 2 .18 3 .24 4 .14 5 .17 6 .09 7 .03

14. The probability of x = 3 is:

a. .57 b. .24 c. .43 d. .18

15. The probability that x is less than 5 is:

a. .71 b. .88 c. .14 d. .17

16. The probability that x is greater than 3 is:

a. .67 b. .24 c. .57 d. .43

17. The probability that x is less than or equal to 5 is:

a. .88 b. .71 c. .12 d. .29

18. The probability that x is greater than or equal to 4 is:

a. .29 b. .14 c. .43 d. .57

19. The probability that x assumes a value from 2 to 5 is:

a. .17 b. .73 c. .38 d. .12

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20. If X and Y are random variables with V(X) = 7.5, V(Y) = 6 , then V(2X+3Y) is: a. 33 b. 37 c. 84 d. 132

Following table lists the probability distribution of the number of computers owned by all families in a city. x P(x) 0 .02 1 .65 2 .26 3 .07

21. The probability that a randomly selected family owns exactly two computers is:

a. .07 b. .93 c. .26 d. .33

22. The probability that a randomly selected family owns at most one computer is:

a. .65 b. .67 c. .98 d. .33

23. The probability that a randomly selected family owns at least two computers is:

a. .33 b. .26 c. .93 d. .67

24. The probability that a randomly selected family owns less than two computers is:

a. .93 b. .33 c. .26 d. .67

25. The probability that a randomly selected family owns more than one computer is:

a. .65 b. .33 c. .98 d. .67

26. If X and Y are random variables with E(X) = 5 and E(Y) = 8, then E(2X+3Y) is:

a. 34

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b. 13 c. 18 d. 40

27. The mean of a discrete random variable is the mean of its

a. frequency distribution b. percentage distribution c. probability distribution d. all of these

28. The mean of a discrete random variable is also called its:

a. box-and-whisker measure b. expected value c. second quartile d. upper hinge

29. The mean of a discrete random variable is obtained by using the formula:

a. Σ (x – µ)P(x) b. ΣyP(x) c. Σmƒ d. ΣxP(x)

30. If X and Y are any random variables, which of the following identities is not true?

a. E(X+Y) = E(X) + E(Y) b. E(X-Y) = E(X) - E(Y) c. V(X+Y) = V(X) + V(Y) d. V(X-Y) = V(X) - V(Y)

31. The standard deviation of a discrete random variable is the standard deviation of its:

a. frequency distribution b. percentage distribution c. probability distribution d. all of these

The following table lists the probability distribution of a discrete random variable x (number of children) of a family in a small city.

x P(x) 0 .04 1 .11 2 .18 3 .24 4 .14 5 .17 6 .09 7 .03

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32. The mean of the random variable x is: a. 3.70 b. 5.47 c. 3.35 d. 1.89

33. The standard deviation of the random variable x is approximately:

a. 2.97 b. 14.19 c. 6.91 d. 1.72

34. If X and Y are independent random variables, which of the following identities is

always true? a. E(2X+3Y) = E(X) + E(Y) b. V(2X+3Y) = 2 V(X) + 3 V(Y) c. V(2X+3Y) = 4 V(X) + 5 V(Y) d. E(2X+3Y) = 2 E(X) + 3 E(Y)

The following table lists the probability distribution of the number of TV sets owned by all families in a city. X P(x) 0 .07 1 .38 2 .31 3 .16 4 .08

35. The mean number of TV sets owned by these families is:

a. 4.34 b. 1.80 c. 2.08 d. 3.25

36. The standard deviation of the number of TV sets owned by these families is

approximately: a. 4.34 b. 2.08 c. 1.80 d. 1.05

37. In general, "n factorial" represents:

a. the product of any n numbers b. the sum of all integers from n to 1 c. the product of all integers from n to 1 d. n – 1

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38. The factorial of zero is:

a. zero b. 1 c. 10 d. cannot be determined

39. The factorial of 8 is:

a. 36 b. 40,320 c. 5040 d. 35

40. The factorial of (14 – 8) is:

a. 720 b. 21 c. 120 d. 20

41. The factorial of (8 – 8) is:

a. zero b. 362,880 c. 1 d. 81

42. The factorial of (4 – 0) is:

a. 24 b. 1 c. zero d. 10

43. Which of the following is not a characteristic of a binomial experiment?

a. There is a sequence of identical trials b. Each trial results in two or more outcomes. c. The trials are independent of each other. d. Probability of success p is the same from one trial to another.

44. The number of combinations for selecting 7 elements from 10 distinct elements is:

a. 70 b. 120 c. 3 d. 100

45. The number of combinations for selecting zero elements from 7 distinct elements is: a. 1

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b. 5040 c. 7 d. zero

46. The number of combinations for selecting 9 elements from 9 distinct elements is:

a. zero b. 1 c. 81 d. 362,880

47. A jury of five persons will be randomly selected from a group of 15 persons. The total

number of combinations are: a. 3,628,800 b. 5005 c. 3003 d. 120

48. An investor will randomly select six stocks from 18 stocks for an investment purpose.

The total number of combinations are: a. 18,564 b. 479,001,600 c. 720 d. 8,568

49. A Bernoulli trial is:

a. the trial of a court case b. a repetition of a binomial experiment c. a repetition of a probability distribution d. the trial of a probability distribution

50. Which of the following is not a condition of the binomial experiment?

a. There are only two trials. b. Each trial has two and only two outcomes. c. p is the probability of success, q is the probability of failure, and p + q = 1. d. The trials are independent.

51. In binomial experiments, the outcome called a "success" is an outcome:

a. that is always beneficial b. that is linked to success c. to which the question refers d. that is favorable

52. The expected value of a binomial probability distribution is:

a. n + p b. npq c. np

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d. n + p + q

53. The parameters of the binomial probability distribution are: a. n, p, and q b. n, p, q, and x c. n, p, and x d. n and p

54. The binomial probability distribution is symmetric if:

a. p is equal to .25 b. p is equal to .50 c. p is less than .50 d. p is greater than .50

55. The binomial probability distribution is skewed to the right if:

a. p is .25 or smaller b. p is .50 c. p is less than .50 d. p is greater than .50

56. The binomial probability distribution is skewed to the left if:

a. p is .25 or greater b. p is equal to .50 c. p is less than .50 d. p is greater than .50

57. The mean of a binomial distribution is:

a. npq b. np c. square of npq d. square root of npq

58. The standard deviation of a binomial distribution is:

a. npq b. np c. square of npq d. square root of npq

59. Which of the following is an example of a binomial experiment?

a. Rolling a die 10 times and observing for a number b. Selecting five persons and observing whether they are in favor of an issue,

against it, or have no opinion c. Tossing a coin 20 times and observing for a head or a tail d. Drawing three marbles from a box that contains red, blue, and yellow marbles

60. Which of the following is not an example of a binomial experiment?

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a. Rolling a die 25 times and observing for an even or an odd number b. Selecting 50 items from the production line and observing if they are good or

defective c. Rolling a die 20 times and observing for a number that is less than or equal to 4

or greater than 4 d. Selecting 50 adults and observing if they are in favor of an issue, against it, or

have no opinion

61. Eight percent of all college graduates hired by companies stay with the same company for more than five years. The probability that in a random sample of 12 such college graduates hired recently by companies, exactly two will stay with the same company for more than five years is:

a. .1294 b. .2301 c. .1835 d. .3305

62. Thirty-two percent of adults did not visit their physicians' offices last year. The

probability that in a random sample of eight adults, exactly four will say they did not visit their physicians' offices last year is:

a. .3484 b. .1927 c. .7813 d. .1569

63. Forty-four percent of customers who visit a department store make a purchase. The

probability that in a random sample of 10 customers who will visit this department store, exactly six will make a purchase is:

a. .3182 b. .1499 c. .1073 d. .3711

64. Five percent of all credit card holders eventually become delinquent. The probability

that in a random sample of 10 credit card holders, exactly three will eventually become delinquent is:

a. .3151 b. .0746 c. .0105 d. .2481

65. Sixty percent of all children in a school do not have cavities. From the binomial

probability distribution table, the probability that in a random sample of nine children selected from this school, at least six will not have cavities is:

a. .2508 b. .5174

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c. .7492 d. .4826

66. Thirty percent of law students who sit for a bar examination pass it the first time. From

the binomial probability distribution table, the probability that in a random sample of 15 law students who will sit for the bar examination, at most three will pass it the first time is:

a. .2968 b. .1700 c. .7032 d. .8300

67. Thirty-two percent of adults did not visit their physician's offices last year. Let x be the

number of adults in a random sample of 15 adults who did not visit their physicians' offices last year. The mean of the probability distribution of x is:

a. 32.0 b. 6.2 c. 4.8 d. 3.2

68. Sixty percent of children in a school do not have cavities. Let x be the number of

children in a random sample of 20 children selected from this school who do not have cavities. The mean of the probability distribution of x is:

a. 8 b. 18 c. 12 d. 10

69. Thirty-two percent of adults did not visit their physicians' offices last year. Let x be the

number of adults in a random sample of 15 adults who did not visit their physician's office last year. The standard deviation of the probability distribution of x is approximately:

a. 3.26 b. 4.80 c. 3.20 d. 1.81

70. Sixty percent of children in a school do not have cavities. Let x be the number of

children in a random sample of 20 children selected from this school who do not have cavities. The standard deviation of the probability distribution of x is approximately:

a. 2.19 b. 4.80 c. 12.00 d. 10.00

71. Which of the following is not a condition to apply the Poisson probability distribution?

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a. x is a discrete random variable. b. there are n identical occurrences. c. the occurrences are random. d. the occurrences are independent.

72. The parameter(s) of the Poisson probability distribution is(are):

a. n, x, and µ b. n and µ c. µ d. µ and x

73. In Poisson, for µ = 5.5, the probability of x = 2 is:

a. .0225 b. .0618 c. .0884 d. .1377

74. In Poisson, for µ = 4.2, the probability P(x < 2) is:

a. .1323 b. .2103 c. .0780 d. .8677

75. In Poisson, for µ = 3.4, the probability P(x > 7) is:

a. .0579 b. .0348 c. .9421 d. .0231

76. The Poisson random variable is a: a. discrete random variable with infinitely many possible values b. discrete random variable with finite number of possible values c. continuous random variable with infinitely many possible values d. continuous random variable with finite number of possible values

A manufacturer packages bolts in boxes containing 100 each. Each box of 100 bolts contains on average 4 defective bolts. A box is randomly selected at the end of the day from an entire production run (Use table).

77. What is the probability that the box will contain exactly six defective bolts?

a. .2615 b. .1042 c. .2319 d. .1824

78. What is the probability that the box will contain at most three defective bolts?

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a. .1954 b. .4335 c. .6665 d. .2381

79. What is the probability that the box will contain less than three defective bolts?

a. .4335 b. .1954 c. .0733 d. .2381

80. The standard deviation of a Poisson distribution, where µ is the average number of successes occurring in a specified interval, is

a. µ b. µ2 c. √(µ) d. 1

Historical data indicates that AirTran Airlines receives an average of 2.5 complaints per day (Use table). 81. What is the probability that on a given day, AirTran Airlines will receive no

complaints? a. .1417 b. .0009 c. .2052 d. .0821

82. What is the probability that on a given day, AirTran Airlines will receive at least seven

complaints? a. .0141 b. .0099 c. .0042 d. .1106

83. What is the probability that on a given day, AirTran Airlines will receive less than four

complaints? a. .8912 b. .7576 c. .1336 d. .2424

84. Which probability distribution is appropriate when the events on interest occur

randomly, independently of one another, and rarely? a. binomial distribution b. poisson distribution c. any discrete distribution

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d. any continuous distribution

85. Which of the following is an example of a continuous random variable? a. the number of times the telephone rings in one day b. the interest rate earned by a particular mutual fund c. the total number achieved on one throw of a pair of dice d. the number of wheels on a vehicle at a filling station

86. The expected number of heads in 100 tosses of an unbiased coin is

a. 30 b. 40 c. 50 d. 60

87. Which of the following cannot generate a Poisson distribution?

a. the number of children watching a movie b. the number of telephone calls received by a switchboard in a specified time

period c. the number of customers arriving at a gas station in christmas day d. the number of bacteria found in a cubic yard of soil

88. A Poisson distribution with µ = .60 is

a. symmetrical distribution b. negatively skewed distribution (skewed to the left) c. positively skewed distribution (skewed to the right) d. binomial distribution

The following table lists the probability distribution of the number of cans of soda consumed in a day by 25 employees at a large garage.

Number of Cans Probability

0 .12 1 .48 2 .28 3 .04 4 .04 5 .04

89. The mean number of cans of soda consumed on any particular day by these employees

is: a. 1.20 b. 1.32 c. 1.44 d. 1.52

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90. All 10 of the orangutans at a certain zoo contract a very serious disease which claims 80% of its victims (if an orangutan contracts the disease, the probability that it will die is .80). What is the probability that exactly two of the orangutans at this zoo will survive?

a. .200 b. .302 c. .698 d. .800

91. The number of small air bubbles per 3 feet by 3 feet plastic sheet has a Poisson

distribution with a mean number of two per sheet. What percent of these sheets have no air bubbles?

a. 27.07% b. 86.47% c. 13.53% d. 18.04%

92. The number of accidents to occur at a busy intersection during a 24-hour period has a

Poisson distribution. If the probability of no accidents during a 24-hour period is .1353, what is the mean number of accidents per 24-hour period?

a. 0 b. 1 c. 2 d. 3

93. A binomial distribution for which the number of trials n is large can well be

approximated by a Poisson distribution when the probability of success, p, is: a. larger than .95 b. larger than .50 c. between .25 and .50 d. smaller than .05

94. The variance of a binomial distribution for which n = 100 and p = .20 is:

a. 100 b. 80 c. 20 d. 19

ANSWER KEY:

1.b 2.a 3.b 4.a 5.a 6.d

33. d 34. d 35. b 36. d 37. c 38. b

65. d 66. a 67. c 68. c 69. d 70. a

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7.c 8.b 9.b 10. c 11. d 12. d 13. b 14. b 15. a 16. d 17. a 18. c 19. b 20. c 21. c 22. b 23. a 24. d 25. b 26. a 27. c 28. b 29. d 30. d 31. c 32. c

39. b 40. a 41. c 42. a 43. b 44. b 45. a 46. b 47. c 48. a 49. b 50. a 51. c 52. c 53. d 54. b 55. c 56. d 57. b 58. d 59. c 60. d 61. c 62. d 63. b 64. c

71. b 72. c 73. b 74. c 75. d 76. a 77. b 78. b 79. d 80. c 81. d 82. a 83. b 84. b 85. b 86. c 87. a 88. c 89. d 90. b 91. c 92. c 93. d 94. d