a.
0.9082
b.
0.9962
c.
0.6568
65. If Z is a standard normal random variable, find the value z for which:
a.
the area between 0 and z is 0.3729
b.
the area to the right of z is 0.7123
c.
the area to the left of z is 0.1736
d.
the area between −z and z is 0.6630
a.
1.14
b.
c.
d.
0.96
66. If Z is a standard normal random variable, find the following probabilities:
a.
P(Z −1.77)
b.
P(Z −1.96)
c.
P(0.35 Z 0.85)
d.
P(−2.88 Z −2.15)
e.
P(Z 1.45)
a.
0.0384
b.
0.9750
c.
0.1655
d.
0.0138
0.9265
Calculus Scores
Scores of high school students on a national calculus exam were normally distributed with a mean of
86 and a standard deviation of 4. (Total possible points = 100.)
67. {Calculus Scores Narrative} What is the probability that a randomly selected student will have a score
of 80 or higher?
68. {Calculus Scores Narrative} What is the probability that a randomly selected student will have a score
between 80 and 90?
69. {Calculus Scores Narrative} What is the probability that a randomly selected student will have a score
of 94 or lower?
Checking Accounts
A bank has determined that the monthly balances of the checking accounts of its customers are
normally distributed with an average balance of $1,200 and a standard deviation of $250.
70. {Checking Accounts Narrative} What proportion of customers have monthly balances less than
$1,000?
ANS:
71. {Checking Accounts Narrative} What proportion of customers have monthly balances more than
$1,125?
72. {Checking Accounts Narrative} What proportion of customers have monthly balances between $950
and $1,075?
IT Graduates Salary
The recent average starting salary for new college graduates in IT systems is $47,500. Assume salaries
are normally distributed with a standard deviation of $4,500.
73. {IT Graduates Salary Narrative} What is the probability of a new graduate receiving a salary between
$45,000 and $50,000?
74. {IT Graduates Salary Narrative} What is the probability of a new graduate getting a starting salary in
excess of $55,000?
75. {IT Graduates Salary Narrative} What percent of starting salaries are no more than $42,250?
76. {IT Graduates Salary Narrative} What is the cutoff for the bottom 5% of the salaries?
77. {IT Graduates Salary Narrative} What is the cutoff for the top 3% of the salaries?
78. A worker earns $16 per hour at a plant and is told that only 5% of all workers make a higher wage. If
the wage is assumed to be normally distributed and the standard deviation of wage rates is $5 per hour,
find the average wage for the plant workers per hour.
79. The mean and the variance of an exponential distribution are equal to each other.
80. The exponential distribution is suitable to model the length of time that elapses before the first
telephone call is received by a switchboard.
81. The mean and standard deviation of an exponential random variable are equal to each other.
82. In the exponential distribution, X takes on an infinite number of possible values in the given range.
83. If the mean of an exponential distribution is 2, then the value of the parameter
is 2.0.
84. If the random variable X is exponentially distributed and the parameter of the distribution
= 4, then
P(X 1) = 0.25.
85. If the random variable X is exponentially distributed with parameter
= 5, then the variance of X,
2 =
V(X) = 0.04.
86. If the random variable X is exponentially distributed with parameter
= 0.05, then the variance of X,
2 = V(X) = 20.
87. If the random variable X is exponentially distributed with parameter
= 0.05, then the probability P(X
> 20) = 0.3679.
88. If the random variable X is exponentially distributed with parameter
= 0.05, then the probability P(X
< 5) = .2865.
89. If the random variable X is exponentially distributed with parameter
= 2, then the probability that X
is between 1 and 2 equals the probability that X is between 2 and 3.
90. Which of the following is true for an exponential distribution with parameter
?
a.
= 1/
b.
= 1/
c.
The Y-intercept of f(x) is
.
d.
All of these choices are true.
91. If the random variable X is exponentially distributed with parameter
= 3, then the probability P(X
2) equals:
a.
0.3333
b.
0.5000
c.
0.6667
d.
0.0025
92. If the random variable X is exponentially distributed with parameter
= 1.5, then the probability P(2
X 4), up to 4 decimal places, is
a.
0.6667
b.
0.0473
c.
0.5000
d.
0.2500
93. If the random variable X is exponentially distributed with parameter
= 4, then the probability P(X
0.25), up to 4 decimal places, is
a.
0.6321
b.
0.3679
c.
0.2500
d.
None of these choices.
94. Which of the following can have an exponential distribution?
a.
Time between phone calls coming in to a technical support desk.
b.
Time until the first customer arrives at the bank in the morning.
c.
Lifetime of a new battery.
d.
All of these choices are true.
95. The exponential density function f(x):
a.
is bell-shaped.
b.
is symmetrical.
c.
approaches infinity as x approaches zero.
d.
approaches zero as x approaches infinity.
96. If the random variable X is exponentially distributed, then the mean of X will be:
a.
greater than the median.
b.
less than the median.
c.
equal to the median.
d.
Cannot tell; the answer depends on what
is.
97. If the mean of an exponential distribution is 2, then the value of the parameter
is
a.
0
b.
2.0
c.
0.5
d.
1.0
98. If the parameter of an exponential distribution is 1, then which of the following is true?
a.
The density function is e−x for x 0.
b.
The mean is equal to 1.
c.
The standard deviation and variance are both equal to 1.
d.
All of these choices are true.
99. A random variable with density function e−x for x 0 has an exponential distribution with
=
____________________.
100. A random variable with density function e−x for x 0 has an exponential distribution whose mean is
____________________.
101. A random variable with density function 0.01e−x/100 for x 0 has an exponential distribution whose
mean is ____________________.
102. The shape of the density function for an exponential distribution is ____________________.
103. The mean of an exponential random variable is ____________________ the median.
104. If X has an exponential distribution, the possible values of X are from ____________________ to
infinity.
105. An exponential random variable is an example of a(n) ____________________ random variable.
106. If X has an exponential distribution with parameter
, then f(0) = ____________________.
107. If X has an exponential distribution with parameter
, then the mean of X is ______________.
108. If X has an exponential distribution, then f(x) approaches ____________________ as x approaches
infinity.
109. The y-intercept of the density function for an exponential distribution with parameter 10 is
____________________.
110. If X has an exponential distribution, its ____________________ is equal to its
____________________.
111. Let X be an exponential random variable with
= 1.50. Find the following:
a.
P(X 2)
b.
P(X 4)
c.
P(1 X 3)
d.
P(X = 1)
a.
b.
0.9975
0.2120
d.
0
112. Let X be an exponential random variable with
= 1.50. Find the following:
a.
f(x)
b.
The y-intercept of f(x)
a.
b.
(0, 1.50)
113. Suppose X has an exponential distribution with mean 2. Find f(x).
Truck Salesman
A used truck salesman in a small town states that, on the average, it takes him 5 days to sell a truck.
Assume that the probability distribution of the length of time between sales is exponentially
distributed.
114. {Truck Salesman Narrative} What is the probability that he will have to wait at least 8 days before
making another sale?
115. {Truck Salesman Narrative} What is the probability that he will have to wait between 6 and 10 days
before making another sale?
ANS:
Repair Time
The time it takes a technician to fix a telephone problem is exponentially distributed with a mean of 15
minutes.
116. {Repair Time Narrative} What is the probability density function for the time it takes a technician to
fix a telephone problem?
117. {Repair Time Narrative} What is the probability that it will take a technician less than 10 minutes to
fix a telephone problem?
118. {Repair Time Narrative} What is the variance of the time it takes a technician to fix a telephone
problem?
119. {Repair Time Narrative} What is the probability that it will take a technician between 10 to 15 minutes
to fix a telephone problem?
Light Bulb Lifetime
The lifetime of a light bulb (in hours) is exponentially distributed with
= 0.008.
120. {Light Bulb Lifetime Narrative} What is the mean and standard deviation of the light bulb’s lifetime?
121. {Light Bulb Lifetime Narrative} Find the probability that a light bulb will last between 120 and 140
hours.
122. {Light Bulb Lifetime Narrative} Find the probability that a light bulb will last for:
a.
more than 125 hours.
b.
at most 125 hours.
c.
no more than 125 hours.
d.
exactly 125 hours.
e.
less than 125 hours.
f.
at least 125 hours.
g.
no less than 125 hours.
ANS:
a.
d.
Counter Sales
Suppose that customers arrive at a counter at an average rate of three customers per minute and that
their arrivals follow the Poisson model.
123. {Counter Sales Narrative} Write the probability density function of the distribution of the time that
will elapse before the next customer arrives.
124. {Counter Sales Narrative} Use the appropriate exponential distribution to find the probability that the
next customer will arrive within 1.5 minutes.
125. {Counter Sales Narrative} Use the appropriate exponential distribution to find the probability that the
next customer will not arrive within the next 2 minutes.
Phone Orders
The L. L. Bean catalog department that receives the majority of its orders by telephone conducted a
study to determine how long customers were willing to wait on hold before ordering a product. The
length of time was found to be a random variable best approximated by an exponential distribution
with a mean equal to 3 minutes.
126. {Phone Orders Narrative} What is the value of
, the parameter of the exponential distribution in this
situation?