Business Statistics A Decision-Making Approach Groebner Shannon

Business Statistics
Groebner
Shannon
Fry
9 781292 023359
9e
ISBN 978-1-29202-335-9
Business Statistics
A Decision-Making Approach
Groebner Shannon Fry
Ninth Edition
Pearson Education Limited
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Harlow
Essex CM20 2JE
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ISBN 10: 1-292-02335-X
ISBN 13: 978-1-292-02335-9
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library
Printed in the United States of America
I n t rod u c t i o n t o Co n t i n u o u s Pro b a b i l i t y Di s t ri b u t i o n s
6-16.
6-17.
6-18.
6-19.
deviation of the filling process. What would the
standard deviation need to be so that no more than
2% of all filled bags weigh more than 52 kilograms?
LaCrosse Technology is one of many manufacturers
of atomic clocks. It makes an atomic digital watch
that is radio controlled and that maintains its accuracy
by reading a radio signal from a WWVB radio signal
from Colorado. It neither loses nor gains a second
in 20 million years. It is powered by a 3-volt lithium
battery expected to last three years. Suppose the life of
the battery has a standard deviation of 0.3 years and is
normally distributed.
a. Determine the probability that the watch’s battery
will last longer than 3.5 years.
b. Calculate the probability that the watch’s battery
will last more than 2.75 years.
c. Compute the length-of-life value for which 10% of
the watch’s batteries last longer.
The average number of acres burned by forest and
range fires in a large Wyoming county is 4,300 acres
per year, with a standard deviation of 750 acres. The
distribution of the number of acres burned is normal.
a. Compute the probability that more than 5,000 acres
will be burned in any year.
b. Determine the probability that fewer then 4,000
acres will be burned in any year.
c. What is the probability that between 2,500 and
4,200 acres will be burned?
d. In those years when more than 5,500 acres are
burned, help is needed from eastern-region fire
teams. Determine the probability help will be
needed in any year.
An Internet retailer stocks a popular electronic toy at
a central warehouse that supplies the eastern United
States. Every week, the retailer makes a decision
about how many units of the toy to stock. Suppose that
weekly demand for the toy is approximately normally
distributed with a mean of 2,500 units and a standard
deviation of 300 units.
a. If the retailer wants to limit the probability of being
out of stock of the electronic toy to no more than
2.5% in a week, how many units should the central
warehouse stock?
b. If the retailer has 2,750 units on hand at the start
of the week, what is the probability that weekly
demand will be greater than inventory?
c. If the standard deviation of weekly demand for
the toy increases from 300 units to 500 units, how
many more toys would have to be stocked to ensure
that the probability of weekly demand exceeding
inventory is no more than 2.5%?
F&G Industries manufactures a wash-down motor
that is used in the food processing industry. The motor
is marketed with a warranty that guarantees it will
be replaced free of charge if it fails within the first
13,000 hours of operation. On average, F&G washdown motors operate for 15,000 hours with a standard
deviation of 1,250 hours before failing. The number
of operating hours before failure is approximately
normally distributed.
a. What is the probability that a wash-down motor will
have to be replaced free of charge?
b. What percentage of F&G wash-down motors can be
expected to operate for more than 17,500 hours?
c. If F&G wants to design a wash-down motor so that
no more than 1% are replaced free of charge, what
would the average hours of operation before failure
have to be if the standard deviation remains at 1,250
hours?
6-20. A private equity firm is evaluating two alternative
investments. Although the returns are random, each
investment’s return can be described using a normal
distribution. The first investment has a mean return of
$2,000,000 with a standard deviation of $125,000. The
second investment has a mean return of $2,275,000
with a standard deviation of $500,000.
a. How likely is it that the first investment will return
$1,900,000 or less?
b. How likely is it that the second investment will
return $1,900,000 or less?
c. If the firm would like to limit the probability of a
return being less than $1,750,000, which investment
should it make?
6-21. L.J. Raney & Associates is a financial planning group
in Kansas City, Missouri. The company specializes in
financial planning for schoolteachers in the Kansas
City area. As such, it administers a 403(b) tax shelter
annuity program in which public schoolteachers can
participate. The teachers can contribute up to $20,000
per year on a pretax basis to the 403(b) account. Very
few teachers have incomes sufficient to allow them
to make the maximum contribution. The lead analyst
at L.J. Raney & Associates has recently analyzed
the company’s 403(b) clients and determined that
the annual contribution is approximately normally
distributed with a mean equal to $6,400. Further, he
has determined that the probability a customer will
contribute more than $13,000 is 0.025. Based on
this information, what is the standard deviation of
contributions to the 403(b) program?
6-22. No Leak Plumbing and Repair provides customers with
firm quotes for a plumbing repair job before actually
starting the job. To be able to do this, No Leak has been
very careful to maintain time records over the years.
For example, it has determined that the time it takes to
remove a broken sink disposal and to install a new unit
is normally distributed with a mean equal to 47 minutes
and a standard deviation equal to 12 minutes. The
company bills at $75.00 for the first 30 minutes and
$2.00 per minute for anything beyond 30 minutes.
Suppose the going rate for this procedure by other
plumbing shops in the area is $85.00, not including
the cost of the new equipment. If No Leak bids the
disposal job at $85, on what percentage of such jobs
will the actual time required exceed the time for which
it will be getting paid?
I n t rod u c t i o n t o Co n t i n u o u s Pro b a b i l i t y Di s t ri b u t i o n s
6-23. According to Business Week, Maternity Chic, a
purveyor of designer maternity wear, sells dresses and
pants priced around $150 each for an average total sale
of $1,200. The total sale has a normal distribution with
a standard deviation of $350.
a. Calculate the probability that a randomly selected
customer will have a total sale of more than $1,500.
b. Compute the probability that the total sale will be
within {2 standard deviations of the mean total sales.
c. Determine the median total sale.
6-24. The Aberdeen Coca-Cola Bottling plant located in
Aberdeen, North Carolina, is the bottler and distributor
for Coca-Cola products in the Aberdeen area. The
company’s product line includes 12-ounce cans of
Coke products. The cans are filled by an automated
filling process that can be adjusted to any mean fill
volume and that will fill cans according to a normal
distribution. However, not all cans will contain the
same volume due to variation in the filling process.
Historical records show that regardless of what the
mean is set at, the standard deviation in fill will be
0.035 ounces. Operations managers at the plant know
that if they put too much Coke in a can, the company
loses money. If too little is put in the can, customers
are short changed, and the North Carolina Department
of Weights and Measures may fine the company.
a. Suppose the industry standards for fill volume call
for each 12-ounce can to contain between 11.98
and 12.02 ounces. Assuming that the Aberdeen
manager sets the mean fill at 12 ounces, what is the
probability that a can will contain a volume of Coke
product that falls in the desired range?
b. Assume that the Aberdeen manager is focused on an
upcoming audit by the North Carolina Department
of Weights and Measures. She knows the process
is to select one Coke can at random and that if it
contains less than 11.97 ounces, the company will
be reprimanded and potentially fined. Assuming
that the manager wants at most a 5% chance of this
happening, at what level should she set the mean fill
level? Comment on the ramifications of this step,
assuming that the company fills tens of thousands of
cans each week.
6-25. MP-3 players, and most notably the Apple iPod, have
become an industry standard for people who want to
have access to their favorite music and videos in a
portable format. The iPod can store massive numbers
of songs and videos with its 120-GB hard drive.
Although owners of the iPod have the potential to store
lots of data, a recent study showed that the actual disk
storage being used is normally distributed with a mean
equal to 1.95 GB and a standard deviation of 0.48 GB.
Suppose a competitor to Apple is thinking of entering
the market with a low-cost iPod clone that has only 1.0
GB of storage. The marketing slogan will be “Why Pay
for Storage Capacity that You Don’t Need?”
Based on the data from the study of iPod owners,
what percentage of owners, based on their current
usage, would have enough capacity with the new 1-GB
player?
6-26. According to the Federal Reserve Board, the average
credit card debt per U.S. household was $8,565 in
2008. Assume that the distribution of credit card debt
per household has a normal distribution with a standard
deviation of $3,000.
a. Determine the percentage of households that have a
credit card debt of more than $13,000.
b. One household has a credit card debt that is at the
95th percentile. Determine its credit card debt.
c. If four households were selected at random,
determine the probability that at least half of them
would have credit card debt of more than $13,000.
6-27. Georgia-Pacific is a major forest products company
in the United States. In addition to timberlands, the
company owns and operates numerous manufacturing
plants that make lumber and paper products. At one
of their plywood plants, the operations manager
has been struggling to make sure that the plywood
thickness meets quality standards. Specifically, all
sheets of their 3/4-inch plywood must fall within the
range 0.747 to 0.753 inches in thickness. Studies have
shown that the current process produces plywood that
has thicknesses that are normally distributed with a
mean of 0.751 inches and a standard deviation equal
to 0.004 inches.
a. Use Excel to determine the proportion of plywood
sheets that will meet quality specifications (0.747 to
0.753), given how the current process is performing.
b. Referring to part a, suppose the manager is
unhappy with the proportion of product meeting
specifications. Assuming that he can get the mean
adjusted to 0.75 inches, what must the standard
deviation be if he is going to have 98% of his
product meet specifications?
6-28. A senior loan officer for Whitney National Bank has
recently studied the bank’s real estate loan portfolio
and found that the distribution of loan balances is
approximately normally distributed with a mean of
$155,600 and a standard deviation equal to $33,050.
As part of an internal audit, bank auditors recently
randomly selected 100 real estate loans from the
portfolio of all loans and found that 80 of these
loans had balances below $170,000. The senior loan
officer is concerned that the sample selected by the
auditors is not representative of the overall portfolio.
In particular, he is interested in knowing the expected
proportion of loans in the portfolio that would have
balances below $170,000. You are asked to conduct
an appropriate analysis and write a short report to the
senior loan officers with your conclusion about the
sample.
Computer Database Exercises
6-29. The PricewaterhouseCoopers Human Capital Index
Report indicated that the average cost for an American
company to fill a job vacancy during the study period
I n t rod u c t i o n t o Co n t i n u o u s Pro b a b i l i t y Di s t ri b u t i o n s
was $3,270. Sample data similar to those used in the
study are in a file titled Hired.
a. Produce a relative frequency histogram for these
data. Does it seem plausible the data were sampled
from a normally distributed population?
b. Calculate the mean and standard deviation of the
cost of filling a job vacancy.
c. Determine the probability that the cost of filling a
job vacancy would be between $2,000 and $3,000.
d. Given that the cost of filling a job vacancy was
between $2,000 and $3,000, determine the
probability that the cost would be more than $2,500.
6-30. A recent article in USA Today discussed prices for
the 200 brand-name drugs most commonly used by
Americans over age 50. Atrovent, a treatment for lung
conditions such as emphysema, was one of the drugs.
The file titled Drug$ contains daily cost data similar to
those obtained in the research.
a. Produce a relative frequency histogram for these
data. Does it seem plausible the data were sampled
from a population that was normally distributed?
b. Compute the mean and standard deviation for the
sample data in the file Drug$.
c. Assuming the sample came from a normally
distributed population and the sample standard
deviation is a good approximation for the population
standard deviation, determine the probability that a
randomly chosen transaction would yield a price of
$2.12 or smaller even though the population mean
was $2.51.
6-31. USA Today’s annual survey of public flagship
universities (Arienne Thompson and Breanne
Gilpatrick, “Double-Digit Hikes Are Down,” October
5, 2005) indicates that the median increase in in-state
tuition was 7% for the 2005–2006 academic year. A
file titled Tuition contains the percentage change for
the 67 flagship universities.
a. Produce a relative frequency histogram for these
data. Does it seem plausible that the data are from a
population that has a normal distribution?
b. Suppose the decimal point of the three largest
numbers had inadvertently been moved one
place to the right in the data. Move the decimal
point one place to the left and reconstruct the
relative frequency histogram. Now does it seem
plausible that the data have an approximate normal
distribution?
c. Use the normal distribution of part b to approximate
the proportion of universities that raised their
in-state tuition more than 10%. Use the appropriate
parameters obtained from this population.
d. Use the normal distribution of part b to approximate
the fifth percentile for the percent of tuition
increase.
END EXERCISES 6-1
2
Other Continuous Probability
Distributions
This section introduces two additional continuous probability distributions that are used in
business decision making: the uniform distribution and the exponential distribution.
Chapter Outcome 4.
Uniform Probability Distribution
The uniform distribution is sometimes referred to as the distribution of little information,
because the probability over any interval of the continuous random variable is the same as for
any other interval of the same width.
Equation 3 defines the continuous uniform density function.
Continuous Uniform Density Function
1
f (x)
b
a
0
if a
x
b
(3)
otherwise
where:
f1 x2 = Value of the density function at any x-value
a = The smallest value assumed by the uniform random variable of interest
b = The largest value assumed by the uniform random variable of interest
I n t rod u c t i o n t o Co n t i n u o u s Pro b a b i l i t y Di s t ri b u t i o n s
FIGURE 12
| f(x)
f (x)
f (x) = 1 = 1 = 0.33 for 2 < x < 5
5–2 3
Uniform Distributions
0.50
0.50
0.25
0.25
2
a
f (x) = 1 = 1 = 0.2 for 3 < x < 8
8–3 5
x
5
b
3
a
(a)
8
b
x
(b)
Figure 12 illustrates two examples of uniform probability distributions with different a to
b intervals. Note the height of the probability density function is the same for all values of
x between a and b for a given distribution. The graph of the uniform distribution is a rectangle.
EXAMPLE 3
USING THE UNIFORM DISTRIBUTION
Georgia Pacific The Georgia Pacific Company owns and operates several tree farms in different parts of the United States and South America. The lead botanist for the company has
stated that pine trees on one parcel of land will increase in diameter between one and four
inches per year according to a uniform distribution. Suppose the company is interested in the
probability that a given tree will have an increased diameter of more than 2 inches. The probability can be determined using the following steps:
Step 1 Define the density function.
The height of the probability rectangle, f(x), for the tree growth interval of one
to four inches is determined using Equation 3, as follows:
f (x)
f (x)
1
b
a
1
4 1
1
3
0.33
Step 2 Define the event of interest.
The botanist is specifically interested in a tree that has increased by more than
two inches in diameter. This event of interest is x 7 2.0.
Step 3 Calculate the required probability.
We determine the probability as follows:
P1x 7 2.02 =
=
=
=
=
1 - P1x … 2.02
1 - f1x212.0 - 1.02
1 - 0.3311.02
1 - 0.33
0.67
Thus, there is a 0.67 probability that a tree will increase by more than two inches in diameter.
>>
END EXAMPLE
TRY PROBLEM 32
Like the normal distribution, the uniform distribution can be further described by
specifying the mean and the standard deviation. These values are computed using
Equations 4 and 5.
I n t rod u c t i o n t o Co n t i n u o u s Pro b a b i l i t y Di s t ri b u t i o n s
Mean and Standard Deviation of a Uniform Distribution
Mean (Expected Value):
a
E(x)
b
(4)
2
Standard Deviation:
(b
a) 2
12
(5)
where:
a = The smallest value assumed by the uniform random variable of interest
b = The largest value assumed by the uniform random variable of interest
araraadt/Fotolia
EXAMPLE 4
THE MEAN AND STANDARD DEVIATION
OF A UNIFORM DISTRIBUTION
Surgery Recovery The chief administrator of a San
Francisco–area surgical center has analyzed data from a large
number of shoulder surgeries conducted by her center and others
in a medical association in California. The analysis shows that
the recovery time for shoulder surgery ranges between 15 and 45
weeks. Without any further information, the administrator will
apply a uniform distribution to surgery times. Based on this, she
can determine the mean and standard deviation for the recovery duration using the following
steps:
Step 1 Define the density function.
Equation 3 can be used to define the distribution:
1
1
1
f (x)
0.0333
b a 45 15 30
Step 2 Compute the mean of the probability distribution using Equation 4.
a
b
2
15
45
2
30
Thus, the mean recovery time is 30 weeks.
Step 3 Compute the standard deviation using Equation 5.
(b
a) 2
12
(45 15)2
12
75
8.66
The standard deviation is 8.66 weeks.
>>
END EXAMPLE
TRY PROBLEM 34
Chapter Outcome 5.
The Exponential Probability Distribution
Another continuous probability distribution frequently used in business situations is the
exponential distribution. The exponential distribution is used to measure the time that
elapses between two occurrences of an event, such as the time between “hits” on an Internet
home page. The exponential distribution might also be used to describe the time between
arrivals of customers at a bank drive-in teller window or the time between failures of an
electronic component. Equation 6 shows the probability density function for the exponential distribution.