Name:
Class:
Date:
Indicate whether the statement is true or false.
1. The number of loan defaults per month at a bank is Poisson distributed.
a.
True
b.
False
2. Using the standard normal distribution, the Z– score representing the 99th percentile is 2.326.
a.
True
b.
False
3. The binomial distribution is a discrete distribution that deals with a sequence of identical trials, each of which has only
two possible outcomes.
a.
True
b.
False
4. The Poisson distribution is characterized by a single parameter , which must be positive.
a.
True
b.
False
5. For a given probability of success p that is not too close to 0 or 1, the binomial distribution tends to take on more of a
symmetric bell shape as the number of trials n increases.
a.
True
b.
False
6. The Poisson probability distribution is one of the most commonly used continuous probability distributions.
a.
True
b.
False
7. A binomial distribution with n number of trials, and probability of success p can be approximated well by a normal
distribution with mean np and variance
a.
True
b.
False
8. If the random variable X is normally distributed with mean and standard deviation , then the random variable Z
defined by is also normally distributed with mean 0 and standard deviation 1.
a.
True
b.
False
9. 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.
a.
True
b.
False
10. The variance of a binomial distribution for which n = 50 and p = 0.20 is 8.0.
a.
True
b.
False
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Class:
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11. A random variable X is normally distributed with a mean of 175 and a standard deviation of 50. Given that X = 150, its
corresponding Z– score is –0.50.
a.
True
b.
False
12. The binomial distribution deals with consecutive trials, each of which has two possible outcomes.
a.
True
b.
False
13. The variance of a binomial distribution is given by the formula , where n is the number of trials, and p
is the probability of success in any trial.
a.
True
b.
False
14. The Poisson random variable is a discrete random variable with infinitely many possible values.
a.
True
b.
False
15. Using the standard normal distribution, the Z–score representing the 5th percentile is 1.645.
a.
True
b.
False
16. The binomial distribution is a continuous distribution that is not far behind the normal distribution in order of
importance.
a.
True
b.
False
17. A random variable X is standardized when each value of X has the mean of X subtracted from it, and the difference is
divided by the standard deviation of X.
a.
True
b.
False
18. The binomial random variable represents the number of successes that occur in a specific period of time.
a.
True
b.
False
19. Much of the study of probabilistic inventory models, queuing models, and reliability models relies heavily on the
Poisson and Exponential distributions.
a.
True
b.
False
20. Using the standard normal curve, the Z– score representing the 75th percentile is 0.674.
a.
True
b.
False
21. Using the standard normal curve, the Z– score representing the 10th percentile is 1.28.
a.
True
Name:
Class:
Date:
b.
False
22. An exponential distribution with parameter = 0.2 has mean and standard deviation both equal to 5.
a.
True
b.
False
23. The mean and standard deviation of a normally distributed random variable which has been “standardized” are zero
and one, respectively.
a.
True
b.
False
24. Poisson distribution is appropriate to determine the probability of a given number of defective items in a shipment.
a.
True
b.
False
25. The total area under the normal distribution curve is equal to one.
a.
True
b.
False
Indicate the answer choice that best completes the statement or answers the question.
26. The binomial probability distribution is used with
a.
a discrete random variable
b.
a continuous random variable
c.
either a discrete or a continuous random variable, depending on the variance
d.
either a discrete or a continuous random variable, depending on the sample size
27. If the random variable X is exponentially distributed with parameter = 1.5, then P(2 X 4), up to 4 decimal places,
is
a.
0.6667
b.
0.0473
c.
0.5000
d.
0.2500
28. Sampling done without replacement means that
a.
only certain members of the population can be sampled
b.
each member of the population can be sampled repeatedly
c.
each member of the population can be sampled only once
d.
each member of the population can be sampled twice
29. Which of the following equations shows the process of standardizing?
a.
b.
c.
d.
30. A continuous probability distribution is characterized by:
a.
a list of possible values
b.
counts
c.
an array of individual values
d.
a continuum of possible values
Name:
Class:
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31. If the value of the standard normal random variable Z is positive, then the original score is where in relationship to the
mean?
a.
equal to the mean
b.
to the left of the mean
c.
to the right of the mean
d.
None of the these choices
32. The standard deviation of a probability distribution is a:
a.
measure of variability of the distribution
b.
measure of central location
c.
measure of relative likelihood
d.
measure of skewness of the distribution
33. We assume that the outcomes of successive trials in a binomial experiment are:
a.
probabilistically independent
b.
probabilistically dependent
c.
identical from trial to trial
d.
random number between 0 and 1
34. Given that Z is a standard normal variable, the value z for which P(Z z) = 0.2580 is
a.
0.70
b.
0.758
c.
–0.65
d.
0.242
35. Tossing a coin is an example of a (n)
a.
binomial distribution
b.
normal distribution
c.
exponential distribution
d.
Poisson distribution
36. The mean of a probability distribution is a:
a.
measure of variability of the distribution
b.
measure of central location
c.
measure of relative likelihood
d.
measure of skewness of the distribution
37. The Poisson and Exponential distributions are commonly used in which of the following applications
a.
Inventory models
b.
Financial models
c.
Failure analysis models
d.
All of these options
38. A Poisson distribution is:
a.
relevant when we sample from a population with only two types of members.
b.
relevant when we perform a series of independent, identical experiments with only two possible outcomes.
c.
usually relevant when we are interested in the number of events that occur over a given interval of time
d.
the cornerstone of statistical theory
e.
All of these choices
39. The variance of a binomial distribution for which n = 100 and p = 0.20 is:
a.
100
b.
80
c.
20
d.
16
40. One reason for standardizing random variables is to measure variables with:
a.
different means and standard deviations on a non-standard scale
b.
different means and standard deviations on a single scale
c.
dissimilar means and standard deviations in like terms
d.
similar means and standard deviations on two scales
Name:
Class:
Date:
41. Which probability distribution applies to the number of events occurring within a specified period of time or space
a.
Binomial distribution
b.
Poisson distribution
c.
Any discrete probability distribution
d.
Any continuous probability distribution
42. If X is a normal random variable with a standard deviation of 10, then 3X has a standard deviation equal to
a.
10
b.
13
c.
30
d.
90
43. If the random variable X is exponentially distributed with parameter = 3, then P(X 2) , up to 4 decimal places , is
a.
0.3333
b.
0.5000
c.
0.6667
d.
0.0025
44. The higher the value of the density function f(x),
a.
the less likely the value x
b.
the more likely the value x
c.
the less likely the distribution is normal
d.
None of these choices
45. 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
46. Which of the following distributions is appropriate to measure the length of time between arrivals at a grocery checkout
counter?
a.
Uniform
b.
Normal
c.
Exponential
d.
Poisson
47. The standard normal distribution has a mean and a standard deviation respectively equal to
a.
0 and 0
b.
1 and 1
c.
1 and 0
d.
0 and 1
48. The normal distribution is:
a.
a discrete distribution with two parameters
b.
a binomial distribution with only one parameter
c.
a density function of a discrete random variable
d.
a continuous distribution with two parameters
49. If the mean of an exponential distribution is 2, then the value of the parameter is
a.
4
b.
2
c.
1
d.
0.5
50. Given that Z is a standard normal random variable, P(-1.0 Z1.5) is
a.
0.7745
b.
0.8413
c.
0.0919
d.
0.9332
51. Which of the following might not be appropriately modeled with a normal distribution?
Name:
Class:
Date:
a.
The daily low temperature in Anchorage, Alaska
b.
The returns on a stock
c.
The daily change in inventory at a computer manufacturer
d.
The salaries of employees at a large company
52. Given that the random variable X is normally distributed with a mean of 80 and a standard deviation of 10, P(85 X
90) is
a.
0.5328
b.
0.3413
c.
0.1915
d.
0.1498
53. If we plot a continuous probability distribution f(x), the total probability under the curve is
a.
-1
b.
0
c.
1
d.
100
Suppose that the number of customers arriving each hour at the only checkout counter at a local convenience store is
approximately Poisson distributed with an expected arrival rate of 30 customers per hour. Let X represent the number of
customers arriving per hour. The probabilities associated with X are shown below.
P(X < 5) = 0.0000, P(X < 10) = 0.0000, P(X < 15) = 0.0009,
P(X < 20) = 0.0219, P(X < 25) = 0.1572, P(X < 30) = 0.4757
P(X = 30) = 0.0726, P(X = 31) = 0.0703, P(X = 32) = 0.0659,
P(X = 33) = 0.0599, P(X = 34) = 0.0529, P(X = 35) = 0.0453
54. What is the probability that the number of customers who arrive at this checkout counter in a given hour will be greater
than 35?
The height of a typical American male adult is normally distributed with a mean of 68 inches and a standard deviation of 5
inches. We observe the heights of 12 American male adults.
55. What is the probability that exactly half the male adults will be less than 62 inches tall?
Consider a binomial random variable X with n = 5 and p = 0.40.
56. Find the probability distribution of X.
The number of arrivals at a local gas station between 3:00 and 5:00 P.M. has a Poisson distribution with a mean of 12.
57. Find the probability that the number of arrivals between 3:30 and 4:00 P.M. is at least 10.
Wendy’s fast-food restaurant sells hamburgers and chicken sandwiches. On a typical weekday, the demand for
hamburgers is normally distributed with a mean of 450 and standard deviation of 80 and the demand for chicken
sandwiches is normally distributed with a mean of 120 and standard deviation of 30. Use this information to answer the
following questions.
58. If the restaurant stocks 600 hamburgers and 150 chicken sandwiches for a given day, what is the probability that it will
run out of hamburgers or chicken sandwiches (or both) that day? Assume that the demands for hamburgers and chicken
sandwiches are probabilistically independent.
The weekly demand for a particular automobile manufacturer follows a normal distribution with a mean of 40,000 cars and
a standard deviation of 10,000. Below you will find probability and percentile calculations related to the customer purchase
Name:
Class:
Date:
amounts. Use this information to answer the following questions.
Probability Calculations
P(Sales < 2,000,000) = 0.134, P(Sales < 2,050,000) = 0.339
P(Sales < 2,100,000) = 0.609, P(Sales < 2,150,000) = 0.834
Percentiles Calculations
1st percentile = 1,912,245, 5th percentile = 1,961,388
95th percentile = 2,198,612, 99th percentile = 2,247,755
59. What is the probability that this company will sell between 2.0 and 2.15 million cars next year?
Wendy’s fast-food restaurant sells hamburgers and chicken sandwiches. On a typical weekday, the demand for
hamburgers is normally distributed with a mean of 450 and standard deviation of 80 and the demand for chicken
sandwiches is normally distributed with a mean of 120 and standard deviation of 30. Use this information to answer the
following questions.
60. Why is the independence assumption in Question 74 probably not realistic? Using a more realistic assumption, do you
think the probability in Question 74 would increase or decrease?
The time it takes a technician to fix a computer problem is exponentially distributed with a mean of 15 minutes.
61. What is the probability density function for the time it takes a technician to fix a computer problem?
The weekly demand for a particular automobile manufacturer follows a normal distribution with a mean of 40,000 cars and
a standard deviation of 10,000. Below you will find probability and percentile calculations related to the customer purchase
amounts. Use this information to answer the following questions.
Probability Calculations
P(Sales < 2,000,000) = 0.134, P(Sales < 2,050,000) = 0.339
P(Sales < 2,100,000) = 0.609, P(Sales < 2,150,000) = 0.834
Percentiles Calculations
1st percentile = 1,912,245, 5th percentile = 1,961,388
95th percentile = 2,198,612, 99th percentile = 2,247,755
62. What number of cars, equidistant from the mean, such that 98% of car sales are between these values?
The service manager for a new appliances store reviewed sales records of the past 20 sales of new microwaves to
determine the number of warranty repairs he will be called on to perform in the next 90 days. Corporate reports indicate
that the probability any one of their new microwaves needs a warranty repair in the first 90 days is 0.05. The manager
assumes that calls for warranty repair are independent of one another and is interested in predicting the number of
warranty repairs he will be called on to perform in the next 90 days for this batch of 20 new microwaves sold.
63. What is the standard deviation of the number of the new microwaves sold that will require a warranty repair in the first
90 days?
The weekly demand for a particular automobile manufacturer follows a normal distribution with a mean of 40,000 cars and
a standard deviation of 10,000. Below you will find probability and percentile calculations related to the customer purchase
amounts. Use this information to answer the following questions.
Probability Calculations
P(Sales < 2,000,000) = 0.134, P(Sales < 2,050,000) = 0.339
P(Sales < 2,100,000) = 0.609, P(Sales < 2,150,000) = 0.834
Name:
Class:
Date:
Percentiles Calculations
1st percentile = 1,912,245, 5th percentile = 1,961,388
95th percentile = 2,198,612, 99th percentile = 2,247,755
64. What number of cars, equidistant from the mean, such that 90% of car sales are between these values?
The service manager for a new appliances store reviewed sales records of the past 20 sales of new microwaves to
determine the number of warranty repairs he will be called on to perform in the next 90 days. Corporate reports indicate
that the probability any one of their new microwaves needs a warranty repair in the first 90 days is 0.05. The manager
assumes that calls for warranty repair are independent of one another and is interested in predicting the number of
warranty repairs he will be called on to perform in the next 90 days for this batch of 20 new microwaves sold.
65. What is the probability that between two and four (inclusive) of the 20 new microwaves sold will require a warranty
repair in the first 90 days?
Suppose that the number of customers arriving each hour at the only checkout counter at a local convenience store is
approximately Poisson distributed with an expected arrival rate of 30 customers per hour. Let X represent the number of
customers arriving per hour. The probabilities associated with X are shown below.
P(X < 5) = 0.0000, P(X < 10) = 0.0000, P(X < 15) = 0.0009,
P(X < 20) = 0.0219, P(X < 25) = 0.1572, P(X < 30) = 0.4757
P(X = 30) = 0.0726, P(X = 31) = 0.0703, P(X = 32) = 0.0659,
P(X = 33) = 0.0599, P(X = 34) = 0.0529, P(X = 35) = 0.0453
66. What is the probability that at least 20 customers, but fewer than 30 customers arrive at this checkout counter in a
given hour?
A continuous random variable X has the probability density function: f(x) = 2 , 0
67. Find the mean and standard deviation of X.
The height of a typical American male adult is normally distributed with a mean of 68 inches and a standard deviation of 5
inches. We observe the heights of 12 American male adults.
68. Let Y be the number of the 12 male adults who are less than 62 inches tall. Determine the mean and standard
deviation of Y.
A continuous random variable X has the probability density function: f(x) = 2 , 0
69. What is the probability that X is between 1 and 3?
The service manager for a new appliances store reviewed sales records of the past 20 sales of new microwaves to
determine the number of warranty repairs he will be called on to perform in the next 90 days. Corporate reports indicate
that the probability any one of their new microwaves needs a warranty repair in the first 90 days is 0.05. The manager
assumes that calls for warranty repair are independent of one another and is interested in predicting the number of
warranty repairs he will be called on to perform in the next 90 days for this batch of 20 new microwaves sold.
70. What is the probability that at most two of the 20 new microwaves sold will require a warranty repair in the first 90
days?
The time it takes a technician to fix a computer problem is exponentially distributed with a mean of 15 minutes.
Name:
Class:
Date:
71. What is the variance of the time it takes a technician to fix a computer problem?
A continuous random variable X has the probability density function: f(x) = 2 , 0
72. What is the distribution of X and what are the parameters?
Consider a binomial random variable X with n = 5 and p = 0.40.
73. Find P(X < 3).
A recent survey in Michigan 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.
74. Find P(4 < X < 9).
A popular retail store knows that the distribution of purchase amounts by its customers is approximately normal with a
mean of $30 and a standard deviation of $9. Below you will find normal probability and percentile calculations related to
the customer purchase amounts.
Probability Calculations
P(Sales < $ 15.00) = 0.048, P(Sales < $ 20.00) = 0.133,
P(Sales < $ 25.00) = 0.289, P(Sales < $ 35.00) = 0.711
Percentiles Calculations
1st Percentile = $9.06, 5th Percentile = $15.20,
95th Percentile = $44.80, 99th Percentile = $50.94
75. What two dollar amounts, equidistant from the mean of $30, such that 90% of all customer purchases are between
these values?
The service manager for a new appliances store reviewed sales records of the past 20 sales of new microwaves to
determine the number of warranty repairs he will be called on to perform in the next 90 days. Corporate reports indicate
that the probability any one of their new microwaves needs a warranty repair in the first 90 days is 0.05. The manager
assumes that calls for warranty repair are independent of one another and is interested in predicting the number of
warranty repairs he will be called on to perform in the next 90 days for this batch of 20 new microwaves sold.
76. What is the expected number of the new microwaves sold that will require a warranty repair in the first 90 days?
The weekly demand for General Motors (GM) car sales follows a normal distribution with a mean of 40,000 cars and a
standard deviation of 12,000 cars.
77. What is the probability that GM will sell between 2.0 and 2.3 million cars during the next year?
The weekly demand for a particular automobile manufacturer follows a normal distribution with a mean of 40,000 cars and
a standard deviation of 10,000. Below you will find probability and percentile calculations related to the customer purchase
amounts. Use this information to answer the following questions.
Probability Calculations
P(Sales < 2,000,000) = 0.134, P(Sales < 2,050,000) = 0.339
P(Sales < 2,100,000) = 0.609, P(Sales < 2,150,000) = 0.834
Percentiles Calculations
1st percentile = 1,912,245, 5th percentile = 1,961,388
Name:
Class:
Date:
95th percentile = 2,198,612, 99th percentile = 2,247,755
78. There is a 1% chance that this company will sell more than what number of cars during the next year?
Suppose that the number of customers arriving each hour at the only checkout counter at a local convenience store is
approximately Poisson distributed with an expected arrival rate of 30 customers per hour. Let X represent the number of
customers arriving per hour. The probabilities associated with X are shown below.
P(X < 5) = 0.0000, P(X < 10) = 0.0000, P(X < 15) = 0.0009,
P(X < 20) = 0.0219, P(X < 25) = 0.1572, P(X < 30) = 0.4757
P(X = 30) = 0.0726, P(X = 31) = 0.0703, P(X = 32) = 0.0659,
P(X = 33) = 0.0599, P(X = 34) = 0.0529, P(X = 35) = 0.0453
79. What is the probability that fewer than 33 customers arrive at this checkout counter in a given hour?
The service manager for a new appliances store reviewed sales records of the past 20 sales of new microwaves to
determine the number of warranty repairs he will be called on to perform in the next 90 days. Corporate reports indicate
that the probability any one of their new microwaves needs a warranty repair in the first 90 days is 0.05. The manager
assumes that calls for warranty repair are independent of one another and is interested in predicting the number of
warranty repairs he will be called on to perform in the next 90 days for this batch of 20 new microwaves sold.
80. What is the probability that none of the 20 new microwaves sold will require a warranty repair in the first 90 days?
A recent survey in Michigan 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.
81. Find P(X = 2).
The time it takes a technician to fix a computer problem is exponentially distributed with a mean of 15 minutes.
82. What is the probability that it will take a technician between 10 to 15 minutes to fix a computer problem?
The weekly demand for a particular automobile manufacturer follows a normal distribution with a mean of 40,000 cars and
a standard deviation of 10,000. Below you will find probability and percentile calculations related to the customer purchase
amounts. Use this information to answer the following questions.
Probability Calculations
P(Sales < 2,000,000) = 0.134, P(Sales < 2,050,000) = 0.339
P(Sales < 2,100,000) = 0.609, P(Sales < 2,150,000) = 0.834
Percentiles Calculations
1st percentile = 1,912,245, 5th percentile = 1,961,388
95th percentile = 2,198,612, 99th percentile = 2,247,755
83. What is the probability that this company will sell more than 2 million cars next year?
Wendy’s fast-food restaurant sells hamburgers and chicken sandwiches. On a typical weekday, the demand for
hamburgers is normally distributed with a mean of 450 and standard deviation of 80 and the demand for chicken
sandwiches is normally distributed with a mean of 120 and standard deviation of 30. Use this information to answer the
following questions.
84. How many hamburgers must the restaurant stock to be 99% sure of not running out on a given day?
Name:
Class:
Date:
A recent survey in Michigan 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.
85. Describe the probability distribution of X.
Consider a binomial random variable X with n = 5 and p = 0.40.
86. Find the mean and the variance of X.
The service manager for a new appliances store reviewed sales records of the past 20 sales of new microwaves to
determine the number of warranty repairs he will be called on to perform in the next 90 days. Corporate reports indicate
that the probability any one of their new microwaves needs a warranty repair in the first 90 days is 0.05. The manager
assumes that calls for warranty repair are independent of one another and is interested in predicting the number of
warranty repairs he will be called on to perform in the next 90 days for this batch of 20 new microwaves sold.
87. What is the probability that more than one of the 20 new microwaves sold will require a warranty repair in the first 90
days?
A used car salesman in a small town states that, on the average, it takes him 5 days to sell a car. Assume that the
probability distribution of the length of time between sales is exponentially distributed.
88. What is the probability that he will have to wait at least 8 days before making another sale?
The service manager for a new appliances store reviewed sales records of the past 20 sales of new microwaves to
determine the number of warranty repairs he will be called on to perform in the next 90 days. Corporate reports indicate
that the probability any one of their new microwaves needs a warranty repair in the first 90 days is 0.05. The manager
assumes that calls for warranty repair are independent of one another and is interested in predicting the number of
warranty repairs he will be called on to perform in the next 90 days for this batch of 20 new microwaves sold.
89. What is the probability that less than two of the 20 new microwaves sold will require a warranty repair in the first 90
days?
A popular retail store knows that the distribution of purchase amounts by its customers is approximately normal with a
mean of $30 and a standard deviation of $9. Below you will find normal probability and percentile calculations related to
the customer purchase amounts.
Probability Calculations
P(Sales < $ 15.00) = 0.048, P(Sales < $ 20.00) = 0.133,
P(Sales < $ 25.00) = 0.289, P(Sales < $ 35.00) = 0.711
Percentiles Calculations
1st Percentile = $9.06, 5th Percentile = $15.20,
95th Percentile = $44.80, 99th Percentile = $50.94
90. What is the probability that a randomly selected customer will spend between $20 and $35?
91. What is the probability that a randomly selected customer will spend less than $15?
A set of final exam scores in an organic chemistry course was found to be normally distributed, with a mean of 73 and a
standard deviation of 8.
Name:
Class:
Date:
92. What is the probability of getting a score higher than 85 on this exam?
Consider a binomial random variable X with n = 5 and p = 0.40.
93. Find
A recent survey in Michigan 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.
94. Find P(3 X6).
The number of arrivals at a local gas station between 3:00 and 5:00 P.M. has a Poisson distribution with a mean of 12.
95. Find the probability that the number of arrivals between 4:00 and 5:00 P.M. is exactly two.
A recent survey in Michigan 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.
96. Suppose that an highway patrol officer can obtain radar readings on 500 vehicles during a typical shift. How many
traffic violations would be found in a shift?
A large retailer has purchased 10,000 DVDs. The retailer is assured by the supplier that the shipment contains no more
than 1% defective DVDs (according to agreed specifications). To check the supplier’s claim, the retailer randomly selects
100 DVDs and finds six of the 100 to be defective.
97. (A) Assuming the supplier’s claim is true, compute the mean and the standard deviation of the number of defective
DVDs in the sample.
(B) Based on your answer to (A), is it likely that as many as six DVDs would be found to be defective, if the claim is
correct?
(C) Suppose that six DVDs are indeed found to be defective. Based on your answer to (A), what might be a reasonable
inference about the manufacturer’s claim for this shipment of 10,000 DVDs?
The weekly demand for General Motors (GM) car sales follows a normal distribution with a mean of 40,000 cars and a
standard deviation of 12,000 cars.
98. There is a 5% chance that GM will sell more than what number of cars during the next year?
The service manager for a new appliances store reviewed sales records of the past 20 sales of new microwaves to
determine the number of warranty repairs he will be called on to perform in the next 90 days. Corporate reports indicate
that the probability any one of their new microwaves needs a warranty repair in the first 90 days is 0.05. The manager
assumes that calls for warranty repair are independent of one another and is interested in predicting the number of
warranty repairs he will be called on to perform in the next 90 days for this batch of 20 new microwaves sold.
99. What is the probability that at least one of the 20 new microwaves sold will require a warranty repair in the first 90
days?
A popular retail store knows that the distribution of purchase amounts by its customers is approximately normal with a
mean of $30 and a standard deviation of $9. Below you will find normal probability and percentile calculations related to
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the customer purchase amounts.
Probability Calculations
P(Sales < $ 15.00) = 0.048, P(Sales < $ 20.00) = 0.133,
P(Sales < $ 25.00) = 0.289, P(Sales < $ 35.00) = 0.711
Percentiles Calculations
1st Percentile = $9.06, 5th Percentile = $15.20,
95th Percentile = $44.80, 99th Percentile = $50.94
100. What two dollar amounts, equidistant from the mean of $30, such that 98% of all customer purchases are between
these values?
The time it takes a technician to fix a computer problem is exponentially distributed with a mean of 15 minutes.
101. What is the probability that it will take a technician less than 10 minutes to fix a computer problem?
The number of arrivals at a local gas station between 3:00 and 5:00 P.M. has a Poisson distribution with a mean of 12.
102. Find the probability that the number of arrivals between 3:00 and 5:00 P.M. is at least 10.
Suppose that the number of customers arriving each hour at the only checkout counter at a local convenience store is
approximately Poisson distributed with an expected arrival rate of 30 customers per hour. Let X represent the number of
customers arriving per hour. The probabilities associated with X are shown below.
P(X < 5) = 0.0000, P(X < 10) = 0.0000, P(X < 15) = 0.0009,
P(X < 20) = 0.0219, P(X < 25) = 0.1572, P(X < 30) = 0.4757
P(X = 30) = 0.0726, P(X = 31) = 0.0703, P(X = 32) = 0.0659,
P(X = 33) = 0.0599, P(X = 34) = 0.0529, P(X = 35) = 0.0453
103. What is the probability that at least 25 customers arrive at this checkout counter in a given hour?
The service manager for a new appliances store reviewed sales records of the past 20 sales of new microwaves to
determine the number of warranty repairs he will be called on to perform in the next 90 days. Corporate reports indicate
that the probability any one of their new microwaves needs a warranty repair in the first 90 days is 0.05. The manager
assumes that calls for warranty repair are independent of one another and is interested in predicting the number of
warranty repairs he will be called on to perform in the next 90 days for this batch of 20 new microwaves sold.
104. What is the probability that only one of the 20 new microwaves sold will require a warranty repair in the first 90 days?
Wendy’s fast-food restaurant sells hamburgers and chicken sandwiches. On a typical weekday, the demand for
hamburgers is normally distributed with a mean of 450 and standard deviation of 80 and the demand for chicken
sandwiches is normally distributed with a mean of 120 and standard deviation of 30. Use this information to answer the
following questions.
105. How many chicken sandwiches must the restaurant stock to be 99% sure of not running out on a given day?
The service manager for a new appliances store reviewed sales records of the past 20 sales of new microwaves to
determine the number of warranty repairs he will be called on to perform in the next 90 days. Corporate reports indicate
that the probability any one of their new microwaves needs a warranty repair in the first 90 days is 0.05. The manager
assumes that calls for warranty repair are independent of one another and is interested in predicting the number of
warranty repairs he will be called on to perform in the next 90 days for this batch of 20 new microwaves sold.
106. What type of probability distribution will most likely be used to analyze warranty repair needs on new microwaves in
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this situation?
A used car salesman in a small town states that, on the average, it takes him 5 days to sell a car. Assume that the
probability distribution of the length of time between sales is exponentially distributed.
107. What is the probability that he will have to wait between 6 and 10 days before making another sale?
A popular retail store knows that the distribution of purchase amounts by its customers is approximately normal with a
mean of $30 and a standard deviation of $9. Below you will find normal probability and percentile calculations related to
the customer purchase amounts.
Probability Calculations
P(Sales < $ 15.00) = 0.048, P(Sales < $ 20.00) = 0.133,
P(Sales < $ 25.00) = 0.289, P(Sales < $ 35.00) = 0.711
Percentiles Calculations
1st Percentile = $9.06, 5th Percentile = $15.20,
95th Percentile = $44.80, 99th Percentile = $50.94
108. What is the probability that a randomly selected customer will spend $20 or more?
A set of final exam scores in an organic chemistry course was found to be normally distributed, with a mean of 73 and a
standard deviation of 8.
109. What percentage of students scored between 81 and 89 on this exam?
A popular retail store knows that the distribution of purchase amounts by its customers is approximately normal with a
mean of $30 and a standard deviation of $9. Below you will find normal probability and percentile calculations related to
the customer purchase amounts.
Probability Calculations
P(Sales < $ 15.00) = 0.048, P(Sales < $ 20.00) = 0.133,
P(Sales < $ 25.00) = 0.289, P(Sales < $ 35.00) = 0.711
Percentiles Calculations
1st Percentile = $9.06, 5th Percentile = $15.20,
95th Percentile = $44.80, 99th Percentile = $50.94
110. What is the probability that a randomly selected customer will spend $30 or more?
The service manager for a new appliances store reviewed sales records of the past 20 sales of new microwaves to
determine the number of warranty repairs he will be called on to perform in the next 90 days. Corporate reports indicate
that the probability any one of their new microwaves needs a warranty repair in the first 90 days is 0.05. The manager
assumes that calls for warranty repair are independent of one another and is interested in predicting the number of
warranty repairs he will be called on to perform in the next 90 days for this batch of 20 new microwaves sold.
111. What is the probability that between three and six (exclusive) of the 20 new microwaves sold will require a warranty
repair in the first 90 days?
The weekly demand for a particular automobile manufacturer follows a normal distribution with a mean of 40,000 cars and
a standard deviation of 10,000. Below you will find probability and percentile calculations related to the customer purchase
amounts. Use this information to answer the following questions.
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Probability Calculations
P(Sales < 2,000,000) = 0.134, P(Sales < 2,050,000) = 0.339
P(Sales < 2,100,000) = 0.609, P(Sales < 2,150,000) = 0.834
Percentiles Calculations
1st percentile = 1,912,245, 5th percentile = 1,961,388
95th percentile = 2,198,612, 99th percentile = 2,247,755
112. Calculate the mean, variance, and standard deviation for the entire year (assume 52 weeks in the year).
A recent survey in Michigan 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.
113. Find P(X = 10).
The service manager for a new appliances store reviewed sales records of the past 20 sales of new microwaves to
determine the number of warranty repairs he will be called on to perform in the next 90 days. Corporate reports indicate
that the probability any one of their new microwaves needs a warranty repair in the first 90 days is 0.05. The manager
assumes that calls for warranty repair are independent of one another and is interested in predicting the number of
warranty repairs he will be called on to perform in the next 90 days for this batch of 20 new microwaves sold.
114. What is the probability that exactly two of the 20 new microwaves sold will require a warranty repair in the first 90
days?
Suppose that the number of customers arriving each hour at the only checkout counter at a local convenience store is
approximately Poisson distributed with an expected arrival rate of 30 customers per hour. Let X represent the number of
customers arriving per hour. The probabilities associated with X are shown below.
P(X < 5) = 0.0000, P(X < 10) = 0.0000, P(X < 15) = 0.0009,
P(X < 20) = 0.0219, P(X < 25) = 0.1572, P(X < 30) = 0.4757
P(X = 30) = 0.0726, P(X = 31) = 0.0703, P(X = 32) = 0.0659,
P(X = 33) = 0.0599, P(X = 34) = 0.0529, P(X = 35) = 0.0453
115. What is the probability that the number of customers who arrive at this checkout counter in a given hour will be
between 30 and 35 (inclusive)?
Past experience indicates that 20% of all freshman college students taking an intermediate algebra course withdraw from
the class.
116. (A) Using the binomial distribution, find the probability that 6 or more of the 30 students taking this course in a given
semester will withdraw from the class.
(B) Using the normal approximation to the binomial, find the probability that 6 or more of the 30 students taking this course
in a given semester will withdraw from the class.
(C) Compare the results obtained in (A) and (B). Under what conditions will the normal approximation to this binomial
probability become even more accurate?
A set of final exam scores in an organic chemistry course was found to be normally distributed, with a mean of 73 and a
standard deviation of 8.
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117. Only 5% of the students taking the test scored higher than what value?
A continuous random variable X has the probability density function: f(x) = 2 , 0
118. What is the probability that X is at most 2?
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0
1
2
3
4
5
.0778
.2592
.3456
.2304
.0768
.0102
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have a higher defect rate than 1%.
Therefore, 13.59% of students scored between 81 and 89 on this exam.
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