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Chapter 6 - Continuous Probability Distributions
b.
0.0606
c.
0.3935
d.
0.9393
87. Refer to Exhibit 6-7. The probability that x is between 3 and 6 is
a.
0.4512
b.
0.1920
c.
0.2592
d.
0.6065
88. Excel's NORM.S.DIST function can be used to compute
a.
cumulative probabilities for a standard normal z value
b.
the standard normal z value given a cumulative probability
c.
cumulative probabilities for a normally distributed x value
d.
the normally distributed x value given a cumulative probability
89. Excel's NORM.S.INV function can be used to compute
a.
cumulative probabilities for a standard normal z value
b.
the standard normal z value given a cumulative probability
c.
cumulative probabilities for a normally distributed x value
d.
the normally distributed x value given a cumulative probability
90. Excel's NORM.DIST function can be used to compute
a.
cumulative probabilities for a standard normal z value
b.
the standard normal z value given a cumulative probability
c.
cumulative probabilities for a normally distributed x value
d.
the normally distributed x value given a cumulative probability
91. Excel's NORM.INV function can be used to compute
a.
cumulative probabilities for a standard normal z value
b.
the standard normal z value given a cumulative probability
c.
cumulative probabilities for a normally distributed x value
d.
the normally distributed x value given a cumulative probability
92. A continuous probability distribution that is useful in describing the time, or space, between occurrences of an event is
a(n)
a.
normal probability distribution
b.
uniform probability distribution
c.
exponential probability distribution
d.
Poisson probability distribution
93. The exponential probability distribution is used with
a.
a discrete random variable
b.
a continuous random variable
c.
any probability distribution with an exponential term
d.
an approximation of the binomial probability distribution
94. An exponential probability distribution
a.
is a continuous distribution
b.
is a discrete distribution
c.
can be either continuous or discrete
d.
must be normally distributed
95. Excel's EXPON.DIST function can be used to compute
a.
exponents
b.
exponential probabilities
c.
cumulative exponential probabilities
d.
Both exponential probabilities and cumulative exponential probabilities are correct.
96. Excel's EXPON.DIST function has how many inputs?
a.
2
b.
3
c.
4
d.
5
97. When using Excel's EXPON.DIST function, one should choose TRUE for the third input if
a.
a probability is desired
b.
a cumulative probability is desired
Chapter 6 - Continuous Probability Distributions
c.
the expected value is desired
d.
the correct answer is desired
98. Which of the following are continuous random variables?
I.
the weight of an elephant
II.
the time to answer a questionnaire
III.
the number of floors in a skyscraper
IV.
the square feet of countertop in a kitchen
a.
I and II only
b.
III and IV only
c.
I, II and IV
d.
I, II, II, and IV
99. In a Poisson probability problem, the rate of errors is one every two hours. To find the probability of three defects in
four hours,
a.
= 1, x = 4
b.
= 2, x = 3
c.
= 3, x = 2
d.
= 3, x = 6
100. An experiment consists of measuring the speed of automobiles on a highway by the use of radar equipment. The
random variable in this experiment is speed, measured in miles per hour. This random variable is a
a.
discrete random variable
b.
continuous random variable
c.
complex random variable
d.
binary random variable
101. The uniform distribution defined over the interval from 25 to 40 has the probability density function
a.
f(x) = 1/40 for all x
b.
f(x) = 5/8 for 25 x 40 and f(x) = 0 elsewhere
c.
f(x) = 1/25 for 0 x 25 and f(x) = 1/40 for 26 x 4
d.
f(x) = 1/15 for 25 x 40 and f(x) = 0 elsewhere
102. Joe's Record World has two stores and sales at each store follow a normal distribution. For store 1, μ = $2,000 and σ
= $200 per day; for store 2,
= $1,900 and σ = $300 per day. Which store is more likely to have a day’s sales in excess of
Chapter 6 - Continuous Probability Distributions
$2200?
a.
store 1
b.
store 2
c.
store 1 and store 2 are equally likely
d.
more information is needed
Subjective Short Answer
103. A random variable x is uniformly distributed between 45 and 150.
a.
Determine the probability of x = 48.
b.
What is the probability of x ≤ 60?
c.
What is the probability of x ≥ 50?
d.
Determine the expected vale of x and its standard deviation.
104. The price of a bond is uniformly distributed between $80 and $85.
a.
What is the probability that the bond price will be at least $83?
b.
What is the probability that the bond price will be between $81 and $90?
c.
Determine the expected price of the bond.
d.
Compute the standard deviation for the bond price.
105. The price of a stock is uniformly distributed between $30 and $40.
a.
What is the probability that the stock price will be more than $37?
b.
What is the probability that the stock price will be less than or equal to $32?
c.
What is the probability that the stock price will be between $34 and $38?
d.
Determine the expected price of the stock.
e.
Determine the standard deviation for the stock price.
106. The time it takes to hand carve a guitar neck is uniformly distributed between 110 and 190 minutes.
a.
What is the probability that a guitar neck can be carved between 95 and 165 minutes?
b.
What is the probability that the guitar neck can be carved between 120 and 200 minutes?
c.
Determine the expected completion time for carving the guitar neck.
d.
Compute the standard deviation.
Chapter 6 - Continuous Probability Distributions
107. The length of time it takes students to complete a statistics examination is uniformly distributed and varies between
40 and 60 minutes.
a.
Find the mathematical expression for the probability density function.
b.
Compute the probability that a student will take between 45 and 50 minutes to complete the
examination.
c.
Compute the probability that a student will take no more than 40 minutes to complete the
examination.
d.
What is the expected amount of time it takes a student to complete the examination?
e.
What is the variance for the amount of time it takes a student to complete the examination?
33.33 (minutes)2
POINTS:
1
108. The advertised weight on a can of soup is 10 ounces. The actual weight in the cans follows a uniform distribution and
varies between 9.3 and 10.3 ounces.
a.
Give the mathematical expression for the probability density function.
b.
What is the probability that a can of soup will have between 9.4 and 10.3 ounces?
c.
What is the mean weight of a can of soup?
d.
What is the standard deviation of the weight?
109. The length of time patients must wait to see a doctor in a local clinic is uniformly distributed between 15 minutes and
2 1/2 hours.
a.
Define the random variable in words.
b.
What is the probability of a patient waiting exactly 50 minutes?
c.
What is the probability that a patient would have to wait between 45 minutes and 2 hours?
d.
Compute the probability that a patient would have to wait over 2 hours.
e.
Determine the expected waiting time and its standard deviation.
110. For the standard normal distribution, determine the probability of obtaining a z value
a.
greater than zero.
b.
between -2.34 to -2.55
Chapter 6 - Continuous Probability Distributions
c.
less than 1.86.
d.
between -1.95 to 2.7.
e.
between 1.5 to 2.75.
111. Z is a standard normal random variable. Compute the following probabilities.
a.
P(-1.33 ≤ z ≤ 1.67)
b.
P(1.23 ≤ z ≤ 1.55)
c.
P(z ≥ 2.32)
d.
P(z ≥ -2.08)
e.
P(z ≥ -1.08)
112. Z is a standard normal random variable. Compute the following probabilities.
a.
P(-1.23 ≤ z ≤ 2.58)
b.
P(1.83 ≤ z ≤ 1.96)
c.
P(z ≥ 1.32)
d.
P(z ≤ 2.52)
e.
P(z ≥ -1.63)
f.
P(z ≤ -1.38)
g.
P(-2.37 ≤ z ≤ -1.54)
h.
P(z = 2.56)
113. Z is a standard normal variable. Find the value of z in the following.
a.
The area between 0 and z is 0.4678.
b.
The area to the right of z is 0.1112.
c.
The area to the left of z is 0.8554
d.
The area between -z and z is 0.754.
e.
The area to the left of -z is 0.0681.
f.
The area to the right of -z is 0.9803.
Chapter 6 - Continuous Probability Distributions
114. The miles-per-gallon obtained by the 1995 model Q cars is normally distributed with a mean of 22 miles-per-gallon
and a standard deviation of 5 miles-per-gallon.
a.
What is the probability that a car will get between 13.35 and 35.1 miles-per-gallon?
b.
What is the probability that a car will get more than 29.6 miles-per-gallon?
c.
What is the probability that a car will get less than 21 miles-per-gallon?
d.
What is the probability that a car will get exactly 22 miles-per-gallon?
115. The salaries at a corporation are normally distributed with an average salary of $19,000 and a standard deviation of
$4,000.
a.
What is the probability that an employee will have a salary between $12,520 and $13,480?
b.
What is the probability that an employee will have a salary more than $11,880?
c.
What is the probability that an employee will have a salary less than $28,440?
116. A major department store has determined that its customers charge an average of $500 per month, with a standard
deviation of $80. Assume the amounts of charges are normally distributed.
a.
What percentage of customers charges more than $380 per month?
b.
What percentage of customers charges less than $340 per month?
c.
What percentage of customers charges between $644 and $700 per month?
117. The contents of soft drink bottles are normally distributed with a mean of twelve ounces and a standard deviation of
one ounce.
a.
What is the probability that a randomly selected bottle will contain more than ten ounces of
soft drink?
b.
What is the probability that a randomly selected bottle will contain between 9.5 and 11 ounces?
c.
What percentage of the bottles will contain less than 10.5 ounces of soft drink?
118. The life expectancy of computer terminals is normally distributed with a mean of 4 years and a standard deviation of
10 months.
Chapter 6 - Continuous Probability Distributions
a.
What is the probability that a randomly selected terminal will last more than 5 years?
b.
What percentage of terminals will last between 5 and 6 years?
c.
What percentage of terminals will last less than 4 years?
d.
What percentage of terminals will last between 2.5 and 4.5 years?
e.
If the manufacturer guarantees the terminals for 3 years (and will replace them if they
malfunction), what percentage of terminals will be replaced?
119. Scores on a recent national statistics exam were normally distributed with a mean of 80 and a standard deviation of 6.
a.
What is the probability that a randomly selected exam will have a score of at least 71?
b.
What percentage of exams will have scores between 89 and 92?
c.
If the top 2.5% of test scores receive merit awards, what is the lowest score eligible for an
award?
d.
If there were 334 exams with scores of at least 89, how many students took the exam?
120. The average starting salary for this year's graduates at a large university (LU) is $30,000 with a standard deviation of
$8,000. Furthermore, it is known that the starting salaries are normally distributed.
a.
What is the probability that a randomly selected LU graduate will have a starting salary of at
least $30,400?
b.
Individuals with starting salaries of less than $15,600 receive a low income tax break. What
percentage of the graduates will receive the tax break?
c.
What are the minimum and the maximum starting salaries of the middle 95% of the LU
graduates?
d.
If 303 of the recent graduates have salaries of at least $43,120, how many students graduated
this year from this university?
121. The weights of items produced by a company are normally distributed with a mean of 4.5 ounces and a standard
deviation of 0.3 ounces.
a.
What is the probability that a randomly selected item from the production will weigh at least
4.14 ounces?
b.
What percentage of the items weighs between 4.8 and 5.04 ounces?
c.
Determine the minimum weight of the heaviest 5% of all items produced.
d.
If 27,875 of the items of the entire production weigh at least 5.01 ounces, how many items
have been produced?
122. The life expectancy of Timely brand watches is normally distributed with a mean of four years and a standard
deviation of eight months.
a.
What is the probability that a randomly selected watch will be in working condition for more
than five years?
b.
The company has a three-year warranty period on their watches. What percentage of their
watches will be in operating condition after the warranty period?
c.
What is the minimum and the maximum life expectancy of the middle 95% of the watches?
d.
Ninety-five percent of the watches will have a life expectancy of at least how many months?
123. The weights of the contents of cans of tomato paste produced by a company are normally distributed with a mean of
6 ounces and a standard deviation of 0.3 ounces.
a.
What percentage of all cans produced contains more than 6.51 ounces of tomato paste?
b.
What percentage of all cans produced contains less than 5.415 ounces?
c.
What percentage of cans contains between 5.46 and 6.495 ounces?
d.
Ninety-five percent of cans will contain at least how many ounces?
e.
What percentage of cans contains between 6.3 and 6.6 ounces?
124. A professor at a local university noted that the grades of her students were normally distributed with a mean of 78
and a standard deviation of 10.
a.
The professor has informed us that 16.6% of her students received grades of A. What is the
minimum score needed to receive a grade of A?
b.
If 12.1% of her students failed the course and received Fs, what was the maximum score
among those who received an F?
c.
If 33% of the students received grades of B or better (i.e., As and Bs), what is the minimum
score of those who received a B?
125. "DRUGS R US" is a large manufacturer of various kinds of liquid vitamins. The quality control department has
noted that the bottles of vitamins marked 6 ounces vary in content with a standard deviation of 0.3 ounces. Assume the
contents of the bottles are normally distributed.
a.
What percentage of all bottles produced contains more than 6.51 ounces of vitamins?
b.
What percentage of all bottles produced contains less than 5.415 ounces?
c.
What percentage of bottles produced contains between 5.46 and 6.495 ounces?
Chapter 6 - Continuous Probability Distributions
d.
Ninety-five percent of the bottles will contain at least how many ounces?
e.
What percentage of the bottles contains between 6.3 and 6.6 ounces?
126. The daily dinner bills in a local restaurant are normally distributed with a mean of $28 and a standard deviation of
$6.
a.
Define the random variable in words.
b.
What is the probability that a randomly selected bill will be at least $39.10?
c.
What percentage of the bills will be less than $16.90?
d.
What are the minimum and maximum of the middle 95% of the bills?
e.
If twelve of one day's bills had a value of at least $43.06, how many bills did the restaurant
collect on that day?
127. The monthly income of residents of Daisy City is normally distributed with a mean of $3000 and a standard
deviation of $500.
a.
Define the random variable in words.
b.
The mayor of Daisy City makes $2,250 a month. What percentage of Daisy City's residents has
incomes that are more than the mayor's?
c.
Individuals with incomes of less than $1,985 per month are exempt from city taxes. What
percentage of residents is exempt from city taxes?
d.
What are the minimum and the maximum incomes of the middle 95% of the residents?
e.
Two hundred residents have incomes of at least $4,440 per month. What is the population of
Daisy City?
128. The average starting salary of this year's MBA students is $35,000 with a standard deviation of $5,000. Furthermore,
it is known that the starting salaries are normally distributed. What are the minimum and the maximum starting salaries of
the middle 95% of MBA graduates?
129. A local bank has determined that the daily balances of the checking accounts of its customers are normally
distributed with an average of $280 and a standard deviation of $20.
a.
What percentage of its customers has daily balances of more than $275?
b.
What percentage of its customers has daily balances less than $243?
Chapter 6 - Continuous Probability Distributions
c.
What percentage of its customers' balances is between $241 and $301.60?
130. The weekly earnings of bus drivers are normally distributed with a mean of $395. If only 1.1% of the bus drivers
have a weekly income of more than $429.35, what is the value of the standard deviation of the weekly earnings of the bus
drivers?
131. The monthly earnings of computer programmers are normally distributed with a mean of $4,000. If only 1.7 percent
of programmers have monthly incomes of less than $2,834, what is the value of the standard deviation of the monthly
earnings of the computer programmers?
132. The Globe Fishery packs shrimp that weigh more than 1.91 ounces each in packages marked" large" and shrimp that
weigh less than 0.47 ounces each into packages marked "small"; the remainder are packed in "medium" size packages. If a
day's catch showed that 19.77% of the shrimp were large and 6.06% were small, determine the mean and the standard
deviation for the shrimp weights. Assume that the shrimps' weights are normally distributed.
133. In grading eggs into small, medium, and large, the Linda Farms packs the eggs that weigh more than 3.6 ounces in
packages marked "large" and the eggs that weigh less than 2.4 ounces into packages marked "small"; the remainder are
packed in packages marked "medium." If a day's packaging contained 10.2% large and 4.18% small eggs, determine the
mean and the standard deviation for the eggs' weights. Assume that the distribution of the weights is normal.
134. A manufacturing process produces items whose weights are normally distributed. It is known that 22.57% of all the
items produced weigh between 100 grams up to the mean and 49.18% weigh from the mean up to 190 grams. Determine
the mean and the standard deviation.
135. Z is the standard normal random variable. Use Excel to calculate the following:
a.
P(z ≤ 2.5)
b.
P(0 ≤ z ≤ 2.5)
c.
P(-2 ≤ z ≤ 2)
d.
P(z ≤ -0.38)
e.
P(z ≥ 1.62)
f.
z value with .05 in the lower tail
g.
z value with .05 in the upper tail
136. X is a normally distributed random variable with a mean of 50 and a standard deviation of 5. Use Excel to calculate
the following:
a.
P(x ≤ 45)
b.
P(45 ≤ x ≤ 55)
c.
P(x ≥ 55)
d.
x value with .20 in the lower tail
e.
x value with .01 in the upper tail
137. The time it takes a mechanic to change the oil in a car is exponentially distributed with a mean of 5 minutes.
a.
What is the probability density function for the time it takes to change the oil?
b.
What is the probability that it will take a mechanic less than 6 minutes to change the oil?
c.
What is the probability that it will take a mechanic between 3 and 5 minutes to change the oil?
138. The time it takes a worker on an assembly line to complete a task is exponentially distributed with a mean of 8
minutes.
a.
What is the probability density function for the time it takes to complete the task?
b.
What is the probability that it will take a worker less than 4 minutes to complete the task?
c.
What is the probability that it will take a worker between 6 and 10 minutes to complete the
task?
139. The time between arrivals of customers at the drive-up window of a bank follows an exponential probability
distribution with a mean of 10 minutes.
a.
What is the probability that the arrival time between customers will be 7 minutes or less?
b.
What is the probability that the arrival time between customers will be between 3 and 7
minutes?
140. The time required to assemble a part of a machine follows an exponential probability distribution with a mean of 14
minutes.
Chapter 6 - Continuous Probability Distributions
a.
What is the probability that the part can be assembled in 7 minutes or less?
b.
What is the probability that the part can be assembled between 3.5 and 7 minutes?
141. The time it takes to completely tune an engine of an automobile follows an exponential distribution with a mean of
40 minutes.
a.
Define the random variable in words.
b.
What is the probability of tuning an engine in 30 minutes or less?
c.
What is the probability of tuning an engine between 30 and 35 minutes?
142. X is a exponentially distributed random variable with a mean of 10. Use Excel to calculate the following:
a.
P(x ≤ 15)
b.
P(8 ≤ x ≤ 12)
c.
P(x ≥ 8)
143. The Harbour Island Ferry leaves on the hour and at 15-minute intervals. The time, x, it takes John to drive from his
house to the ferry has a uniform distribution with x between 10 and 20 minutes. One morning John leaves his house at
precisely 8:00a.m.
a. What is the probability John will wait less than 5 minutes for the ferry?
b. What is the probability John will wait less than 10 minutes for the ferry?
c. What is the probability John will wait less than 15 minutes for the ferry?
d. What is the probability John will not have to wait for the ferry?
e. Suppose John leaves at 8:05a.m. What is the probability John will wait (1) less than 5 minutes for the ferry; (2) less
than 10 minutes for the ferry?
f. Suppose John leaves at 8:10a.m. What is the probability John will wait (1) less than 5 minutes for the ferry; (2) less than
10 minutes for the ferry?
g. What appears to be the best time for John to leave home if he wishes to maximize the probability of waiting less than
10 minutes for the ferry?
144. Delicious Candy markets a two-pound box of assorted chocolates. Because of imperfections in the candy making
equipment, the actual weight of the chocolate has a uniform distribution ranging from 31.8 to 32.6 ounces.
a. Define a probability density function for the weight of the box of chocolate.
b. What is the probability that a box weighs (1) exactly 32 ounces; (2) more than 32.3 ounces; (3) less than 31.8 ounces?
Chapter 6 - Continuous Probability Distributions
c. The government requires that at least 60% of all products sold weigh at least as much as the stated weight. Is Delicious
violating government regulations?
145. The time at which the mailman delivers the mail to Ace Bike Shop follows a normal distribution with mean 2:00 PM
and standard deviation of 15 minutes.
a. What is the probability the mail will arrive after 2:30 PM?
b. What is the probability the mail will arrive before 1:36 PM?
c. What is the probability the mail will arrive between 1:48 PM and 2:09 PM?
146. The township of Middleton sets the speed limit on its roads by conducting a traffic study and determining the speed
(to the nearest 5 miles per hour) at which 80% of the drivers travel at or below. A study was done on Brown's Dock Road
that indicated driver's speeds follow a normal distribution with a mean of 36.25 miles per hour and a variance of 6.25.
a. What should the speed limit be?
b. What percent of the drivers travel below that speed?
147. A light bulb manufacturer claims its light bulbs will last 500 hours on the average. The lifetime of a light bulb is
assumed to follow an exponential distribution.
a. What is the probability that the light bulb will have to be replaced within 500 hours?
b. What is the probability that the light bulb will last more than 1000 hours?
c. What is the probability that the light bulb will last between 200 and 800 hours.
149. Scores on an endurance test for cardiac patients are normally distributed with
= 182 and
= 24.
a.
What is the probability a patient will score above 190?
b.
What percentage of patients score below 170?
c.
What score does a patient at the 75th percentile receive?
Chapter 6 - Continuous Probability Distributions
150. The time it takes to travel from home to the office is normally distributed with
= 25 minutes and
= 5 minutes.
a.
What is the probability the trip takes more than 20 minutes?
b.
What is the probability the trip takes less than 15 minutes.
c.
What is the probability the trip takes between 30 and 35 minutes?
d.
What is the probability the trip takes more than 40 minutes?
151. Mark Investment Service is currently recommending the purchase of shares of Dollar Department Stores selling at
$18 per share. Mark estimates that in one year the price of the shares will be at x, where x is a random variable which is
approximately normally distributed with mean of $20 and a standard deviation of $2.
a) What is the probability that in a year the shares will be selling for (1) exactly $20; (2) more than $20; (3) less than $20;
and (4) less than $18.
b) What is the expected profit per share within a year?
