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2 Editorial Director: Sally Yagan Editor in Chief: Donna Battista Senior Acquisitions Editor: Chuck Synovec Senior Acquisitions Editor, Global Edition: Steven Jackson Editor, Global Edition: Leandra Paoli Senior Editorial Project Manager: Mary Kate Murray Editorial Assistant: Ashlee Bradbury Director of Marketing: Maggie Moylan Executive Marketing Manager: Anne Fahlgren Marketing Manager, International: Dean Erasmus Senior Managing Editor: Judy Leale Production Project Manager: Jacqueline A. Martin Senior Operations Supervisor: Arnold Vila Operations Specialist: Cathleen Petersen Art Director: Steve Frim Cover Designer: Jodi Notowitz Cover Art: Zoe - Fotolia.com Media Project Manager: John Cassar Associate Media Project Manager: Sarah Peterson Full-Service Project Management: PreMediaGlobal, Inc. Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at: Pearson Education Limited 2013 The rights of Paul Newbold, William L. Carlson and Betty Thorne to be identified as authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act Authorised adaptation from the United States edition, entitled Statistics for Business and Economics, 8 th Edition, ISBN: by Paul Newbold, William L. Carlson and Betty Thorne, published by Pearson Education, Inc., All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without either the prior written permission of the publisher or a licence permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, Saffron House, 6 10 Kirby Street, London EC1N 8TS. All trademarks used herein are the property of their respective owners. The use of any trademark in this text does not vest in the author or publisher any trademark ownership rights in such trademarks, nor does the use of such trademarks imply any affiliation with or endorsement of this book by such owners. Microsoft and Windows are registered trademarks of the Microsoft Corporation in the U.S.A. and other countries. This book is not sponsored or endorsed by or affiliated with the Microsoft Corporation. Microsoft and/or its respective suppliers make no representations about the suitability of the information contained in the documents and related graphics published as part of the services for any purpose. All such documents and related graphics are provided as is without warranty of any kind. Microsoft and/or its respective suppliers hereby disclaim all warranties and conditions with regard to this information, including all warranties and conditions of merchantability, whether express, implied or statutory, fitness for a particular purpose, title and non-infringement. In no event shall Microsoft and/or its respective suppliers be liable for any special, indirect or consequential damages or any damages whatsoever resulting from loss of use, data or profits, whether in an action of contract, negligence or other tortious action, arising out of or in connection with the use or performance of information available from the services. The documents and related graphics contained herein could include technical inaccuracies or typographical errors. Changes are periodically added to the information herein. Microsoft and/or its respective suppliers may make improvements and/or changes in the product(s) and/or the program(s) described herein at any time. Partial screen shots may be viewed in full within the software version specified. Credits and acknowledgments borrowed from other sources and reproduced, with permission, in this textbook appear on the appropriate page within the text. ISBN 13: ISBN 10: British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Typeset in Palatino LT Std by PreMediaGlobal, Inc. Printed and bound by Courier Kendallville in The United States of America The publisher s policy is to use paper manufactured from sustainable forests.

3 EXERCISES Basic Exercises 5.61 A random variable X is normally distributed with a mean of 100 and a variance of 100, and a random variable coefficient equal to 0.5. Find the mean and variance of the random variable: W = 5X + 4Y 5.62 A random variable X is normally distributed with a mean of 100 and a variance of 100, and a random variable coefficient equal to Find the mean and variance of the random variable: W = 5X + 4Y 5.63 A random variable X is normally distributed with a mean of 100 and a variance of 100, and a random variable coefficient equal to 0.5. Find the mean and variance of the random variable: W = 5X - 4Y 5.64 A random variable X is normally distributed with a mean of 500 and a variance of 100, and a random variable coefficient equal to 0.5. Find the mean and variance of the random variable: W = 5X - 4Y 5.65 A random variable X is normally distributed with a mean of 100 and a variance of 500, and a random variable coefficient equal to Find the mean and variance of the random variable: W = 5X - 4Y Application Exercises 5.66 An investor plans to divide $200,000 between two investments. The first yields a certain profit of 10%, whereas the second yields a profit with expected value 18% and standard deviation 6%. If the investor divides the money equally between these two investments, find the mean and standard deviation of the total profit A homeowner has installed a new energy-efficient furnace. It is estimated that over a year the new furnace will reduce energy costs by an amount that can be regarded as a random variable with a mean of $200 and a standard deviation of $60. Stating any assumptions you need to make, find the mean and standard deviation of the total energy cost reductions over a period of 5 years A consultant is beginning work on three projects. The expected profits from these projects are $50,000, $72,000, and $40,000. The associated standard deviations are $10,000, $12,000, and $9,000. Assuming independence of outcomes, find the mean and standard deviation of the consultant s total profit from these three projects A consultant has three sources of income from teaching short courses, from selling computer software, and from advising on projects. His expected annual incomes from these sources are $20,000, $25,000, and $15,000, and the respective standard deviations are $2,000, $5,000, and $4,000. Assuming independence, find the mean and standard deviation of his total annual income Five inspectors are employed to check the quality of components produced on an assembly line. For each inspector the number of components that can be checked in a shift can be represented by a random variable with mean 120 and standard deviation 15. Let X represent the number of components checked by an inspector in a shift. Then the total number checked is 5X, which has a mean of 600 and a standard deviation of 80. What is wrong with this argument? Assuming that inspectors performances are independent of one another, find the mean and standard deviation of the total number of components checked in a shift It is estimated that in normal highway driving, the number of miles that can be covered by automobiles of a particular model on 1 gallon of gasoline can be represented by a random variable with mean 28 and standard deviation 2.4. Sixteen of these cars, each with 1 gallon of gasoline, are driven independently under highway conditions. Find the mean and standard deviation of the average number of miles that will be achieved by these cars Shirley Johnson, portfolio manager, has asked you to analyze a newly acquired portfolio to determine its mean value and variability. The portfolio consists of 50 shares of Xylophone Music and 40 shares of Yankee Workshop. Analysis of past history indicates that the share price of Xylophone Music has a mean of 25 and a variance of 121. A similar analysis indicates that Yankee has a mean share price of 40 with a variance of 225. Your best evidence indicates that the share prices have a correlation of a. Compute the mean and variance of the portfolio. b. Suppose that the correlation between share prices was actually Now what are the mean and variance of the portfolio? 5.73 Prairie Flower Cereal has annual sales revenue of $400,000,000. George Severn, a 58-year-old senior vice president, is responsible for production and sales of Nougy 93 Fruity cereal. Daily production in cases is normally distributed, with a mean of 100 and a variance of 625. Daily sales in cases are also normally distributed, with a mean of 100 and a standard deviation Exercises 237

4 of 8. Sales and production have a correlation of The selling price per case is $10. The variable production cost per case is $7. The fixed production costs per day are $250. a. What is the probability that total revenue is greater than total costs on any day? b. Construct a 95% acceptance interval for total sales revenue minus total costs The nation of Olecarl, located in the South Pacific, has asked you to analyze international trade patterns. You first discover that each year it exports 10 units and imports 10 units of wonderful stuff. The price of exports is a random variable with a mean of 100 and a variance of 100. The price of imports is a random variable with a mean of 90 and a variance of 400. In addition, you discover that the prices of imports and exports have a correlation of r = The prices of both exports and imports follow a normal probability density function. Define the balance of trade as the difference between the total revenue from exports and the total cost of imports. a. What are the mean and variance of the balance of trade? b. What is the probability that the balance of trade is negative? 5.75 You have been asked to determine the probability that the contribution margin for a particular product line exceeds the fixed cost of $2,000. The total number of units sold is a normally distributed random variable with a mean of 400 and a variance of 900, X N1400, The selling price per unit is $10. The total number of units produced is a normally distributed random variable with a mean of 400 and a variance of 1,600, Y N1400, 1,6002. The variable production cost is $4 per unit. Production and sales have a positive correlation of The nation of Waipo has recently created an economic development plan that includes expanded exports and imports. It has completed a series of extensive studies of the world economy and Waipo s economic capability, following Waipo s extensive 10-year educational-enhancement program. The resulting model indicates that in the next year exports will be normally distributed with a mean of 100 and a variance of 900 (in billions of Waipo yuan). In addition, imports are expected to be normally distributed with a mean of 105 and a variance of 625 in the same units. The correlation between exports and imports is expected to be Define the trade balance as exports minus imports. a. Determine the mean and variance of the trade balance (exports minus imports) if the model parameters given above are true. b. What is the probability that the trade balance will be positive? KEY WORDS correlation, 229 covariance, 229 cumulative distribution function, 198 cumulative distribution function of the normal distribution, 208 differences between pairs of random variables, 230 expected value, 203 exponential probability distribution, 225 joint cumulative distribution function, 228 linear combinations of random variables, 232 marginal distribution, 229 mean of X, 204 probability density function, 199 probability density function of the normal distribution, 207 properties of the normal distribution, 207 range probabilities for normal random variables, 209 standard deviation, 204 standard normal distribution, 209 sums of random variables, 229 uniform probability distribution, 198 variance, 204 DATA FILES Return on Stock Price 60 month, 235, 242 Stock Price File, 241, 242 CHAPTER EXERCISES AND APPLICATIONS 5.77 A consultant knows that it will cost him $10,000 to fulfill a particular contract. The contract is to be put out for bids, and he believes that the lowest bid, excluding his own, can be represented by a distribution that is uniform between $8,000 and $20,000. Therefore, if the random variable X denotes the lowest of all other bids (in thousands of dollars), its probability density function is as follows: f1x2 = e 1>12 for 8 6 x for all other values of x 238 Chapter 5 Continuous Probability Distributions

5 a. What is the probability that the lowest of the other bids will be less than the consultant s cost estimate of $10,000? b. If the consultant submits a bid of $12,000, what is the probability that he will secure the contract? c. The consultant decides to submit a bid of $12,000. What is his expected profit from this strategy? d. If the consultant wants to submit a bid so that his expected profit is as high as possible, discuss how he should go about making this choice The ages of a group of executives attending a convention are uniformly distributed between 35 and 65 years. If the random variable X denotes ages in years, the probability density function is as follows: 1>30 for 35 6 x 6 65 f1x2 = e 0 for all other values of x a. Graph the probability density function for X. b. Find and graph the cumulative distribution function for X. c. Find the probability that the age of a randomly chosen executive in this group is between 40 and 50 years. d. Find the mean age of the executives in the group The random variable X has probability density function as follows: x for 0 6 x 6 1 f1x2 = 2 - x for 1 6 x for all other values of x a. Graph the probability density function for X. b. Show that the density has the properties of a proper probability density function. c. Find the probability that X takes a value between 0.5 and An investor puts $2,000 into a deposit account with a fixed rate of return of 10% per year. A second sum of $1,000 is invested in a fund with an expected rate of return of 16% and a standard deviation of 8% per year. a. Find the expected value of the total amount of money this investor will have after a year. b. Find the standard deviation of the total amount after a year A hamburger stand sells hamburgers for $1.45 each. Daily sales have a distribution with a mean of 530 and a standard deviation of 69. a. Find the mean daily total revenues from the sale of hamburgers. b. Find the standard deviation of total revenues from the sale of hamburgers. c. Daily costs (in dollars) are given by C = X where X is the number of hamburgers sold. Find the mean and standard deviation of daily profits from sales An analyst forecasts corporate earnings, and her record is evaluated by comparing actual earnings with predicted earnings. Define the following: actual earnings = predicted earnings + forecast error If the predicted earnings and forecast error are independent of each other, show that the variance of predicted earnings is less than the variance of actual earnings Let X 1 and X 2 be a pair of random variables. Show that the covariance between the random variables Y 1 = 1X 1 + X 2 2 and Y 2 = 1X 1 - X 2 2 is 0 if and only if X 1 and X 2 have the same variance Grade point averages of students on a large campus follow a normal distribution with a mean of 2.6 and a standard deviation of 0.5. a. One student is chosen at random from this campus. What is the probability that this student has a grade point average higher than 3.0? b. One student is chosen at random from this campus. What is the probability that this student has a grade point average between 2.25 and 2.75? c. What is the minimum grade point average needed for a student s grade point average to be among the highest 10% on this campus? d. A random sample of 400 students is chosen from this campus. What is the probability that at least 80 of these students have grade point averages higher than 3.0? e. Two students are chosen at random from this campus. What is the probability that at least one of them has a grade point average higher than 3.0? 5.85 A company services home air conditioners. It is known that times for service calls follow a normal distribution with a mean of 60 minutes and a standard deviation of 10 minutes. a. What is the probability that a single service call takes more than 65 minutes? b. What is the probability that a single service call takes between 50 and 70 minutes? c. The probability is that a single service call takes more than how many minutes? d. Find the shortest range of times that includes 50% of all service calls. e. A random sample of four service calls is taken. What is the probability that exactly two of them take more than 65 minutes? 5.86 It has been found that times taken by people to complete a particular tax form follow a normal distribution with a mean of 100 minutes and a standard deviation of 30 minutes. a. What is the probability that a randomly chosen person takes less than 85 minutes to complete this form? b. What is the probability that a randomly chosen person takes between 70 and 130 minutes to complete this form? c. Five percent of all people take more than how many minutes to complete this form? Chapter Exercises and Applications 239

6 d. Two people are chosen at random. What is the probability that at least one of them takes more than an hour to complete this form? e. Four people are chosen at random. What is the probability that exactly two of them take longer than an hour to complete this form? f. For a randomly chosen person, state in which of the following ranges (expressed in minutes) the time to complete the form is most likely to lie , , , g. For a randomly chosen person, state in which of the following ranges (expressed in minutes) the time to complete the form is least likely to lie , , , A pizza delivery service delivers to a campus dormitory. Delivery times follow a normal distribution with a mean of 20 minutes and a standard deviation of 4 minutes. a. What is the probability that a delivery will take between 15 and 25 minutes? b. The service does not charge for the pizza if delivery takes more than 30 minutes. What is the probability of getting a free pizza from a single order? c. During final exams, a student plans to order pizza five consecutive evenings. Assume that these delivery times are independent of each other. What is the probability that the student will get at least one free pizza? d. Find the shortest range of times that includes 40% of all deliveries from this service. e. For a single delivery, state in which of the following ranges (expressed in minutes) the delivery time is most likely to lie , 19921, 20922, f. For a single delivery, state in which of the following ranges (expressed in minutes) the delivery time is least likely to lie , 19921, 20922, A video-rental chain estimates that annual expenditures of members on rentals follow a normal distribution with a mean of $100. It was also found that 10% of all members spend more than $130 in a year. What percentage of members spends more than $140 in a year? 5.89 It is estimated that amounts of money spent on gasoline by customers at a gas station follow a normal distribution with a standard deviation of $2.50. It is also found that 10% of all customers spent more than $25. What percentage of customers spent less than $20? 5.90 A market research organization has found that 40% of all supermarket shoppers refuse to cooperate when questioned by its pollsters. If 1,000 shoppers are approached, what is the probability that fewer than 500 will refuse to cooperate? 5.91 An organization that gives regular seminars on sales motivation methods determines that 60% of its clients have attended previous seminars. From a sample of 400 clients what is the probability that more than half have attended previous seminars? 5.92 An ambulance service receives an average of 15 calls per day during the time period 6 p.m. to 6 a.m. for assistance. For any given day what is the probability that fewer than 10 calls will be received during the 12-hour period? What is the probability that more than 17 calls during the 12-hour period will be received? 5.93 In a large department store a customer-complaints office handles an average of six complaints per hour about the quality of service. The distribution is Poisson. a. What is the probability that in any hour exactly six complaints will be received? b. What is the probability that more than 20 minutes will elapse between successive complaints? c. What is the probability that fewer than 5 minutes will elapse between successive complaints? d. The store manager observes the complaints office for a 30-minute period, during which no complaints are received. He concludes that a talk he gave to his staff on the theme the customer is always right has obviously had a beneficial effect. Suppose that, in fact, the talk had no effect. What is the probability of the manager observing the office for a period of 30 minutes or longer with no complaints? 5.94 A fish market in Hong Kong offers a large variety of fresh fish on its stands. You have found out that the average chunk of tuna sushi on sale has a weight of 3.2 grams, with a standard deviation of 0.8 gram. Assuming the weights of tuna sushi are normally distributed, what is the probability that a randomly selected piece of sushi will weigh more than 4.4 grams? 5.95 In a Godiva Chocolate Shop, there are different sizes and weights of boxes of truffles. a. Find the probability that a box of truffles weighs between 283 and grams. The mean weight of a box is 283 grams and the standard deviation is 1.6 grams. b. After a more careful check, the standard deviation was found to be 2.2 grams. Find the new probability A management consultant found that the amount of time per day spent by executives performing tasks that could be done equally well by subordinates followed a normal distribution with a mean of 2.4 hours. It was also found that 10% of executives spent over 3.5 hours per day on tasks of this type. For a random sample of 400 executives, find the probability that more than 80 spend more than 3 hours per day on tasks of this type Financial Managers, Inc., buys and sells a large number of stocks routinely for the various accounts that it manages. Portfolio manager Andrea Colson has asked 240 Chapter 5 Continuous Probability Distributions

7 for your assistance in the analysis of the Johnson Fund. A portion of this portfolio consists of 10 shares of stock A and 8 shares of stock B. The price of A has a mean of 10 and a variance of 16, while the price of B has a mean of 12 and a variance of 9. The correlation between prices is 0.3. a. What are the mean and variance of the portfolio value? b. Andrea has been asked to reduce the variance (risk) of the portfolio. She offers to trade the 10 shares of stock A and receives two offers, from which she can select one: 10 shares of stock 1 with a mean price of 10, a variance of 25, and a correlation with the price of stock B equal to -0.2; or 10 shares of stock 2 with a mean price of 10, a variance of 9, and a correlation with the price of stock B equal to Which offer should she select? 5.98 Financial Managers, Inc., buys and sells a large number of stocks routinely for the various accounts that it manages. Portfolio manager Sarah Bloom has asked for your assistance in the analysis of the Burde Fund. A portion of this portfolio consists of 10 shares of stock A and 8 shares of stock B. The price of A has a mean of 12 and a variance of 14, while the price of B has a mean of 10 and a variance of 12. The correlation between prices is 0.5. a. What are the mean and variance of the portfolio value? b. Sarah has been asked to reduce the variance (risk) of the portfolio. She offers to trade the 10 shares of stock A and receives two offers from which she can select one: 10 shares of stock 1 with a mean price of 12, a variance of 25, and a correlation with the price of stock B equal to -0.2; or 10 shares of stock 2 with a mean price of 10, a variance of 9, and a correlation with the price of stock B, equal to Which offer should she select? 5.99 Big Nail Construction Inc. is building a large, new student center for a famous Midwestern liberal arts college. During the project Christine Buildumbig, the project manager, requests that a pile of sand weighing between 138,000 pounds and 141,000 pounds be placed on the newly constructed driveway. You have been asked to determine the probability that the delivered sand satisfies Christine s request. You have ordered that one big truck and one small truck be used to deliver the sand. Sand loads in the big truck are normally distributed with a mean of 80,000 and a variance of 1,000,000, and sand loads in the small truck are also normally distributed with a mean weight of 60,000 pounds and a variance of 810,000. From past experience with the sand-loading facility, you know that the weight of sand in the two trucks has a correlation of What is the probability that the resulting pile of sand has a weight that is between 138,000 and 141,000 pounds? An investment portfolio in Singapore specializes in airline stocks and contains two of them. One is Singapore Airlines (mean: 0.12; standard deviation: 0.02), and it accounts for 30% of the portfolio shares. The other airline present in the portfolio is AirAsia (mean: 0.25; standard deviation: 0.15), a higher-risk, higherreturn investment. a. What is the expected value and the standard deviation of the portfolio if the coefficient of correlation of the two stocks is 0.5? b. What will they be if the correlation is 0.2 instead? PORTFOLIO MINI CASE STUDIES Visit or editions.com/newbold to access the data files. The following exercises, or case studies, provide the opportunity to prepare small stock portfolios and to analyze their characteristics in terms of growth and risk. These require considerably more work than other exercises, but they do provide important insights into portfolio computations and analysis. We have deliberately selected stock performance data from before the 2008 crash to avoid the major additional complexities that occur in a major financial collapse. So you will be working with real data on real stocks, but avoiding the situation where it is very difficult if not impossible to predict long term performance from the data Shirley Johnson is developing a new mutual fund portfolio and in the process has asked you to develop the mean and variance for the stock price that consists of 10 shares of stocks from each of the following firms: Alcoa Inc., Reliant Energy, and Sea Container. Using the data file Stock Price File, compute the mean and variance for this portfolio. Prepare the analysis by using means, variances, and covariances for individual stocks following the methods used in Examples 5.16 and 5.17 then confirm your results by obtaining the portfolio price for each year using the computer. Assuming that the portfolio price is normally distributed, determine the narrowest interval that contains 95% of the distribution of portfolio value Zafer Toprak is a developing a new mutual fund portfolio and in the process has asked you to develop the mean and variance for the stock price that consists of 10 shares of stocks from Alcoa Inc., 20 shares from AB Volvo, 10 shares from TCF Financial, Portfolio Mini Case Studies 241