– 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?