BSTAT2 2nd Edition by Gerald Keller - Test Bank

BSTAT2 2nd Edition by Gerald Keller – Test Bank

 

Instant Download – Complete Test Bank With Answers

 

 

Sample Questions Are Posted Below

 

CHAPTER 6:  RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTIONS

 

TRUE/FALSE

 

1. The time required to drive from New York to New Mexico is a discrete random variable.

 

ANS:   F                      NAT:   Analytic; Probability Distributions

 

2. A random variable is a function or rule that assigns a number to each outcome of an experiment.

 

ANS:   T                      NAT:   Analytic; Probability Distributions

 

3. The number of home insurance policy holders is an example of a discrete random variable

 

ANS:   T                      NAT:   Analytic; Probability Distributions

 

4. The mean of a discrete probability distribution for X is the sum of all possible values of X, divided by the number of possible values of X.

 

ANS:   F                      NAT:   Analytic; Probability Distributions

 

5. The length of time for which an apartment in a large complex remains vacant is a discrete random variable.

 

ANS:   F                      NAT:   Analytic; Probability Distributions

 

6. The number of homeless people in Boston is an example of a discrete random variable.

 

ANS:   T                      NAT:   Analytic; Probability Distributions

 

7. A continuous variable may take on any value within its relevant range even though the measurement device may not be precise enough to record it.

 

ANS:   T                      NAT:   Analytic; Probability Distributions

 

8. Given that X is a discrete random variable, then the laws of expected value and variance can be applied to show that E(X + 5) = E(X) + 5, and V(X + 5) = V(X) + 25.

 

ANS:   F                      NAT:   Analytic; Probability Distributions

 

9. A table, formula, or graph that shows all possible values a random variable can assume, together with their associated probabilities, is referred to as discrete probability distribution.

 

ANS:   T                      NAT:   Analytic; Probability Distributions

 

10. Faculty rank (professor, associate professor, assistant professor, and lecturer) is an example of a discrete random variable.

 

ANS:   F                      NAT:   Analytic; Probability Distributions

 

11. For a random variable X, if V(cX) = 4V(X), where V refers to the variance, then c must be 2.

 

ANS:   T                      NAT:   Analytic; Probability Distributions

 

12. The amount of milk consumed by a baby in a day is an example of a discrete random variable.

 

ANS:   F                      NAT:   Analytic; Probability Distributions

 

13. Another name for the mean of a probability distribution is its expected value.

 

ANS:   T                      NAT:   Analytic; Probability Distributions

 

14. For a random variable X, E(X + 2) – 5 = E(X) – 3, where E refers to the expected value.

 

ANS:   T                      NAT:   Analytic; Probability Distributions

 

15. For a random variable X, V(X + 3) = V(X + 6), where V refers to the variance.

 

ANS:   T                      NAT:   Analytic; Probability Distributions

 

16. The binomial random variable is the number of successes that occur in a fixed period of time.

 

ANS:   F                      NAT:   Analytic; Probability Distributions

 

17. The binomial probability distribution is a discrete probability distribution.

 

ANS:   T                      NAT:   Analytic; Probability Distributions

 

18. The binomial distribution deals with consecutive trials, each of which has two possible outcomes.

 

ANS:   T                      NAT:   Analytic; Probability Distributions

 

19. The binomial distribution deals with consecutive trials, each of which has two possible outcomes.

 

ANS:   T                      NAT:   Analytic; Probability Distributions

 

20. The variance of a binomial distribution for which n = 50 and p = 0.20 is 8.0.

 

ANS:   T                      NAT:   Analytic; Probability Distributions

 

21. The expected number of heads in 250 tosses of an unbiased coin is 125.

 

ANS:   T                      NAT:   Analytic; Probability Distributions

 

22. If X is a binomial random variable with n = 25, and p = 0.25, then P(X = 25) = 1.0.

 

ANS:   F                      NAT:   Analytic; Probability Distributions

 

23. The standard deviation of a binomial random variable X is given by the formula s² = np(1 – p), where n is the number of trials, and p is the probability of success.

 

ANS:   F                      NAT:   Analytic; Probability Distributions

 

24. The number of female customers out of a random sample of 100 customers arriving at a department store has a binomial distribution.

 

ANS:   T                      NAT:   Analytic; Probability Distributions

 

25. If the probability of success p remains constant in a binomial distribution, an increase in n will increase the variance.

 

ANS:   T                      NAT:   Analytic; Probability Distributions

 

26. If the probability of success p remains constant in a binomial distribution, an increase in n will not change the mean.

 

ANS:   F                      NAT:   Analytic; Probability Distributions

 

27. The Poisson probability distribution is a continuous probability distribution.

 

ANS:   F                      NAT:   Analytic; Probability Distributions

 

28. In a Poisson distribution, the mean and variance are equal.

 

ANS:   T                      NAT:   Analytic; Probability Distributions

 

29. The Poisson random variable is a discrete random variable with infinitely many possible values.

 

ANS:   T                      NAT:   Analytic; Probability Distributions

 

30. The mean of a Poisson distribution, where m is the average number of successes occurring in a specified interval, is m.

 

ANS:   T                      NAT:   Analytic; Probability Distributions

 

31. The number of accidents that occur at a busy intersection in one month is an example of a Poisson random variable.

 

ANS:   T                      NAT:   Analytic; Probability Distributions

 

32. The number of customers arriving at a department store in a 5-minute period has a Poisson distribution.

 

ANS:   T                      NAT:   Analytic; Probability Distributions

 

33. The number of customers making a purchase out of 30 randomly selected customers has a Poisson distribution.

 

ANS:   F                      NAT:   Analytic; Probability Distributions

 

34. The largest value that a Poisson random variable X can have is n.

 

ANS:   F                      NAT:   Analytic; Probability Distributions

 

35. The Poisson distribution is applied to events for which the probability of occurrence over a given span of time, space, or distance is very small.

 

ANS:   T                      NAT:   Analytic; Probability Distributions

 

36. In a Poisson distribution, the variance and standard deviation are equal.

 

ANS:   F                      NAT:   Analytic; Probability Distributions

 

37. In a Poisson distribution, the mean and standard deviation are equal.

 

ANS:   F                      NAT:   Analytic; Probability Distributions

 

MULTIPLE CHOICE

 

38. A table, formula, or graph that shows all possible values a random variable can assume, together with their associated probabilities, is called a(n):

a.	discrete probability distribution.
b.	discrete random variable.
c.	expected value of a discrete random variable.
d.	None of these choices.

 

 

ANS:   A                     NAT:   Analytic; Probability Distributions

 

39. A function or rule that assigns a numerical value to each simple event of an experiment is called:

a.	a sample space.
b.	a probability distribution.
c.	a random variable.
d.	None of these choices.

 

 

ANS:   C                     NAT:   Analytic; Probability Distributions

 

40. The weighted average of the possible values that a random variable X can assume, where the weights are the probabilities of occurrence of those values, is referred to as the:

a.	variance.
b.	standard deviation.
c.	expected value.
d.	None of these choices.

 

 

ANS:   C                     NAT:   Analytic; Probability Distributions

 

41. The number of accidents that occur annually on a busy stretch of highway is an example of:

a.	a discrete random variable.
b.	a continuous random variable.
c.	expected value of a discrete random variable.
d.	expected value of a continuous random variable.

 

 

ANS:   A                     NAT:   Analytic; Probability Distributions

 

42. Which of the following are required conditions for the distribution of a discrete random variable X that can assume values x_i?

a.	0 £ p(xi) £ 1     for all xi
b.
c.	Both a and b are required conditions.
d.	Neither a nor b are required conditions.

 

 

ANS:   C                     NAT:   Analytic; Probability Distributions

 

43. Which of the following is not a required condition for the distribution of a discrete random variable X that can assume values x_i?

a.	0 £ p(xi) £ 1     for all xi
b.
c.	p(xi) > 1     for all xi
d.	All of these choices are true.

 

 

ANS:   C                     NAT:   Analytic; Probability Distributions

 

44. A lab at the DeBakey Institute orders 150 rats a week for each of the 52 weeks in the year for experiments that the lab conducts. Suppose the mean cost of rats used in lab experiments turned out to be $20.00 per week. Interpret this value.

a.	Most of the weeks resulted in rat costs of $20.00
b.	The median cost for the distribution of rat costs is $20.00
c.	The expected or average costs for all weekly rat purchases is $20.00
d.	The rat cost that occurs more often than any other is $20.00

 

 

ANS:   C                     NAT:   Analytic; Probability Distributions

 

45. In the notation below, X is the random variable, c is a constant, and V refers to the variance. Which of the following laws of variance is not true?

a.	V(c) = 0
b.	V(X + c) = V(X) + c
c.	V(cX) = c2 V(X)
d.	None of these choices.

 

 

ANS:   B                     NAT:   Analytic; Probability Distributions

 

46. Which of the following is a discrete random variable?

a.	The Dow Jones Industrial average.
b.	The volume of water in Michigan Lakes.
c.	The time it takes you to drive to school.
d.	The number of employees of a soft drink company.

 

 

ANS:   D                     NAT:   Analytic; Probability Distributions

 

47. Which of the following is a continuous random variable?

a.	The number of employees of an automobile company.
b.	The amount of milk produced by a cow in one 24-hour period.
c.	The number of gallons of milk sold at Albertson’s grocery store last week.
d.	None of these choices.

 

 

ANS:   B                     NAT:   Analytic; Probability Distributions

 

48. In the notation below, X is the random variable, E and V refer to the expected value and variance, respectively. Which of the following is false?

a.	E(3X) = 3E(X)
b.	V(2) = 0
c.	E(X + 1) = E(X) + 1
d.	All of these choices are true.

 

 

ANS:   D                     NAT:   Analytic; Probability Distributions

 

49. Which of the following about the binomial distribution is not a true statement?

a.	The probability of success must be constant from trial to trial.
b.	The random variable of interest is continuous.
c.	Each outcome may be classified as either “success” or “failure”.
d.	Each outcome is independent of the other.

 

 

ANS:   B                     NAT:   Analytic; Probability Distributions

 

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

a.	25
b.	50
c.	75
d.	100

 

 

ANS:   C                     NAT:   Analytic; Probability Distributions

 

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

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

 

 

ANS:   A                     NAT:   Analytic; Probability Distributions

 

52. The variance of a binomial distribution for which n = 100 and p = 0.20 is:

a.	100
b.	80
c.	20
d.	16

 

 

ANS:   D                     NAT:   Analytic; Probability Distributions

 

53. If n = 10 and p = 0.60, then the mean of the binomial distribution is

a.	0.06
b.	2.65
c.	6.00
d.	5.76

 

 

ANS:   C                     NAT:   Analytic; Probability Distributions

 

54. If n= 20 and p = 0.70, then the standard deviation of the binomial distribution is

a.	0.14
b.	2.05
c.	14.0
d.	14.7

 

 

ANS:   B                     NAT:   Analytic; Probability Distributions

 

55. The expected value, E(X), of a binomial probability distribution with n trials and probability p of success is:

a.	n + p
b.	np(1 – p)
c.	np
d.	n + p – 1

 

 

ANS:   C                     NAT:   Analytic; Probability Distributions

 

56. Which of the following cannot have a Poisson distribution?

a.	The length of 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.

 

 

ANS:   A                     NAT:   Analytic; Probability Distributions

 

57. The Sutton police department must write, on average, 6 tickets a day to keep department revenues at budgeted levels. Suppose the number of tickets written per day follows a Poisson distribution with a mean of 6.5 tickets per day. Interpret the value of the mean.

a.	The mean has no interpretation.
b.	The expected number of tickets written would be 6.5 per day.
c.	Half of the days have less than 6.5 tickets written and half of the days have more than 6.5 tickets written.
d.	The number of tickets that is written most often is 6.5 tickets per day.

 

 

ANS:   B                     NAT:   Analytic; Probability Distributions

 

58. 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.

 

 

ANS:   A                     NAT:   Analytic; Probability Distributions

 

59. Given a Poisson random variable X, where the average number of successes occurring in a specified interval is 1.8, then P(X = 0) is:

a.	1.8
b.	1.3416
c.	0.1653
d.	6.05

 

 

ANS:   C                     NAT:   Analytic; Probability Distributions

 

60. In a Poisson distribution, the:

a.	mean equals the standard deviation.
b.	median equals the standard deviation.
c.	mean equals the variance.
d.	None of these choices.

 

 

ANS:   C                     NAT:   Analytic; Probability Distributions

 

61. On the average, 1.6 customers per minute arrive at any one of the checkout counters of Sunshine food market. What type of probability distribution can be used to find out the probability that there will be no customers arriving at a checkout counter in 10 minutes?

a.	Poisson distribution
b.	Normal distribution
c.	Binomial distribution
d.	None of these choices.

 

 

ANS:   A                     NAT:   Analytic; Probability Distributions

 

62. A community college has 150 word processors. The probability that any one of them will require repair on a given day is 0.025. To find the probability that exactly 25 of the word processors will require repair, one will use what type of probability distribution?

a.	Normal distribution
b.	Poisson distribution
c.	Binomial distribution
d.	None of these choices.

 

 

ANS:   C                     NAT:   Analytic; Probability Distributions

 

COMPLETION

 

63. A motorcycle insurance company evaluates many numerical variables about a person before deciding on an appropriate rate for motorcycle insurance. How long a person has been a licensed rider is an example of a(n) ____________________ random variable.

 

ANS:   continuous

 

NAT:   Analytic; Probability Distributions

 

64. An auto insurance company evaluates many numerical variables about a person before deciding on an appropriate rate for automobile insurance. The number of claims a person has made in the last 3 years is an example of a(n) ____________________ random variable.

 

ANS:   discrete

 

NAT:   Analytic; Probability Distributions

 

65. An auto insurance company evaluates many numerical variables about a person before deciding on an appropriate rate for automobile insurance. A person’s age is an example of a(n) ____________________ random variable.

 

ANS:   continuous

 

NAT:   Analytic; Probability Distributions

 

66. A motorcycle insurance company evaluates many numerical variables about a person before deciding on an appropriate rate for motorcycle insurance. The number of tickets a person has received in the last 3 years is an example of a(n) ____________________ random variable.

 

ANS:   discrete

 

NAT:   Analytic; Probability Distributions

 

67. A motorcycle insurance company evaluates many numerical variables about a person before deciding on an appropriate rate for motorcycle insurance. The distance a person rides in a year is an example of a(n) ____________________ random variable.

 

ANS:   continuous

 

NAT:   Analytic; Probability Distributions

 

68. The dean of students conducted a survey on campus. Grade point average (GPA) is an example of a(n) ____________________ random variable.

 

ANS:   continuous

 

NAT:   Analytic; Probability Distributions

 

69. The amount of time that a microcomputer is used per week is an example of a(n) ____________________ random variable.

 

ANS:   continuous

 

NAT:   Analytic; Probability Distributions

 

70. The number of days that a microcomputer goes without a breakdown is an example of a(n) ____________________ random variable.

 

ANS:   discrete

 

NAT:   Analytic; Probability Distributions

 

71. A(n) ____________________ random variable is one whose values are uncountable.

 

ANS:   continuous

 

NAT:   Analytic; Probability Distributions

 

72. A(n) ____________________ random variable is one whose values are countable.

 

ANS:   discrete

 

NAT:   Analytic; Probability Distributions

 

73. A binomial experiment consists of a(n) ____________________ number of trials, n.

 

ANS:   fixed

 

NAT:   Analytic; Probability Distributions

 

74. In each trial of a binomial experiment, there are ____________________ possible outcomes.

 

ANS:

two

2

 

NAT:   Analytic; Probability Distributions

 

75. The probability of a success in a binomial experiment is denoted by ____________________.

 

ANS:   p

 

NAT:   Analytic; Probability Distributions

 

76. The probability of a failure in a binomial experiment is denoted by ____________________.

 

ANS:   1 – p

 

NAT:   Analytic; Probability Distributions

 

77. The trials in a binomial experiment are ____________________, meaning the outcome of one trial does not affect the outcomes of any other trials.

 

ANS:   independent

 

NAT:   Analytic; Probability Distributions

 

78. The mean of a binomial distribution is equal to ____________________.

 

ANS:   np

 

NAT:   Analytic; Probability Distributions

 

79. The variance of a binomial distribution is equal to ____________________.

 

ANS:   np(1 – p)

 

NAT:   Analytic; Probability Distributions

 

80. The probability P(X £ x) is called a(n) ____________________ probability. The binomial table reports these probabilities.

 

ANS:   cumulative

 

NAT:   Analytic; Probability Distributions

 

81. To find the probability that X is at least 10, you should find the probability that X is 10 or ____________________.

 

ANS:   more

 

NAT:   Analytic; Probability Distributions

 

82. To find the probability that X is at most 10, you should find the probability that X is 10 or ____________________.

 

ANS:   less

 

NAT:   Analytic; Probability Distributions

 

83. In a Poisson experiment, the number of successes that occur in any interval of time is ____________________ of the number of success that occur in any other interval.

 

ANS:   independent

 

NAT:   Analytic; Probability Distributions

 

84. In a(n) ____________________ experiment, the probability of a success in an interval is the same for all equal-sized intervals.

 

ANS:   Poisson

 

NAT:   Analytic; Probability Distributions

 

85. In a Poisson experiment, the probability of a success in an interval is ____________________ to the size of the interval.

 

ANS:   proportional

 

NAT:   Analytic; Probability Distributions

 

86. In Poisson experiment, the probability of more than one success in an interval approaches ____________________ as the interval becomes smaller.

 

ANS:

zero

0

 

NAT:   Analytic; Probability Distributions

 

87. A Poisson random variable is the number of successes that occur in a period of ____________________ or an interval of ____________________ in a Poisson experiment.

 

ANS:   time; space

 

NAT:   Analytic; Probability Distributions

 

88. The ____________________ of a Poisson distribution is the rate at which successes occur for a given period of time or interval of space.

 

ANS:

mean

expected value

 

NAT:   Analytic; Probability Distributions

 

89. In the Poisson distribution, the mean is equal to the ____________________.

 

ANS:   variance

 

NAT:   Analytic; Probability Distributions

 

90. In the Poisson distribution, the ____________________ is equal to the variance.

 

ANS:   mean

 

NAT:   Analytic; Probability Distributions

 

91. The possible values of a Poisson random variable start at ____________________.

 

ANS:

zero

0

 

NAT:   Analytic; Probability Distributions

 

92. A Poisson random variable is a(n) ____________________ random variable.

 

ANS:   discrete

 

NAT:   Analytic; Probability Distributions

 

SHORT ANSWER

 

93. For each of the following random variables, indicate whether the variable is discrete or continuous, and specify the possible values that it can assume.

 

a.	X = the number of traffic accidents in Albuquerque on a given day.
b.	X = the amount of weight lost in a month by a randomly selected dieter.
c.	X = the average number of children per family in a random sample of 175 families.
d.	X = the number of households out of 10 surveyed that own a convection oven.
e.	X = the time in minutes required to obtain service in a restaurant.

 

 

ANS:

 

a.	discrete; x = 0, 1, 2, 3, . . .
b.	continuous; -¥ < x < ¥
c.	continuous; x ³ 0
d.	discrete; x = 0, 1, 2, . . . , 10
e.	continuous; x > 0

 

 

NAT:   Analytic; Probability Distributions

 

Number of Motorcycles

 

The probability distribution of a discrete random variable X is shown below, where X represents the number of motorcycles owned by a family.

 

x	0	1	2	3
p(x)	0.25	0.40	0.20	0.15

 

 

94. {Number of Motorcycles Narrative} Find the following probabilities:

 

a.	P(X > 1)
b.	P(X £ 2)
c.	P(1 £ X £ 2)
d.	P(0 < X < 1)
e.	P(1 £ X < 3)

 

 

ANS:

 

a.	0.35
b.	0.85
c.	0.60
d.	0.00
e.	0.60

 

 

NAT:   Analytic; Probability Distributions

 

95. {Number of Motorcycles Narrative} Find the expected value of X.

 

ANS:

E(X) = 1.25 cars

 

NAT:   Analytic; Probability Distributions

 

96. {Number of Motorcycles Narrative} Find the standard deviation of X.

 

ANS:

s = 0.9937 cars

 

NAT:   Analytic; Probability Distributions

 

97. {Number of Motorcycles Narrative} Apply the laws of expected value to find the following:

 

a.	E(X2)
b.	E(2X2 + 5)
c.	E(X – 2)2

 

 

ANS:

 

a.	2.55
b.	10.1
c.	1.55

 

 

NAT:   Analytic; Probability Distributions

 

98. {Number of Motorcycles Narrative} Apply the laws of expected value and variance to find the following:

 

a.	V(3X)
b.	V(3X – 2)
c.	V(3)
d.	V(3X) – 2

 

 

ANS:

 

a.	8.89
b.	8.89
c.	0
d.	8.89

 

 

NAT:   Analytic; Probability Distributions

 

Number of Horses

 

The random variable X represents the number of horses per family in a rural area in Iowa, with the probability distribution: p(x) = 0.05x, x = 2, 3, 4, 5, or 6.

 

99. {Number of Horses Narrative} Express the probability distribution in tabular form.

 

ANS:

 

x	2	3	4	5	6
p(x)	0.10	0.15	0.20	0.25	0.30

 

 

NAT:   Analytic; Probability Distributions

 

100. {Number of Horses Narrative} Find the expected number of horses per family.

 

ANS:

E(X) = 4.5

 

NAT:   Analytic; Probability Distributions

 

101. {Number of Horses Narrative} Find the variance and standard deviation of X.

 

ANS:

s² = 1.75, and s = 1.323

 

NAT:   Analytic; Probability Distributions

 

102. {Number of Horses Narrative} Find the following probabilities:

 

a.	P(X ³ 4)
b.	P(X > 4)
c.	P(3 £ X £ 5)
d.	P(2 < X < 4)
e.	P(X = 4.5)

 

 

ANS:

 

a.	0.75
b.	0.55
c.	0.60
d.	0.15
e.	0.00

 

 

NAT:   Analytic; Probability Distributions

 

103. Determine which of the following are not valid probability distributions, and explain why not.

 

a.	x	0	1	2	3
	p(x)	0.15	0.25	0.35	0.45
b.	x	2	3	4	5
	p(x)	-0.10	0.40	0.50	0.25
c.	x	-2	-1	0	1	2
	p(x)	0.10	0.20	0.40	0.20	0.10

 

 

ANS:

 

a.	This is not a valid probability distribution because the probabilities don’t sum to one.
b.	This is not a valid probability distribution because it contains a negative probability.
c.	This is a valid probability distribution.

 

 

NAT:   Analytic; Probability Distributions

 

Blackjack

 

The probability distribution of a random variable X is shown below, where X represents the amount of money (in $1,000s) gained or lost in a particular game of Blackjack.

 

x	-4	0	4	8
p(x)	0.15	0.25	0.20	0.40

 

 

104. {Blackjack Narrative} Find the following probabilities:

 

a.	P(X £ 0)
b.	P(X > 3)
c.	P(0 £ X £ 4)
d.	P(X = 5)

 

 

ANS:

 

a.	0.40
b.	0.60
c.	0.45
d.	0.00

 

 

NAT:   Analytic; Probability Distributions

 

105. {Blackjack Narrative} Find the following values, and indicate their units.

 

a.	E(X)
b.	V(X)
c.	Standard deviation of X

 

 

ANS:

 

a.	$3.40
b.	19.64 (dollars squared)
c.	$4.43

 

 

NAT:   Analytic; Probability Distributions

 

Gym Visits

 

Let X represent the number of times a student visits a gym in a one month period. Assume that the probability distribution of X is as follows:

 

x	0	1	2	3
p(x)	0.05	0.25	0.50	0.20

 

 

106. {Gym Visits Narrative} Find the mean m and the standard deviation s of this distribution.

 

ANS:

m_x =1.85, and s_x = 0.792

 

NAT:   Analytic; Probability Distributions

 

107. {Gym Visits Narrative} Find the mean and the standard deviation of Y = 2X – 1.

 

ANS:

m_y = 2.70, and s_y = 1.584

 

NAT:   Analytic; Probability Distributions

 

108. {Gym Visits Narrative} What is the probability that the student visits the gym at least once in a month?

 

ANS:

P(1) + P(2) + P(3) = 0.95

 

NAT:   Analytic; Probability Distributions

 

109. {Gym Visits Narrative} What is the probability that the student visits the gym at most twice in a month?

 

ANS:

P(0) + P(1) + P(2) = 0.80

 

NAT:   Analytic; Probability Distributions

 

110. The monthly sales at a Gas Station have a mean of $50,000 and a standard deviation of $6,000. Profits are calculated by multiplying sales by 40% and subtracting fixed costs of $12,000. Find the mean and standard deviation of monthly profits.

 

ANS:

Let P = profit, and X = sales. Then P = 0.40X – 12,000.

E(P) = E(0.40X – 12,000) = 0.40 E(X) – 12,000 = 0.40($50,000) – $12,000 = $8,000

V(P) = V(0.40X – 12,000) = (0.40)² V(X) = (0.40)² (6,000)² = 5,760,000.

Thus, the mean and standard deviation of monthly profits are $8,000 and $2,400, respectively.

 

NAT:   Analytic; Probability Distributions

 

Shopping Outlet

 

A shopping outlet estimates the probability distribution of the number of stores shoppers actually enter as shown in the table below.

 

x	0	1	2	3	4
p(x)	0.05	0.35	0.25	0.20	0.15

 

 

111. {Shopping Outlet Narrative} Find the expected value of the number of stores entered.

 

ANS:

E(X) = 2.05

 

NAT:   Analytic; Probability Distributions

 

112. {Shopping Outlet Narrative} Find the variance and standard deviation of the number of stores entered.

 

ANS:

 

NAT:   Analytic; Probability Distributions

 

113. {Shopping Outlet Narrative} Suppose Y = 2X + 1 for each value of X. What is the probability distribution of Y?

 

ANS:

 

y	1	3	5	7	9
P(y)	0.05	0.35	0.25	0.20	0.15

 

 

NAT:   Analytic; Probability Distributions

 

114. {Shopping Outlet Narrative} Calculate the expected value of Y directly from the probability distribution of Y.

 

ANS:

E(Y) = 5.10

 

NAT:   Analytic; Probability Distributions

 

115. {Shopping Outlet Narrative} Use the laws of expected value to calculate the mean of Y from the probability distribution of X.

 

ANS:

E(Y) = E(2X + 1) = 2E(X) + 1 = 2(2.05) + 1 = 5.10

 

NAT:   Analytic; Probability Distributions

 

116. {Shopping Outlet Narrative} Calculate the variance and standard deviation of Y directly from the probability distribution of Y.

 

ANS:

 

NAT:   Analytic; Probability Distributions

 

117. {Shopping Outlet Narrative} Use the laws of variance to calculate the variance and standard deviation of Y from the probability distribution of X.

 

ANS:

V(Y) = V(2X + 1) = 4V(X) = 4(1.3475) = 5.39; SD(Y) = ÖV(X) = Ö5.39 = 2.32

 

NAT:   Analytic; Probability Distributions

 

118. {Shopping Outlet Narrative} What did you notice about the mean, variance, and standard deviation of Y = 2X + 1 in terms of the mean, variance, and standard deviation of X?

 

ANS:

E(Y) = 2E(X) + 1, V(Y) = 4V(X), and SD(Y) = 2SD(X).

 

NAT:   Analytic; Probability Distributions

 

Retries

 

The following table contains the probability distribution for X = the number of retries necessary to successfully transmit a 1024K data package through a double satellite media.

 

x	0	1	2	3
p(x)	0.35	0.35	0.25	0.05

 

 

119. {Retries Narrative} What is the probability of no retries?

 

ANS:

p(0) = 0.35

 

NAT:   Analytic; Probability Distributions

 

120. {Retries Narrative} What is the probability of a least one retry?

 

ANS:

p(1) + p(2) + p(3) = 0.65

 

NAT:   Analytic; Probability Distributions

 

121. {Retries Narrative} What is the mean or expected value for the number of retries?

 

ANS:

E(X) = 1.0

 

NAT:   Analytic; Probability Distributions

 

122. {Retries Narrative} What is the variance for the number of retries?

 

ANS:

V(X) = 0.80

 

NAT:   Analytic; Probability Distributions

 

123. {Retries Narrative} What is the standard deviation of the number of retries?

 

ANS:

s = 0.894

 

NAT:   Analytic; Probability Distributions

 

124. Evaluate the following binomial coefficients.

a.
b.
c.
d.

 

 

ANS:

 

a.	15
b.	20
c.	35
d.	35

 

 

NAT:   Analytic; Probability Distributions

 

Stress

 

Consider a binomial random variable X with n = 5 and p = 0. 40, where X represents the number of times in the final exam week a student with 18 credit hours may feel stressed.

 

125. {Stress Narrative} Find the probability distribution of X.

 

ANS:

 

x	0	1	2	3	4	5
p(x)	.0778	.2592	.3456	.2304	.0768	.0102

 

 

NAT:   Analytic; Probability Distributions

 

126. {Stress Narrative} Find P(X < 3).

 

ANS:

0.6826

 

NAT:   Analytic; Probability Distributions

 

127. {Stress Narrative} Find P(2 £ X £ 4).

 

ANS:

0.6528

 

NAT:   Analytic; Probability Distributions

 

128. {Stress Narrative} Find the expected number of times a student may feel stressed during the final exam week.

 

ANS:

E(X) = 2

 

NAT:   Analytic; Probability Distributions

 

129. {Stress Narrative} Find the variance and standard deviation.

 

ANS:

s² = 1.2 and s = 1.095

 

NAT:   Analytic; Probability Distributions

 

130. Given a binomial random variable with n = 20 and p = 0.60, find the following probabilities using the binomial table.

 

a.	P(X £ 13)
b.	P(X ³ 15)
c.	P(X = 17)
d.	P(11 £ X £ 14)
e.	P(11 < X < 14)

 

 

ANS:

 

a.	0.75
b.	0.126
c.	0.012
d.	0.629
e.	0.346

 

 

NAT:   Analytic; Probability Distributions

 

Montana Highways

 

A recent survey in Montana revealed that 60% of the vehicles traveling on highways, where speed limits are posted at 70 miles per hour, were exceeding the limit. Suppose you randomly record the speeds of ten vehicles traveling on US 131 where the speed limit is 70 miles per hour. Let X denote the number of vehicles that were exceeding the limit.

 

131. {Montana Highways Narrative} What is the distribution of X?

 

ANS:

X is a binomial random variable with n = 10 and p = 0.60.

 

NAT:   Analytic; Probability Distributions

 

132. {Montana Highways Narrative} Find P(X = 10).

 

ANS:

0.006

 

NAT:   Analytic; Probability Distributions

 

133. {Montana Highways Narrative} Find P(4 < X < 9).

 

ANS:

0.788

 

NAT:   Analytic; Probability Distributions

 

134. {Montana Highways Narrative} Find P(X = 2).

 

ANS:

0.01

 

NAT:   Analytic; Probability Distributions

 

135. {Montana Highways Narrative} Find P(3 £ X £ 6).

 

ANS:

0.606

 

NAT:   Analytic; Probability Distributions

 

136. {Montana Highways Narrative} Find the expected number of vehicles that are traveling on Montana highways and exceeding the speed limit.

 

ANS:

E(X) = 6

 

NAT:   Analytic; Probability Distributions

 

137. {Montana Highways Narrative} Find the standard deviation of number of vehicles that are traveling on Montana highways and exceeding the speed limit.

 

ANS:

s = 1.549

 

NAT:   Analytic; Probability Distributions

 

Online Bankers

 

An official from the securities commission estimates that 75% of all online bankers have profited from the use of insider information. Assume that 15 online bankers are selected at random from the commission’s registry.

 

138. {Online Bankers Narrative} Find the probability that at most 10 have profited from insider information.

 

ANS:

0.314

 

NAT:   Analytic; Probability Distributions

 

139. {Online Bankers Narrative} Find the probability that at least 6 have profited from insider information.

 

ANS:

0.999

 

NAT:   Analytic; Probability Distributions

 

140. {Online Bankers Narrative} Find the probability that all 15 have profited from insider information.

 

ANS:

0.013

 

NAT:   Analytic; Probability Distributions

 

141. {Online Bankers Narrative} What is the expected number of Online bankers who have profited from the use of insider information?

 

ANS:

E(X) = 11.25

 

NAT:   Analytic; Probability Distributions

 

142. {Online Bankers Narrative} Find the variance and standard deviation of the number of Online bankers who have profited from the use of insider information.

 

ANS:

s² = 2.8125, and s = 1.677

 

NAT:   Analytic; Probability Distributions

 

143. Let X be a binomial random variable with n = 25 and p = 0.01.

 

a.	Use the binomial table to find P(X = 0), P(X = 1), and P(X = 2).
b.	Find the variance and standard deviation of X.

 

 

ANS:

 

a.	P(X = 0) = 0.778, P(X = 1) = 0.196, and P(X = 2) = 0.024
b.	s2 = np(1 – p) = 25(0.01)(0.99) = 0.2475, and s = 0.4975

 

 

NAT:   Analytic; Probability Distributions

 

144. A remedial program evenly enrolls tradition and non-traditional students. If a random sample of 4 students is selected from the program to be interviewed about the introduction of a new on-line class, what is the probability that all 4 students selected are traditional students?

 

ANS:

.0625

 

NAT:   Analytic; Probability Distributions

 

145. If X has a binomial distribution with n = 4 and p = 0.3, find P(X = 0).

 

ANS:

0.4116

 

NAT:   Analytic; Probability Distributions

 

146. If X has a binomial distribution with n = 4 and p = 0.3, find P(X > 1).

 

ANS:

0.3483

 

NAT:   Analytic; Probability Distributions

 

147. If X has a binomial distribution with n = 4 and p = 0.3, find the probability that X is at most one.

 

ANS:

.6517

 

NAT:   Analytic; Probability Distributions

 

148. If X has a binomial distribution with n = 4 and p = 0.3, find the probability that X is at least one.

 

ANS:

.7599

 

NAT:   Analytic; Probability Distributions

 

Sports Fans

 

Suppose that past history shows that 5% of college students are sports fans. A sample of 10 students is to be selected.

 

149. {Sports Fans Narrative} Find the probability that exactly 1 student is a sports fan.

 

ANS:

0.315

 

NAT:   Analytic; Probability Distributions

 

150. {Sports Fans Narrative} Find the probability that at least 1 student is a sports fan.

 

ANS:

0.401

 

NAT:   Analytic; Probability Distributions

 

151. {Sports Fans Narrative} Find the probability that less than 1 student is a sports fan.

 

ANS:

0.599

 

NAT:   Analytic; Probability Distributions

 

152. {Sports Fans Narrative} Find the probability that at most 1 student is a sports fan.

 

ANS:

0.914

 

NAT:   Analytic; Probability Distributions

 

153. {Sports Fans Narrative} Find the probability that more than 1 student is a sports fan.

 

ANS:

0.086

 

NAT:   Analytic; Probability Distributions

 

154. {Sports Fans Narrative} A sample of 100 students is to be selected. What is the average number that you would expect to sports fan?

 

ANS:

5

 

NAT:   Analytic; Probability Distributions

 

155. {Sports Fans Narrative} A sample of 100 students is to be selected. What is the standard deviation of the number of sports fans you expect?

 

ANS:

2.18

 

NAT:   Analytic; Probability Distributions

 

156. Compute the following Poisson probabilities (to 4 decimal places) using the Poisson formula:

 

a.	P(X = 3), if m = 2.5
b.	P(X £ 1), if m = 2.0
c.	P(X ³ 2), if m = 3.0

 

 

ANS:

 

a.	0.2138
b.	0.4060
c.	0.8009

 

 

NAT:   Analytic; Probability Distributions

 

157. Let X be a Poisson random variable with m = 6. Use the table of Poisson probabilities to calculate:

 

a.	P(X £ 8)
b.	P(X = 8)
c.	P(X ³ 5)
d.	P(6 £ X £ 10)

 

 

ANS:

 

a.	0.847
b.	0.103
c.	0.715
d.	0.511

 

 

NAT:   Analytic; Probability Distributions

 

158. Let X be a Poisson random variable with m = 8. Use the table of Poisson probabilities to calculate:

 

a.	P(X £ 6)
b.	P(X = 4)
c.	P(X ³ 3)
d.	P(9 £ X £ 14)

 

 

ANS:

 

a.	0.313
b.	0.058
c.	0.986
d.	0.390

 

 

NAT:   Analytic; Probability Distributions

 

911 Phone Calls

 

911 phone calls arrive at the rate of 30 per hour at the local call center.

 

159. {911 Phone Calls Narrative} Find the probability of receiving two calls in a five-minute interval of time.

 

ANS:

m = 5(30/60) = 2.5; P(X = 2) = 0.2565

 

NAT:   Analytic; Probability Distributions

 

160. {911 Phone Calls Narrative} Find the probability of receiving exactly eight calls in 15 minutes.

 

ANS:

m = 15(30/60) = 7.5; P(X = 8) = 0.1373

 

NAT:   Analytic; Probability Distributions

 

161. {911 Phone Calls Narrative} If no calls are currently being processed, what is the probability that the desk employee can take four minutes break without being interrupted?

 

ANS:

m = 4(30/60) = 2.0; P(X = 0) = 0.1353

 

NAT:   Analytic; Probability Distributions

 

Classified Department Phone Calls

 

A classified department receives an average of 10 telephone calls each afternoon between 2 and 4 P.M. The calls occur randomly and independently of one another.

 

162. {Classified Department Phone Calls Narrative} Find the probability that the department will receive 13 calls between 2 and 4 P.M. on a particular afternoon.

 

ANS:

m = 10; P(X = 13) = 0.072

 

NAT:   Analytic; Probability Distributions

 

163. {Classified Department Phone Calls Narrative} Find the probability that the department will receive seven calls between 2 and 3 P.M. on a particular afternoon.

 

ANS:

m  = 5; P(X = 7) = 0.105

 

NAT:   Analytic; Probability Distributions

 

164. {Classified Department Phone Calls Narrative} Find the probability that the department will receive at least five calls between 2 and 4 P.M. on a particular afternoon.

 

ANS:

m = 10; P(X ³ 5) = 0.971

 

NAT:   Analytic; Probability Distributions

 

Post Office

 

The number of arrivals at a local post office between 3:00 and 5:00 P.M. has a Poisson distribution with a mean of 12.

 

165. {Post Office Narrative} Find the probability that the number of arrivals between 3:00 and 5:00 P.M. is at least 10.

 

ANS:

m =12; P(X ³ 10) = 0.758

 

NAT:   Analytic; Probability Distributions

 

166. {Post Office Narrative} Find the probability that the number of arrivals between 3:30 and 4:00 P.M. is at least 10.

 

ANS:

m = 3; P(X ³ 10) = 0.001

 

NAT:   Analytic; Probability Distributions

 

167. {{Post Office Narrative} Find the probability that the number of arrivals between 4:00 and 5:00 P.M. is exactly two.

 

ANS:

m = 6; P(X = 2) = 0.045

 

NAT:   Analytic; Probability Distributions

 

168. Suppose that the number of buses arriving at a Depot per minute is a Poisson process. If the average number of buses arriving per minute is 3, what is the probability that exactly 6 buses arrive in the next minute?

 

ANS:

0.0504

 

NAT:   Analytic; Probability Distributions

 

 

Unsafe Levels of Radioactivity

 

The number of incidents at a nuclear power plant has a Poisson distribution with a mean of 6 incidents per year.

 

169. {Unsafe Levels of Radioactivity Narrative} Find the probability that there will be exactly 3 incidents in a year.

 

ANS:

0.0892

 

NAT:   Analytic; Probability Distributions

 

170. {Unsafe Levels of Radioactivity Narrative} Find the probability that there will be at least 3 incidents in a year.

 

ANS:

0.9380

 

NAT:   Analytic; Probability Distributions

 

171. {Unsafe Levels of Radioactivity Narrative} Find the probability that there will be at least 1 incident in a year.

 

ANS:

0.9975

 

NAT:   Analytic; Probability Distributions

 

172. {Unsafe Levels of Radioactivity Narrative} Find the probability that there will be no more than 1 incident in a year.

 

ANS:

0.0174

 

NAT:   Analytic; Probability Distributions

 

173. {Unsafe Levels of Radioactivity Narrative} Find the variance of the number of incidents in one year.

 

ANS:

6

 

NAT:   Analytic; Probability Distributions

 

174. {Unsafe Levels of Radioactivity Narrative} Find the standard deviation of the number of incidents is in one year.

 

ANS:

2.45

 

NAT:   Analytic; Probability Distributions