Extract - Pearson

– New! Introductory chapter “Let’s Get Started: Big Things to Learn
First” defines business analytics and big data and explains how they
are changing the face of statistics.
– New! Continuing end-of-chapter cases help students to apply theory
into practice.
SEVENTH
EDITION
The seventh edition of Statistics for Managers Using Microsoft® Excel
focuses on making statistics even more relevant to the business world
today. Students are encouraged to see the relevance of statistics in their
own careers by providing examples drawn from the areas in which they
may be specializing.
Using Microsoft ® Excel
– Updated! Microsoft Windows and OS X Excel-Based Solutions
guides are comprehensive and easy to use.
Statistics for Managers
This Global Edition has been edited to include enhancements making it
more relevant to students outside the United States. The editorial team
at Pearson has worked closely with educators around the globe
to include:
Levine
Stephan
Szabat
This is a special edition of an established title widely
used by colleges and universities throughout the world.
Pearson published this exclusive edition for the benefit
of students outside the United States and Canada. If you
purchased this book within the United States or Canada
you should be aware that it has been imported without
the approval of the Publisher or Author.
Pearson International Edition
GLOBAL
EDITION
GLOBAL
EDITION
GLOBAL
EDITION
Statistics for Managers
Using Microsoft ® Excel
SEVENTH EDITION
David M. Levine • David F. Stephan • Kathryn A. Szabat
5.5 Hypergeometric Distribution
237
equation (5.16) defines the mean of the hypergeometric distribution, and equation (5.17)
defines the standard deviation.
mEan of THE HyPERgEomETRiC DiSTRibuTion
nA
m = E1X2 =
N
(5.16)
STanDaRD DEViaTion of THE HyPERgEomETRiC DiSTRibuTion
s =
A
nA1 N - A2
N
2
N - n
AN - 1
(5.17)
N - n
is a finite population correction factor that
AN - 1
results from sampling without replacement from a finite population.
To illustrate the hypergeometric distribution, suppose that you are forming a team of 8
managers from different departments within your company. Your company has a total of 30
managers, and 10 of these managers are from the finance department. If you are to randomly
select members of the team, what is the probability that the team will contain 2 managers
from the finance department? Here, the population of N = 30 managers within the company is
finite. In addition, A = 10 are from the finance department. A team of n = 8 members is to
be selected.
Using equation (5.15),
In equation (5.17), the expression
10 20
ba b
2
6
P1X = 2 n = 8, N = 30, A = 102 =
30
a b
8
1202!
10!
a
ba
b
2!182! 162!1142!
=
30!
a
b
8!1222!
= 0.298
a
Thus, the probability that the team will contain two members from the finance department is
0.298, or 29.8%.
Computing hypergeometric probabilities can be tedious, especially as N gets large. Figure
5.5 shows how the worksheet HYpGeOM.DIST function can compute hypergeometric probabilities for the team formation example.
FigURE 5.5
worksheet
for computing
hypergeometric
probabilities for the
team formation problem
Figure 5.5 displays the
COMPUTE worksheet of
Hypergeometric workbook
that the Section EG5.5
instructions use.
238
CHApTeR 5
Discrete probability Distributions
example 5.7 shows an application of the hypergeometric distribution in portfolio selection.
ExamPlE 5.7
computing
Hypergeometric
Probabilities
You are a financial analyst facing the task of selecting mutual funds to purchase for a client’s
portfolio. You have narrowed the funds to be selected to 10 different funds. In order to diversify
your client’s portfolio, you will recommend the purchase of 4 different funds. Six of the funds
are growth funds. What is the probability that of the 4 funds selected, 3 are growth funds?
solUtion Using equation (5.15) with X = 3, n = 4, N = 10, and A = 6,
6 4
a ba b
3 1
P1X = 3 n = 4, N = 10, A = 62 =
10
a b
4
142!
6!
a
ba
b
3!132! 112!132!
=
10!
a
b
4!162!
= 0.3810
The probability that of the 4 funds selected, 3 are growth funds, is 0.3810, or 38.10%.
Problems for section 5.5
lEaRning tHE Basics
5.42 Determine the following:
a. If n = 4, N = 10, and A = 5, find P1X = 32 .
b. If n = 4, N = 6, and A = 3, find P1X = 12.
c. If n = 5, N = 12, and A = 3, find P1X = 02.
d. If n = 3, N = 10, and A = 3, find P1X = 32.
5.43 Referring to problem 5.42, compute the mean and
standard deviation for the hypergeometric distributions described in (a) through (d).
aPPlying tHE concEPts
SELF 5.44 An auditor for the Internal Revenue Service
Test is selecting a sample of 6 tax returns for an audit.
If 2 or more of these returns are “improper,” the entire population of 100 tax returns will be audited. What is the probability that the entire population will be audited if the true
number of improper returns in the population is
c. 5?
a. 25?
d. 10?
b. 30?
e. Discuss the differences in your results, depending on the
true number of improper returns in the population.
5.45 KSDLDS-pros, an IT project management consulting
firm, is forming an IT project management team of 5 professionals. In the firm of 50 professionals, 8 are considered to
be data analytics specialists. If the professionals are selected
at random, what is the probability that the team will include
a. no data analytics specialist?
b. at least one data analytics specialist?
c. no more than two data analytics specialists?
d. What is your answer to (a) if the team consists of
7 members?
5.46 From an inventory of 30 cars being shipped to a local
automobile dealer, 4 are SUVs. What is the probability that
if 4 cars arrive at a particular dealership,
a. all 4 are SUVs?
b. none are SUVs?
c. at least 1 is an SUV?
d. What are your answers to (a) through (c) if 6 cars being
shipped are SUVs?
5.47 As a quality control manager, you are responsible for
checking the quality level of AC adapters for tablet pCs that
your company manufactures. You must reject a shipment if
you find 4 defective units. Suppose a shipment of 40 AC
References
adapters has 8 defective units and 32 nondefective units. If
you sample 12 AC adapters, what’s the probability that
a. there will be no defective units in the shipment?
b. there will be at least 1 defective unit in the shipment?
c. there will be 4 defective units in the shipment?
d. the shipment will be accepted?
5.48 In example 5.7 on page 238, a financial analyst was
facing the task of selecting mutual funds to purchase for a
U s I n g s tat I s t I c s
Monkey Business Images / Shutterstock
239
client’s portfolio. Suppose that the number of funds had
been narrowed to 12 funds instead of the 10 funds (still with
6 growth funds) in example 5.7. What is the probability
that of the 4 funds selected,
a. exactly 1 is a growth fund?
b. at least 1 is a growth fund?
c. 3 are growth fund?
d. Compare the result of (c) to the result of example 5.7.
Events of Interest at Ricknel Home
Centers, Revisited
I
n the Ricknel Home Improvement scenario at the beginning of this
chapter, you were an accountant for the Ricknel Home Improvement Company. The company’s accounting information system automatically reviews order forms from online customers for possible
mistakes. Any questionable invoices are tagged and included in a daily exceptions report. Knowing that the probability that an order will be tagged is 0.10, you were able
to use the binomial distribution to determine the chance of finding a certain number of tagged
forms in a sample of size four. There was a 65.6% chance that none of the forms would be
tagged, a 29.2% chance that one would be tagged, and a 5.2% chance that two or more would
be tagged. You were also able to determine that, on average, you would expect 0.4 forms to be
tagged, and the standard deviation of the number of tagged order forms would be 0.6. Now
that you have learned the mechanics of using the binomial distribution for a known probability
of 0.10 and a sample size of four, you will be able to apply the same approach to any given probability and sample size. Thus, you will be able to make inferences about the online ordering process and, more importantly, evaluate any changes or proposed changes to the process.
sUmmaRy
In this chapter, you have studied the probability distribution
for a discrete variable, the covariance and its application in finance, and three important discrete probability distributions:
the binomial, poisson, and hypergeometric distributions. In
the next chapter, you will study several important continuous
distributions, including the normal distribution.
To help decide which discrete probability distribution
to use for a particular situation, you need to ask the following questions:
• Is there a fixed number of observations, n, each of
which is classified as an event of interest or not an
event of interest? Is there an area of opportunity? If
there is a fixed number of observations, n, each of
which is classified as an event of interest or not an
event of interest, you use the binomial or hypergeometric distribution. If there is an area of opportunity,
you use the poisson distribution.
• In deciding whether to use the binomial or hypergeometric distribution, is the probability of an event of
interest constant over all trials? If yes, you can use the
binomial distribution. If no, you can use the hypergeometric distribution.
REFEREncEs
1. Bernstein, p. L. Against the Gods: The Remarkable
Story of Risk. New York: Wiley, 1996.
2. emery, D. R., J. D. Finnerty, and J. D. Stowe. Corporate
Financial Management, 3rd ed. Upper Saddle River, NJ:
prentice Hall, 2007.
3. Levine, D. M., p. Ramsey, and R. Smidt. Applied Statistics for Engineers and Scientists Using Microsoft
Excel and Minitab. Upper Saddle River, NJ: prentice
Hall, 2001.
4. Microsoft Excel 2010. Redmond, WA: Microsoft Corp.,
2010.
5. Taleb, N. The Black Swan, 2nd ed. New York: Random
House, 2010.
240
CHApTeR 5
Discrete probability Distributions
K E y E q U at I o n s
Combinations
Expected Value, m, of a Discrete Variable
m = E1X2 = a xiP1X = xi2
N
(5.1)
i=1
2
(5.2)
i=1
Covariance
(5.10)
n!
px 11 - p2 n - x
x! 1n - x2!
(5.11)
Mean of the Binomial Distribution
2
a3xi - E1X24 P1X = xi2
N
Ai=1
(5.12)
m = E1X2 = np
Standard Deviation of a Discrete Variable
s = 2s2 =
n!
x! 1 n - x 2!
P1X = x n, p2 =
s = a3xi - E1X24 P1X = xi2
2
=
Binomial Distribution
Variance of a Discrete Variable
N
nCx
(5.3)
Standard Deviation of the Binomial Distribution
s = 2s2 = 2Var 1X2 = 2np 11 - p2
(5.13)
Poisson Distribution
sXY = a 3xi - E1X243yi - E1Y24 P1xiyi2
(5.4)
E1X + Y2 = E1X2 + E1Y2
(5.5)
N
i=1
P1X = x l2 =
e-llx
x!
(5.14)
Hypergeometric Distribution
Expected Value of the Sum of Two Variables
Variance of the Sum of Two Variables
Var1X + Y2 = s2X + Y = s2X + s2Y + 2sXY
(5.6)
Standard Deviation of the Sum of Two Variables
sX + Y = 2s2X + Y
(5.7)
E1P2 = wE1X2 + 11 - w2E1Y2
(5.8)
sp = 2w2s2X + 11 - w2 2s2Y + 2w11 - w2 sXY
(5.9)
Portfolio Expected Return
A N - A
a ba
b
x
n - x
P1X = x n, N, A2 =
N
a b
n
(5.15)
Mean of the Hypergeometric Distribution
m = E1X2 =
nA
N
(5.16)
Standard Deviation of the Hypergeometric Distribution
Portfolio Risk
s =
A
nA1 N - A2
N
2
N - n
AN - 1
(5.17)
KEy tERms
area of opportunity 232
binomial distribution 225
covariance of a probability
distribution (sXY ) 219
expected value 216
expected value of the sum of two
variables 221
finite population correction
factor 237
hypergeometric distribution 236
mathematical model 225
poisson distribution 232
portfolios 221
portfolio expected return 221
portfolio risk 221
probability distribution for a discrete
variable 216
probability distribution function 225
rule of combinations 226
standard deviation of a discrete
variable 217
standard deviation of the sum of two
variables 221
variance of a discrete variable 217
variance of the sum of two
variables 221
Chapter Review problems
241
c H E c K I n g y o U R U n D E R s ta n D I n g
5.49 What is the meaning of the expected value of a probability distribution?
5.51 What are the four properties that must be present in
order to use the poisson distribution?
5.50 What are the four properties that must be present in
order to use the binomial distribution?
5.52 When do you use the hypergeometric distribution instead of the binomial distribution?
cHaPtER REvIEw PRoBlEms
5.53 Darwin Head, a 35-year-old sawmill worker, won
$1 million and a Chevrolet Malibu Hybrid by scoring
15 goals within 24 seconds at the Vancouver Canucks
National Hockey League game (B. Ziemer, “Darwin evolves
into an Instant Millionaire,” Vancouver Sun, February 28,
2008, p. 1). Head said he would use the money to pay off
his mortgage and provide for his children, and he had no
plans to quit his job. The contest was part of the Chevrolet
Malibu Million Dollar Shootout, sponsored by General
Motors Canadian Division. Did GM-Canada risk the
$1 million? No! GM-Canada purchased event insurance from
a company specializing in promotions at sporting events such
as a half-court basketball shot or a hole-in-one giveaway at
the local charity golf outing. The event insurance company
estimates the probability of a contestant winning the contest,
and for a modest charge, insures the event. The promoters pay
the insurance premium but take on no added risk as the insurance company will make the large payout in the unlikely
event that a contestant wins. To see how it works, suppose
that the insurance company estimates that the probability a
contestant would win a Million Dollar Shootout is 0.001, and
that the insurance company charges $4,000.
a. Calculate the expected value of the profit made by the
insurance company.
b. Many call this kind of situation a win–win opportunity
for the insurance company and the promoter. Do you
agree? explain.
5.54 Between 1896 when the Dow Jones Index was created and 2009, the index rose in 64% of the years. (Data
extracted from M. Hulbert, “What the past Can’t Tell
Investors,” The New York Times, January 3, 2010, p. BU2.)
Based on this information, and assuming a binomial distribution, what do you think is the probability that the stock
market will rise
a. next year?
b. the year after next?
c. in four of the next five years?
d. in none of the next five years?
e. For this situation, what assumption of the binomial distribution might not be valid?
5.55 In early 2012, it was reported that 38% of U.S. adult
cellphone owners called a friend for advice about a purchase
while in a store. (Data extracted from “Mobile Advice, Sunday Stats” The Palm Beach Post, February 19, 2012, p. 1F,)
If a sample of 10 U.S. adult cellphone owners is selected,
what is the probability that
a. 6 called a friend for advice about a purchase while in a
store?
b. at least 6 called a friend for advice about a purchase
while in a store?
c. all 10 called a friend for advice about a purchase while in
a store?
d. If you selected the sample in a particular geographical
area and found that none of the 10 respondents called a
friend for advice about a purchase while in a store, what
conclusion might you reach about whether the percentage
of adult cellphone owners who called a friend for advice
about a purchase while in a store in this area was 38%?
5.56 One theory concerning the Dow Jones Industrial
Average is that it is likely to increase during U.S. presidential election years. From 1964 through 2008, the Dow Jones
Industrial Average increased in 9 of the 12 U.S. presidential election years. Assuming that this indicator is a random
event with no predictive value, you would expect that the
indicator would be correct 50% of the time.
a. What is the probability of the Dow Jones Industrial
Average increasing in 9 or more of the 12 U.S. presidential election years if the probability of an increase in the
Dow Jones Industrial Average is 0.50?
b. What is the probability that the Dow Jones Industrial
Average will increase in 9 or more of the 12 U.S. presidential election years if the probability of an increase in
the Dow Jones Industrial Average in any year is 0.75?
5.57 Medical billing errors and fraud are on the rise.
According to Medical Billing Advocates of America, 8
out of 10 times, the medical bills that you get are not right.
(Data extracted from “Services Diagnose, Treat Medical
Billing errors,” USA Today, June 20, 2012.) If a sample of
10 medical bills is selected, what is the probability that
a. 0 medical bills will contain errors?
b. exactly 5 medical bills will contain errors?
c. more than 5 medical bills will contain errors?
d. What are the mean and standard deviation of the probability distribution?