www.freebookslides.com Statistical Techniques in Business & Economics Seventeenth Edition LIND MARCHAL WATHEN www.freebookslides.com Statistical Techniques in BUSINESS & ECONOMICS www.freebookslides.com The McGraw-Hill/Irwin Series in Operations and Decision Sciences SUPPLY CHAIN MANAGEMENT BUSINESS RESEARCH METHODS Benton Purchasing and Supply Chain Management Third Edition Cooper and Schindler Business Research Methods Twelfth Edition Swink, Melnyk, Cooper, and Hartley Managing Operations across the Supply Chain Second Edition BUSINESS FORECASTING PRODUCT DESIGN Wilson, Keating, and John Galt Solutions, Inc Business Forecasting Sixth Edition Ulrich and Eppinger Product Design and Development Fifth Edition LINEAR STATISTICS AND REGRESSION Slater and Wittry Math for Business and Finance: An Algebraic Approach First Edition Bowersox, Closs, Cooper, and Bowersox Supply Chain Logistics Management Fourth Edition Burt, Petcavage, and Pinkerton Supply Management Eighth Edition Johnson, Leenders, and Flynn Purchasing and Supply Management Fourteenth Edition Simchi-Levi, Kaminsky, and Simchi-Levi Designing and Managing the Supply Chain: Concepts, Strategies, Case Studies Third Edition PROJECT MANAGEMENT Brown and Hyer Managing Projects: A Team-Based Approach First Edition Larson and Gray Project Management: The Managerial Process Fifth Edition Kutner, Nachtsheim, and Neter Applied Linear Regression Models Fourth Edition BUSINESS SYSTEMS DYNAMICS Sterman Business Dynamics: Systems Thinking and Modeling for a Complex World First Edition OPERATIONS MANAGEMENT Cachon and Terwiesch Matching Supply with Demand: An Introduction to Operations Management Third Edition SERVICE OPERATIONS MANAGEMENT Finch Interactive Models for Operations and Supply Chain Management First Edition Fitzsimmons and Fitzsimmons Service Management: Operations, Strategy, Information Technology Eighth Edition Jacobs and Chase Operations and Supply Chain Management Fourteenth Edition MANAGEMENT SCIENCE Jacobs and Chase Operations and Supply Chain Management: The Core Third Edition Hillier and Hillier Introduction to Management Science: A Modeling and Case Studies Approach with Spreadsheets Fifth Edition Stevenson and Ozgur Introduction to Management Science with Spreadsheets First Edition MANUFACTURING CONTROL SYSTEMS Jacobs, Berry, Whybark, and Vollmann Manufacturing Planning & Control for Supply Chain Management Sixth Edition Jacobs and Whybark Why ERP? 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Edition www.freebookslides.com Statistical Techniques in BUSINESS & ECONOMICS SEVENTEENTH EDITION DOUGLAS A LIND Coastal Carolina University and The University of Toledo WILLIAM G MARCHAL The University of Toledo SAMUEL A WATHEN Coastal Carolina University www.freebookslides.com STATISTICAL TECHNIQUES IN BUSINESS & ECONOMICS, SEVENTEENTH EDITION Published by McGraw-Hill Education, Penn Plaza, New York, NY 10121 Copyright © 2018 by McGraw-Hill Education All rights reserved Printed in the United States of America Previous editions © 2015, 2012, and 2010 No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written consent of McGrawHill Education, including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning Some ancillaries, including electronic and print components, may not be available to customers outside the United States 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considered to be an extension of the copyright page Library of Congress Cataloging-in-Publication Data Names: Lind, Douglas A., author | Marchal, William G., author | Wathen, Samuel Adam author Title: Statistical techniques in business & economics/Douglas A Lind, Coastal Carolina University and The University of Toledo, William G Marchal, The University of Toledo, Samuel A Wathen, Coastal Carolina University Other titles: Statistical techniques in business and economics Description: Seventeenth Edition | Dubuque, IA : McGraw-Hill Education, [2017] | Revised edition of the authors’ Statistical techniques in business & economics, [2015] Identifiers: LCCN 2016054310| ISBN 9781259666360 (alk paper) | ISBN 1259666360 (alk paper) Subjects: LCSH: Social sciences—Statistical methods | Economics—Statistical methods | Commercial statistics Classification: LCC HA29 M268 2017 | DDC 519.5—dc23 LC record available at https://lccn.loc.gov/2016054310 The Internet addresses listed in the text were accurate at the time of publication The inclusion of a website does not indicate an endorsement by the authors or McGraw-Hill Education, and McGraw-Hill Education does not guarantee the accuracy of the information presented at these sites mheducation.com/highered www.freebookslides.com D E D I CATI O N To Jane, my wife and best friend, and our sons, their wives, and our grandchildren: Mike and Sue (Steve and Courtney), Steve and Kathryn (Kennedy, Jake, and Brady), and Mark and Sarah (Jared, Drew, and Nate) Douglas A Lind To Oscar Sambath Marchal, Julian Irving Horowitz, Cecilia Marchal Nicholson and Andrea William G Marchal To my wonderful family: Barb, Hannah, and Isaac Samuel A Wathen www.freebookslides.com A NOTE FROM THE AUTHORS Over the years, we received many compliments on this text and understand that it’s a favorite among students We accept that as the highest compliment and continue to work very hard to maintain that status The objective of Statistical Techniques in Business and Economics is to provide students majoring in management, marketing, finance, accounting, economics, and other fields of business administration with an introductory survey of descriptive and inferential statistics To illustrate the application of statistics, we use many examples and exercises that focus on business applications, but also relate to the current world of the college student A previous course in statistics is not necessary, and the mathematical requirement is first-year algebra In this text, we show beginning students every step needed to be successful in a basic statistics course This step-by-step approach enhances performance, accelerates preparedness, and significantly improves motivation Understanding the concepts, seeing and doing plenty of examples and exercises, and comprehending the application of statistical methods in business and economics are the focus of this book The first edition of this text was published in 1967 At that time, locating relevant business data was difficult That has changed! Today, locating data is not a problem The number of items you purchase at the grocery store is automatically recorded at the checkout counter Phone companies track the time of our calls, the length of calls, and the identity of the person called Credit card companies maintain information on the number, time and date, and amount of our purchases Medical devices automatically monitor our heart rate, blood pressure, and temperature from remote locations A large amount of business information is recorded and reported almost instantly CNN, USA Today, and MSNBC, for example, all have websites that track stock prices in real time Today, the practice of data analytics is widely applied to “big data.” The practice of data analytics requires skills and knowledge in several areas Computer skills are needed to process large volumes of information Analytical skills are needed to evaluate, summarize, organize, and analyze the information Critical thinking skills are needed to interpret and communicate the results of processing the information Our text supports the development of basic data analytical skills In this edition, we added a new section at the end of each chapter called Data Analytics As you work through the text, this section provides the instructor and student with opportunities to apply statistical knowledge and statistical software to explore several business environments Interpretation of the analytical results is an integral part of these exercises A variety of statistical software is available to complement our text Microsoft Excel includes an add-in with many statistical analyses Megastat is an add-in available for Microsoft Excel Minitab and JMP are stand-alone statistical software available to download for either PC or MAC computers In our text, Microsoft Excel, Minitab, and Megastat are used to illustrate statistical software analyses When a software application is presented, the software commands for the application are available in Appendix C We use screen captures within the chapters, so the student becomes familiar with the nature of the software output Because of the availability of computers and software, it is no longer necessary to dwell on calculations We have replaced many of the calculation examples with interpretative ones, to assist the student in understanding and interpreting the statistical results In addition, we place more emphasis on the conceptual nature of the statistical topics While making these changes, we still continue to present, as best we can, the key concepts, along with supporting interesting and relevant examples vi www.freebookslides.com WHAT’S NEW IN THE SEVENTEENTH EDITION? We have made many changes to examples and exercises throughout the text The section on “Enhancements” to our text details them The major change to the text is in response to user interest in the area of data analytics Our approach is to provide instructors and students with the opportunity to combine statistical knowledge, computer and statistical software skills, and interpretative and critical thinking skills A set of new and revised exercises is included at the end of chapters through 18 in a section titled “Data Analytics.” In these sections, exercises refer to three data sets The North Valley Real Estate sales data set lists 105 homes currently on the market The Lincolnville School District bus data lists information on 80 buses in the school district’s bus fleet The authors designed these data so that students will be able to use statistical software to explore the data and find realistic relationships in the variables The Baseball Statistics for the 2016 season is updated from the previous edition The intent of the exercises is to provide the basis of a continuing case analysis We suggest that instructors select one of the data sets and assign the corresponding exercises as each chapter is completed Instructor feedback regarding student performance is important Students should retain a copy of each chapter’s results and interpretations to develop a portfolio of discoveries and findings These will be helpful as students progress through the course and use new statistical techniques to further explore the data The ideal ending for these continuing data analytics exercises is a comprehensive report based on the analytical findings We know that working with a statistics class to develop a very basic competence in data analytics is challenging Instructors will be teaching statistics In addition, instructors will be faced with choosing statistical software and supporting students in developing or enhancing their computer skills Finally, instructors will need to assess student performance based on assignments that include both statistical and written components Using a mentoring approach may be helpful We hope that you and your students find this new feature interesting and engaging vii www.freebookslides.com H OW A RE C H A P TE RS O RGA N I Z E D TO E N GAG E DESCRIBING DATA: STU D E NTS A N D PRO M OTE LE ADISPLAYING RN I NAND G?EXPLORING DATA 95 INTRODUCTION Chapter began our study of descriptive statistics In order to transform raw or ungrouped data into a meaningful form, we organize the data into a frequency distribution We present the frequency distribution in graphic form as a histogram or a frequency polygon This allowsrecently us to visualize data tend to cluster, the for largest and the MERRILL LYNCH completedwhere a studythe of online investment portfolios a sample Each chapter begins with a set of smallest values, and general in shape of the data these data into a frequency of clients For the 70the participants the study, organize ) distribution (See and LO2-3 In Chapter 3, Exercise we first 43 computed several measures of location, such as the mean, learning objectives designed to promedian, and mode These measures of location allow us to report a typical value in the vide focus for the chapter and motivate set of observations We also computed several measures of dispersion, such as the student learning These objectives, lorange, variance, and standard deviation These measures of dispersion allow us to deLEARNING OBJECTIVES cated in the margins next to the topic, scribe the variation or the spread in a set of observations When you have completed this chapter, you will be able to: We continue our study of descriptive statistics in this chapter We study (1) dot plots, indicate what the student should be Summarize qualitative(3) variables with frequency tables and statistics (2)LO2-1 stem-and-leaf displays, percentiles, and (4)and boxrelative plots.frequency These charts able to after completing each secgive us additional insight into are concentrated as well as the general LO2-2 Display a frequency tablewhere using athe bar values or pie chart tion in the chapter shape of the data Then we consider bivariate data In bivariate data, we observe two LO2-3 Summarize quantitative variables with frequency and relative frequency distributions variables for each individual or observation Examples include the number of hours a LO2-4 studied Display aand frequency distribution using or frequency student the points earned ona histogram an examination; if a polygon sampled product meets quality specifications and the shift on which it is manufactured; or the amount of electricity used in a month by a homeowner and the mean daily high temperature in the region theshows month how These the charts and graphs provide useful as to weause business A representative exercise opens the chapter for and chapter content can be insights applied real-world analytics to enhance our understanding of data situation Chapter Learning Objectives Source: © rido/123RF Chapter Opening Exercise 19 DESCRIBING DATA: FREQUENCY TABLES, FREQUENCY DISTRIBUTIONS, AND GRAPHIC PRESENTATION LO4-1 Construct and interpret a dot plot Introduction to the Topic DOT PLOTS INTRODUCTION Recall for the Applewood Auto Group data, we summarized the profit earned on the The United States automobile retailing industry highlyclasses competitive It is dominated by 180 vehicles sold with a frequency distribution using iseight When we orgamegadealerships that ownwe andlost operate or more franchises, over 10,000 Each chapter starts with a review of nized the data into the eight classes, the 50 exact value of the employ observations A people, and generate several billion dollars in annual sales Many of the top dealerships dot plot, on the other hand, groups theowned datawith as shares little as possible, andYork weStock not lose the important concepts of the previare publicly traded on the New Exchange the identity of an individual observation To develop dot plot, we was display a dot(ticker for Lin66360_ch02_018-050.indd 18 or NASDAQ In 2014, the largestamegadealership AutoNation ous chapter and provides a link to the symbol AN), followedline by Penske Auto Group (PAG), Group Automotive, each observation along a horizontal number indicating the possible values of the Inc (ticker symbol GPI), and the privately Van Tuyl material in the current chapter This data If there are identical observations or the observations areowned too close toGroup be shown These large corporations use statistics and analytics to summarize individually, the dots are “piled” on top of each other This allows us to see theAsshape step-by-step approach increases comand analyze data and information to support their decisions an exof the distribution, the value about which the at data tend to cluster, and Itthe largest and ample, we will look the Applewood Auto group owns four dealerprehension by providing continuity shipsare andmost sells auseful wide range of vehicles the popular smallest observations Dot plots for smaller dataThese sets,include whereas histoacross the concepts brands Kia sets and Hyundai, BMW and sedans luxury grams tend to be most usefulKorean for large data An example willVolvo show howand to conand a full line of Ford and Chevrolet cars and trucks struct and interpret dot plots.SUVs, Ms Kathryn Ball is a member of the senior management team at Applewood Auto Group, which has its corporate offices adjacent to Kane Motors She is responsible for tracking and analyzing vehicle sales and the profitability of those vehicles Kathryn would like to summarize the profit earned on the vehicles sold with tables, charts, and graphs that she would review monthly She E X A M P L E wants to know the profit per vehicle sold, as well as the lowest and highest amount of profit She is also interested in describing the demographics of the buyers What are The service departments at many Tionesta Ford and Sheffield their ages? How vehicles haveLincoln they previously purchasedMotors from oneInc., of thetwo Appleof the four Applewood Auto Group were both open 24 days last wood dealerships? What typedealerships, of vehicle did they purchase? The Applewood Auto Group operates four dealerships: month Listed below is the number of vehicles serviced last month at the two Source: © Justin Sullivan/Getty Images Example/Solution After important concepts are introduced, a solved example is given This example provides a how-to illustration and shows a relevant business application that helps students answer the question, “How can I apply this concept?” dealerships Construct dot plots andsells report summary statistics to compare the • Tionesta Ford Lincoln Ford and Lincoln cars and trucks • Olean Automotive Inc has the Nissan franchise as well as the General Motors two dealerships brands of Chevrolet, Cadillac, and GMC Trucks • Sheffield Motors Inc sells Buick, GMC trucks, Hyundai, and Kia • Kane Motors offers the Chrysler, Dodge, and Jeep line as well as BMW and Volvo Tionesta Ford Lincoln Monday month, Ms Ball collects data from each of the four dealerships Tuesday Every Wednesday Thursday Friday Saturday and enters them into an Excel spreadsheet Last month the Applewood 23 30 CHAPTER294 106 33Auto Group27 28 at the 39 26 sold 180 vehicles four dealerships A copy of the first 32few observations 28 appears 33 35 variables 32collected include: to the left The 25 • Age—the 36 age of the buyer 31 at the 32 27 time of the purchase 32 • Profit—the 35 amount earned 37 36 dealership 30 on the sale of each by the 35 vehicle calculate quartiles Excel and Excel 2016 offer both The Excel function, • 2013 Location—the dealership where themethods vehicle was purchased Quartile.exc, will result the same answer sedan, as Equation 4–1 The or Excel function, Quar • in Vehicle type—SUV, compact, hybrid, truck • Excel tile.inc, will result in the Method answers Previous—the number of vehicles previously purchased at any of the Self-Reviews Self-Reviews are interspersed throughout each chapter and follow Example/Solution sections They help students monLin66360_ch04_094-131.indd 95 itor their progress and provide immediate reinforcement for that particular technique Answers are in Appendix E four Applewood dealerships by the consumer SELF-REVIEW The entire data set is available at the McGraw-Hill website (www.mhhe com/lind17e) and in Appendix A.4 at the end of the text 4–2 The Quality Control department of Plainsville Peanut Company is responsible for checking CONSTRUCTING FREQUENCY TABLES LO2-1 the weight of the 8-ounce jar of peanut butter The weights of a sample of nine jars proSummarize qualitative duced last hour are: Recall from Chapter that techniques used to describe a set of data are called descrip1/10/17 7:41 PM variables with frequency tive statistics Descriptive statistics organize data to show the general pattern of the and relative frequency 7.72where 7.8values 7.86tend 7.90 7.94 7.97 8.06 8.09 data, 7.69 to identify to concentrate, and to expose extreme or unusual tables data values The first technique we discuss is a frequency table (a) What is the median weight? (b) Determine the weights corresponding first anddata thirdinto quartiles. FREQUENCY TABLE A groupingtoofthe qualitative mutually exclusive and collectively exhaustive classes showing the number of observations in each class EXERCISES 11 viii Determine the median and the first and third quartiles in the following data 46 Lin66360_ch02_018-050.indd 19 12 47 49 49 51 53 54 54 55 55 59 Determine the median and the first and third quartiles in the following data 1/6/17 4:52 AM 1/6/17 4:52 AM (c) Are the events in part (a)(i) complementary or mutually ex The probability of passing both is 50 What is the probability of passing at least one? 21 The aquarium at Sea Critters Depot contains 140 fish Eighty of these fish are green www.freebookslides.com swordtails (44 female and 36 male) and 60 are orange swordtails (36 female and The General Rule of Addition 24 males) A fish is randomly captured from the aquarium: a What is the probability the selected fish is a green may not be mutually exclu The outcomes of answordtail? experiment b What is the probability the selected is male? selected a sample of 200 tourists wh Tourist fish Commission c What is the probability the selected fishsurvey is a male green swordtail? year The revealed that 120 tourists went to Disney d What is the probability the selected fish is either a male or a green swordtail? Gardens near Tampa What is the probability that a person 22 A National Park Service survey of visitors to the Rocky Mountain region revealed or Busch Gardens? If the that 50% visit Yellowstone Park,World 40% visit the Tetons, and 35% visitspecial both rule of addition is use a touristwill who to Disney Worldattractions? is 60, found by 120/200 a What is the probability a vacationer visitwent at least one of these tourist going to Busch Gardens is 50 The sum of these p b What is the probability 35 called? however, that this probability cannot be greater than The c Are the events mutually exclusive? Explain Statistics in Action STATISTICS IN ACTION ists visited both attractions and are being counted twice! A c revealed that 60 out of 200 sampled did, in fact, visit both a that you believe that at bility of visiting both Thus: A SURVEY OF PROBABILITY CONCEPTS 145 LO5-4 To answer our question, “What is the probability a se If you wish to get some Calculate probabilities Disney World or Busch Gardens?” (1) add the probability attention at the next gathStatistics in Action articles are usingscattered the rules of throughWorld and the probability he or she visited Busch Garden ering you attend, announce multiplication out the text, usually about two per chapter They In this section, we discuss the rules for computing the likelihood that two events both RULES OF MULTIPLICATION TO CALCULATE PROBABILITY happen, or their joint probability example, 16% of the 2016 tax returns were preprovide unique, interesting applications and hisP(Disney)For = 60 P(Busch) = 50 least two people present P(Disney Busch) What = P(Disney) + P(Busch) − P(bo pared by H&R Block and 75% of those returns showedor a refund is the likelihood torical insights in the field of statistics weretax born on the same a person’s form was prepared by H&R Block and the person received a refund? = 60 + 50 − 30 = 80 64 Definitions Definitions of new terms or terms unique to the study of statistics are set apart from the text and highlighted for easy reference and review They also appear in the Glossary at the end of the book Formulas Formulas that are used for the first time are boxed and numbered for reference In addition, a formula card is bound into the back of the text that lists all the key formulas Exercises date—thatillustrate is, the same Venn diagrams this as the intersection of two events To find the likelihood of day of the year butwe notuse the rules of When two events both the probability two events happening, multiplication There areoccur, two rules of multipli- is called necessarily same ability cation: the specialthe rule andyear the general rule.(.30) that a tourist visits both attractions is an examp If there are 30 people in the room, the of probability of Special Rule Multiplication a duplicate is 706 If there The special rule of multiplication are 60 people3in the room, requires that two events A and B are independent CHAPTER P(Disney and Busch) = 30 Two events are independent if the occurrence of one event does not alter the probabilthe probability is 994 that ity of theatoccurrence of the least two people shareother the event same birthday With as few INDEPENDENCE The one of event has nounemployment effect on therates? probability of a What isoccurrence the arithmeticofmean the Alaska as 23 people the chances JOINT PROBABILITY probability that measures the likelihood two or more the occurrence of another event Find theAthat median are even,b.that is 50, at and the mode for the unemployment rates. events will happen concurrently Compute least two c.people sharethe thearithmetic mean and median for just the winter (Dec–Mar) months Is it much different? birthday Hint: To Onesame way to think about independence to assume events A and Bfor occur 22 Big Orange Trucking is is designing an that information system use at in differ“in-cab” ent times For example, when event B occurs after event A occurs, does A have any this, find the communications It must summarize data from eight sites a region So compute the general rule of addition, which is used to compute thethroughout probability ofeffect twoto on the likelihood that event Bexclusive, occurs? Ifis: the answer no, then measure A and B of are independent typical conditions Compute an is appropriate central location for probability everyone was events that are describe not mutually the variables wind direction, temperature, and pavement events To illustrate independence, The outcome of a coin born on a different day and suppose two coins are©tossed Rostislav Glinsky/Shutterstock.com toss (head tail) is unaffected useorthe complement rule.by the outcome of any other prior coin toss (head or tail) City Wind Direction Temperature Pavement For Try twothis independent events A and B,The the probability that A and B shows will both occur is that are n Venn events in your class GENERAL RULE OF ADDITION P(A or following B) = P(A) + P(B)diagram − P(A and B) two[5–4] West 89the found by multiplying Anniston, the twoALprobabilities This is thetospecial rule ofjoint multiplication events overlap illustrate eventDry that and some people h Atlanta, Northwest 86 Wet is written symbolically as: GA Augusta, GA Southwest 92 Wet For the expression P(A or AL B), the wordSouth or suggests that A may Birmingham, 91 occur or B may Dry occur This also includes the possibility that A and B may This use of or is sometimes SPECIAL RULE OF MULTIPLICATION P(Aoccur and B) =92 P(A)P(B) Jackson, MS Southwest Dry[5–5] called an inclusive You could or B or both) to emphasize that theTrace union of Meridian, MS also write P(A South 92 the events includes the intersection of A and B Monroe, LA Southwest 93 Wet If we compare the general rules of addition, the is Tuscaloosa, AL and special Southwest 93 important difference Trace determining if the events are mutually exclusive If the events are mutually exclusive, then the joint probability DATA: P(A and B) is andMEASURES we could use the special rule of addition OtherDESCRIBING NUMERICAL 79 wise, we must account for the joint probability and use the general rule of addition Software Solution Lin66360_ch05_132-174.indd 144 Exercises are included after sec-147 E X E R C I S E SE X A M P L E Lin66360_ch05_132-174.indd 1/10/17 7:41 PM 47–52, the following: What isFor theExercises probability that a card chosen at random from a standard deck of cards tions within the chapter and at E X A M P L E a Compute sample variance will be either a king orthe a heart? the end of the chapter Section b Determine the sample standard deviation Table 2–4 on page 26 shows the profit on the sales of 180 vehicles at Applewood 47 Consider these values a sample: 7, 2, 6, 2, and 3. exercises cover the material studAuto Group Determine the mean and the median selling price 48 SOLU T IThe O Nfollowing five values are a sample: 11, 6, 10, 6, and ied in the section Many exercises 49 Dave’s Automatic Door, referred to in Exercise 37, installs automatic garage openers on a sample, times, in minutes, required We may bedoor inclined to addBased the probability of afollowing king andare thethe probability of a heart But thisto have data files available to import S O L U T IIfdoor Owe N openers: install 10 28, 32, 44, 40, 54, 38, 32, and creates a problem that, the king24, of 46, hearts is counted with the42. kings and also into statistical software They are Theifsample of eight companies in the aerospace industry, to in with the50 hearts So, we simply addmodal the probability king (there are 4referred in aindeck ofExer52 The mean, median, and amountsofofa profit are reported the following cise 38, was of surveyed as to their return on investment last year The results are indicated with the FILE icon cards) to the probability a heart (there are shot) 13 in a(Reminder: deck of 52The cards) and report 17 output (highlighted in the screen instructions tothat create the 10.6, 12.6, 14.8, 18.2, 12.0, 14.8, 12.2, and 15.6 out of 52 cards meet the requirement, we have counted the king of hearts twice We output appear in the Software Commands in Appendix C.) There are 180 vehicles Answers to the odd-numbered 51 The Houston, Texas, Motel Owner Association conducted a survey regarding need to subtract card 17 theListed king below ofbehearts is counted once inweekday the study, sofrom using aincalculator would tedious and prone tobusiness-class error Thus, motel ratesthe the so area is the room rateonly for exercises are in Appendix D there are 16 cards that are either hearts or kings So the probability is 16/52 = 3077 We can use a statistical software package to find many measures of location guests for a sample of 10 motels Card $101 $97 $103 $110 Probability $78 $87 $101 $80 Explanation $106 $88 A consumer organization is concerned credit card debt A P(A)watchdog = 4/52 kings in a deckabout of 52 cards survey of 10 young debt of more than $2,000 Heart P(B) adults=with 13/52credit card13 hearts in a deck of 52 cards showed they paid an averageP(A of and justB)over $100 per month balances Listed below King of Hearts = 1/52 king ofagainst hearts intheir a deck of 52 cards are the amounts each young adult paid last month 52.King Computer Output $110 $126 $103 $93 $99 $113 $87 The text includes many software examples, using Excel, MegaStat®, and Minitab The software results are LO3-5 for a particular illustrated in the chapters Instructions INTERPRETATION AND USES software example are in AppendixExplain C and apply Chebyshev’s theorem OF THE STANDARD DEVIATION and the Empirical Rule Lin66360_ch05_132-174.indd 145 STATISTICS IN ACTION Most colleges report the “average class size.” This information can be mislead- $101 $109 $100 The standard deviation is commonly used as a measure to compare the spread in two or more sets of observations For example, the standard deviation of the biweekly amounts invested in the Dupree Paint Company profit-sharing plan is computed1/10/17 to be 7:41 PM $7.51 Suppose these employees are located in Georgia If the standard deviation for a group of employees in Texas is $10.47, and the means are about the same, it indicates that the amounts invested by the Georgia employees are not dispersed as much as those in Texas (because $7.51 < $10.47) Since the amounts invested by the Georgia employees are clustered more closely about the mean, the mean for the Georgia emix ployees is a more reliable measure than the mean for the Texas group Chebyshev’s Theorem We have stressed that a small standard deviation for a set of values indicates that these www.freebookslides.com 59 DESCRIBING DATA: NUMERICAL MEASURES The major properties of the median are: It is not affected by extremely large or small values Therefore, the median is a valuable measure of location when such values occur It can be computed for ordinal-level data or higher Recall from Chapter that ordinal-level data can be ranked from low to high The Mode The mode is another measure of location MODE The value of the observation that appears most frequently The mode is especially useful in summarizing nominal-level data As an example of its use for nominal-level data, a company has developed five bath oils The bar chart in Chart 3–1 shows the results of a marketing survey designed to find which bath oil consumers prefer The largest number of respondents favored Lamoure, as evidenced by the highest bar Thus, Lamoure is the mode Number of Responses 400 300 200 100 Amor Lamoure Mode Soothing Smell Nice Far Out Bath oil CHART 3–1 Number of Respondents Favoring Various Bath Oils EXAMPLE Recall the data regarding the distance in miles between exits on I-75 in Kentucky The information is repeated below 11 4 10 4 9 3 8 10 3 14 1 10 3 5 2 2 5 6 1 2 2 3 7 1 3 7 8 10 1 4 7 5 2 2 5 1 1 3 3 1 2 1 What is the modal distance? SOLUTION The first step is to organize the distances into a frequency table This will help us determine the distance that occurs most frequently www.freebookslides.com 60 CHAPTER Distance in Miles between Exits Frequency 10 11 14 8 7 7 3 4 1 3 2 1 4 1 1 Total 42 The distance that occurs most often is mile This happens eight times—that is, there are eight exits that are mile apart So the modal distance between exits is mile Which of the three measures of location (mean, median, or mode) best represents the central location of these data? Is the mode the best measure of location to represent the Kentucky data? No The mode assumes only the nominal scale of measurement and the variable miles is measured using the ratio scale We calculated the mean to be 4.57 miles See page 54 Is the mean the best measure of location to represent these data? Probably not There are several cases in which the distance between exits is large These values are affecting the mean, making it too large and not representative of the distances between exits What about the median? The median distance is miles That is, half of the distances between exits are miles or less In this case, the median of miles between exits is probably a more representative measure of the distance between exits In summary, we can determine the mode for all levels of data—nominal, ordinal, interval, and ratio The mode also has the advantage of not being affected by extremely high or low values The mode does have disadvantages, however, that cause it to be used less frequently than the mean or median For many sets of data, there is no mode because no value appears more than once For example, there is no mode for this set of price data because every value occurs once: $19, $21, $23, $20, and $18 Conversely, for some data sets there is more than one mode Suppose the ages of the individuals in a stock investment club are 22, 26, 27, 27, 31, 35, and 35 Both the ages 27 and 35 are modes Thus, this grouping of ages is referred to as bimodal (having two modes) One would question the use of two modes to represent the location of this set of age data SELF-REVIEW 3–2 A sample of single persons in Towson, Texas, receiving Social Security payments revealed these monthly benefits: $852, $598, $580, $1,374, $960, $878, and $1,130 (a) What is the median monthly benefit? (b) How many observations are below the median? Above it? The number of work stoppages in the United States over the last 10 years are 22, 20, 21, 15, 5, 11, 19, 19, 15, and 11. (a) What is the median number of stoppages? (b) How many observations are below the median? Above it? (c) What is the modal number of work stoppages? www.freebookslides.com 61 DESCRIBING DATA: NUMERICAL MEASURES EXERCISES 13 What would you report as the modal value for a set of observations if there were a total of: a 10 observations and no two values were the same? b observations and they were all the same? c observations and the values were 1, 2, 3, 3, 4, and 4? For Exercises 14–16, determine the (a) mean, (b) median, and (c) mode 14 The following is the number of oil changes for the last days at the Jiffy Lube located at the corner of Elm Street and Pennsylvania Avenue 41 15 39 54 31 15 33 15 The following is the percent change in net income from last year to this year for a sample of 12 construction companies in Denver. 5 1 −10 −6 5 12 7 8 6 5 −1 11 16 The following are the ages of the 10 people in the Java Coffee Shop at the Southwyck Shopping Mall at 10 a.m 21 41 20 23 24 33 37 42 23 29 17 Several indicators of long-term economic growth in the United States and their annual percent change are listed below Economic Indicator Inflation Exports Imports Real disposable income Consumption Percent Change 4.5% 4.7 2.3 2.9 2.7 Economic Indicator Real GNP Investment (residential) Investment (nonresidential) Productivity (total) Productivity (manufacturing) Percent Change 2.9% 3.6 2.1 1.4 5.2 a What is the median percent change? b What is the modal percent change? 18 Sally Reynolds sells real estate along the coastal area of Northern California Below are her total annual commissions between 2005 and 2015 Find the mean, median, and mode of the commissions she earned for the 11 years Year Amount (thousands) 2005 292.16 2006 233.80 2007 206.97 2008 202.67 2009 164.69 2010 206.53 2011 237.51 2012 225.57 2013 255.33 2014 248.14 2015 269.11 19 The accounting firm of Rowatti and Koppel specializes in income tax returns for self-employed professionals, such as physicians, dentists, architects, and lawyers The firm employs 11 accountants who prepare the returns For last year, the number of returns prepared by each accountant was: 58 75 31 58 46 65 60 71 45 58 80 www.freebookslides.com 62 CHAPTER Find the mean, median, and mode for the number of returns prepared by each accountant If you could report only one, which measure of location would you recommend reporting? 20 The demand for the video games provided by Mid-Tech Video Games Inc has exploded in the last several years Hence, the owner needs to hire several new technical people to keep up with the demand Mid-Tech gives each applicant a special test that Dr McGraw, the designer of the test, believes is closely related to the ability to create video games For the general population, the mean on this test is 100 Below are the scores on this test for the applicants 95 105 120 81 90 115 99 100 130 10 The president is interested in the overall quality of the job applicants based on this test Compute the mean and the median scores for the 10 applicants What would you report to the president? Does it seem that the applicants are better than the general population? The Relative Positions of the Mean, Median, and Mode Frequency Refer to the histogram in Chart 3–2 It is a symmetric distribution, which is also moundshaped This distribution has the same shape on either side of the center If the histogram were folded in half, the two halves would be identical For any symmetric distribution, the mode, median, and mean are located at the center and are always equal They are all equal to 30 years in Chart 3–2 We should point out that there are symmetric distributions that are not mound-shaped Mean = 30 Median = 30 Mode = 30 Age CHART 3–2 A Symmetric Distribution The number of years corresponding to the highest point of the curve is the mode (30 years) Because the distribution is symmetrical, the median corresponds to the point where the distribution is cut in half (30 years) Also, because the arithmetic mean is the balance point of a distribution (as shown in the Properties of the Arithmetic Mean section on page 56), and the distribution is symmetric, the arithmetic mean is 30 Logically, any of the three measures would be appropriate to represent the distribution’s center If a distribution is nonsymmetrical, or skewed, the relationship among the three measures changes In a positively skewed distribution, such as the distribution of weekly income in Chart 3–3, the arithmetic mean is the largest of the three measures Why? Because the mean is influenced more than the median or mode by a few extremely high values The median is generally the next largest measure in a positively skewed frequency distribution The mode is the smallest of the three measures If the distribution is highly skewed, the mean would not be a good measure to use The median and mode would be more representative www.freebookslides.com 63 Frequency DESCRIBING DATA: NUMERICAL MEASURES Weekly Income Mode = 25 Median = 29 Mean = 60 CHART 3–3 A Positively Skewed Distribution Frequency Conversely, if a distribution is negatively skewed, such as the distribution of tensile strength in Chart 3–4, the mean is the lowest of the three measures The mean is, of course, influenced by a few extremely low observations The median is greater than the arithmetic mean, and the modal value is the largest of the three measures Again, if the distribution is highly skewed, the mean should not be used to represent the data Tensile Strength Mean = 45 Median = 76 Mode = 80 CHART 3–4 A Negatively Skewed Distribution SELF-REVIEW 3–3 The weekly sales from a sample of Hi-Tec electronic supply stores were organized into a frequency distribution The mean of weekly sales was computed to be $105,900, the median $105,000, and the mode $104,500 (a) Sketch the sales in the form of a smoothed frequency polygon Note the location of the mean, median, and mode on the X-axis. (b) Is the distribution symmetrical, positively skewed, or negatively skewed? Explain. EXERCISES 21 The unemployment rate in the state of Alaska by month is given in the table below: Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 8.7 8.8 8.7 7.8 7.3 7.8 6.6 6.5 6.5 6.8 7.3 7.6 www.freebookslides.com 64 CHAPTER a What is the arithmetic mean of the Alaska unemployment rates? b Find the median and the mode for the unemployment rates. c Compute the arithmetic mean and median for just the winter (Dec–Mar) months 22 Is it much different? Big Orange Trucking is designing an information system for use in “in-cab” communications It must summarize data from eight sites throughout a region to describe typical conditions Compute an appropriate measure of central location for the variables wind direction, temperature, and pavement City Anniston, AL Atlanta, GA Augusta, GA Birmingham, AL Jackson, MS Meridian, MS Monroe, LA Tuscaloosa, AL Wind Direction Temperature Pavement West Northwest Southwest South Southwest South Southwest Southwest 89 86 92 91 92 92 93 93 Dry Wet Wet Dry Dry Trace Wet Trace Software Solution We can use a statistical software package to find many measures of location EXAMPLE Table 2–4 on page 26 shows the profit on the sales of 180 vehicles at Applewood Auto Group Determine the mean and the median selling price SOLUTION The mean, median, and modal amounts of profit are reported in the following output (highlighted in the screen shot) (Reminder: The instructions to create the output appear in the Software Commands in Appendix C.) There are 180 vehicles in the study, so using a calculator would be tedious and prone to error www.freebookslides.com 65 DESCRIBING DATA: NUMERICAL MEASURES The mean profit is $1,843.17 and the median is $1,882.50 These two values are less than $40 apart, so either value is reasonable We can also see from the Excel output that there were 180 vehicles sold and their total profit was $331,770.00 We will describe the meaning of standard error, standard deviation, and other measures reported on the output later in this chapter and in later chapters What can we conclude? The typical profit on a vehicle is about $1,850 Management at Applewood might use this value for revenue projections For example, if the dealership could increase the number of vehicles sold in a month from 180 to 200, this would result in an additional estimated $37,000 of revenue, found by 20($1,850) LO3-2 Compute a weighted mean THE WEIGHTED MEAN The weighted mean is a convenient way to compute the arithmetic mean when there are several observations of the same value To explain, suppose the nearby Wendy’s Restaurant sold medium, large, and Biggie-sized soft drinks for $1.84, $2.07, and $2.40, respectively Of the last 10 drinks sold, were medium, were large, and were Biggiesized To find the mean price of the last 10 drinks sold, we could use formula (3–2) $1.84 + $1.84 + $1.84 + $2.07 + $2.07 + $2.07 + $2.07 + $2.40 + $2.40 + $2.40 10 $21.00 x= = $2.10 10 x= The mean selling price of the last 10 drinks is $2.10 An easier way to find the mean selling price is to determine the weighted mean That is, we multiply each observation by the number of times it occurs We will refer to the weighted mean as x W This is read “x bar sub w.” xw = 3($1.84) + 4($2.07) + 3($2.40) $21.00 = = $2.10 10 10 In this case, the weights are frequency counts However, any measure of importance could be used as a weight In general, the weighted mean of a set of numbers designated x1, x2, x3, , xn with the corresponding weights w1, w2, w3, , wn is computed by: WEIGHTED MEAN xw = w1x1 + w2x2 + w3x3 + … + wnxn w1 + w + w + … + w n This may be shortened to: xw = (3–3) Σ (wx) Σw Note that the denominator of a weighted mean is always the sum of the weights EXAMPLE The Carter Construction Company pays its hourly employees $16.50, $19.00, or $25.00 per hour There are 26 hourly employees, 14 of whom are paid at the $16.50 rate, 10 at the $19.00 rate, and at the $25.00 rate What is the mean hourly rate paid the 26 employees? www.freebookslides.com 66 CHAPTER SOLUTION To find the mean hourly rate, we multiply each of the hourly rates by the number of employees earning that rate From formula (3–3), the mean hourly rate is xw = 14($16.50) + 10($19.00) + 2($25.00) $471.00 = = $18.1154 14 + 10 + 26 The weighted mean hourly wage is rounded to $18.12 SELF-REVIEW 3–4 Springers sold 95 Antonelli men’s suits for the regular price of $400 For the spring sale, the suits were reduced to $200 and 126 were sold At the final clearance, the price was reduced to $100 and the remaining 79 suits were sold (a) What was the weighted mean price of an Antonelli suit? (b) Springers paid $200 a suit for the 300 suits Comment on the store’s profit per suit if a salesperson receives a $25 commission for each one sold. EXERCISES 23 In June, an investor purchased 300 shares of Oracle (an information technology company) stock at $20 per share In August, she purchased an additional 400 shares at $25 per share In November, she purchased an additional 400 shares, but the stock declined to $23 per share What is the weighted mean price per share? 24 The Bookstall Inc is a specialty bookstore concentrating on used books sold via the Internet Paperbacks are $1.00 each, and hardcover books are $3.50 Of the 50 books sold last Tuesday morning, 40 were paperback and the rest were hardcover What was the weighted mean price of a book? 25 The Loris Healthcare System employs 200 persons on the nursing staff Fifty are nurse’s aides, 50 are practical nurses, and 100 are registered nurses Nurse’s aides receive $8 an hour, practical nurses $15 an hour, and registered nurses $24 an hour What is the weighted mean hourly wage? 26 Andrews and Associates specialize in corporate law They charge $100 an hour for researching a case, $75 an hour for consultations, and $200 an hour for writing a brief Last week one of the associates spent 10 hours consulting with her client, 10 hours researching the case, and 20 hours writing the brief What was the weighted mean hourly charge for her legal services? LO3-3 Compute and interpret the geometric mean THE GEOMETRIC MEAN The geometric mean is useful in finding the average change of percentages, ratios, indexes, or growth rates over time It has a wide application in business and economics because we are often interested in finding the percentage changes in sales, salaries, or economic figures, such as the gross domestic product, which compound or build on each other The geometric mean of a set of n positive numbers is defined as the nth root of the product of n values The formula for the geometric mean is written: GEOMETRIC MEAN n GM = √ (x1 ) (x2 ) … (xn ) (3–4) The geometric mean will always be less than or equal to (never more than) the arithmetic mean Also, all the data values must be positive As an example of the geometric mean, suppose you receive a 5% increase in salary this year and a 15% increase next year The average annual percent increase is 9.886%, www.freebookslides.com DESCRIBING DATA: NUMERICAL MEASURES 67 not 10.0% Why is this so? We begin by calculating the geometric mean Recall, for example, that a 5% increase in salary is 105% We will write it as 1.05 GM = √ (1.05) (1.15) = 1.09886 This can be verified by assuming that your monthly earning was $3,000 to start and you received two increases of 5% and 15% Raise = $3,000(.05) = $150.00 Raise = $3,150(.15) = 472.50 Total $622.50 Your total salary increase is $622.50 This is equivalent to: $3,000.00(.09886) = $296.59 $3,296.58(.09886) = 325.91 $622.50 The following example shows the geometric mean of several percentages EXAMPLE The return on investment earned by Atkins Construction Company for four successive years was 30%, 20%, −40%, and 200% What is the geometric mean rate of return on investment? SOLUTION The number 1.3 represents the 30% return on investment, which is the “original” investment of 1.0 plus the “return” of 0.3 The number 0.6 represents the loss of 40%, which is the original investment of 1.0 less the loss of 0.4 This calculation assumes the total return each period is reinvested or becomes the base for the next period In other words, the base for the second period is 1.3 and the base for the third period is (1.3)(1.2) and so forth Then the geometric mean rate of return is 29.4%, found by n 4 GM = √ (x1 ) (x2 ) … (xn ) = √ (1.3) (1.2) (0.6) (3.0) = √ 2.808 = 1.294 The geometric mean is the fourth root of 2.808 So, the average rate of return (compound annual growth rate) is 29.4% Notice also that if you compute the arithmetic mean [(30 + 20 − 40 + 200)/4 = 52.5], you would have a much larger number, which would overstate the true rate of return! A second application of the geometric mean is to find an average percentage change over a period of time For example, if you earned $45,000 in 2004 and $100,000 in 2016, what is your annual rate of increase over the period? It is 6.88% The rate of increase is determined from the following formula RATE OF INCREASE OVER TIME n Value at end of period GM = √ − Value at start of period (3–5) In formula 3-5 above, n is the number of periods An example will show the details of finding the average annual percent increase www.freebookslides.com 68 CHAPTER EXAMPLE During the decade of the 1990s, and into the 2000s, Las Vegas, Nevada, was one of the fastest-growing cities in the United States The population increased from 258,295 in 1990 to 613,599 in 2014 This is an increase of 355,304 people, or a 137.56% increase over the period The population has more than doubled What is the average annual percent increase? SOLUTION There are 24 years between 1990 and 2014, so n = 24 Then the geometric mean formula (3–5) as applied to this problem is: 24 613,599 n Value at end of period GM = √ − 1.0 = √ − 1.0 = 1.0367 − 1.0 = 0367 Value at start of period 258,295 To summarize, the steps to compute the geometric mean are: Divide the value at the end of the period by the value at the beginning of the period Find the nth root of the ratio, where n is the number of periods Subtract one The value of 0367 indicates that the average annual growth over the period was 3.67% To put it another way, the population of Las Vegas increased at a rate of 3.67% per year from 1990 to 2014 SELF-REVIEW 3–5 The percent increase in sales for the last years at Combs Cosmetics were 4.91, 5.75, 8.12, and 21.60 (a) Find the geometric mean percent increase. (b) Find the arithmetic mean percent increase. (c) Is the arithmetic mean equal to or greater than the geometric mean? Production of Cablos trucks increased from 23,000 units in 1996 to 120,520 in 2016 Find the geometric mean annual percent increase. EXERCISES 27 Compute the geometric mean of the following monthly percent increases: 8, 12, 14, 26, and 5. 28 Compute the geometric mean of the following weekly percent increases: 2, 8, 6, 4, 10, 6, 8, and 29 Listed below is the percent increase in sales for the MG Corporation over the last years Determine the geometric mean percent increase in sales over the period. 9.4 13.8 11.7 11.9 14.7 30 In 2001, a total of 40,244,000 taxpayers in the United States filed their individual tax returns electronically By the year 2015, the number increased to 128,653,000 What is the geometric mean annual increase for the period? 31 The Consumer Price Index is reported monthly by the U.S Bureau of Labor Statistics It reports the change in prices for a market basket of goods from one period to another The index for 2000 was 172.2 By 2015, it increased to 236.525 What was the geometric mean annual increase for the period? 32 JetBlue Airways is an American low-cost airline headquartered in New York City Its main base is John F Kennedy International Airport JetBlue’s revenue in 2002 was $635.2 million By 2014, revenue had increased to $5,817.0 million What was the geometric mean annual increase for the period? www.freebookslides.com 69 DESCRIBING DATA: NUMERICAL MEASURES 33 In 2000, there were 720,000 cell phone subscribers worldwide By 2015, the num- ber of cell phone subscribers increased to 752,000,000 What is the geometric mean annual increase for the period? 34 The information below shows the cost for a year of college in public and private colleges in 2002–03 and 2015–16 What is the geometric mean annual increase for the period for the two types of colleges? Compare the rates of increase STATISTICS IN ACTION The U.S Postal Service has tried to become more “user friendly” in the last several years A recent survey showed that customers were interested in more consistency in the time it takes to make a delivery Under the old conditions, a local letter might take only one day to deliver, or it might take several “Just tell me how many days ahead I need to mail the birthday card to Mom so it gets there on her birthday, not early, not late,” was a common complaint The level of consistency is measured by the standard deviation of the delivery times 2002–03 2015–16 Public Private $ 4,960 18,056 $23,893 32,405 WHY STUDY DISPERSION? A measure of location, such as the mean, median, or mode, only describes the center of the data It is valuable from that standpoint, but it does not tell us anything about the spread of the data For example, if your nature guide told you that the river ahead averaged feet in depth, would you want to wade across on foot without additional information? Probably not You would want to know something about the variation in the depth Is the maximum depth of the river 3.25 feet and the minimum 2.75 feet? If that is the case, you would probably agree to cross What if you learned the river depth ranged from 0.50 foot to 5.5 feet? Your decision would probably be not to cross Before making a decision about crossing the river, you want information on both the typical depth and the dispersion in the depth of the river A small value for a measure of dispersion indicates that the data are clustered closely, say, around the arithmetic mean The mean is therefore considered representative of the data Conversely, a large measure of dispersion indicates that the mean is not reliable Refer to Chart 3–5 The 100 employees of Hammond Iron Works Inc., a steel fabricating company, are organized into a histogram based on the number of years of employment with the company The mean is 4.9 years, but the spread of the data is from months to 16.8 years The mean of 4.9 years is not very representative of all the employees 20 Employees LO3-4 Compute and interpret the range, variance, and standard deviation Type of College 10 0 10 Years 20 CHART 3–5 Histogram of Years of Employment at Hammond Iron Works Inc A second reason for studying the dispersion in a set of data is to compare the spread in two or more distributions Suppose, for example, that the new Vision Quest LCD computer monitor is assembled in Baton Rouge and also in Tucson The arithmetic mean hourly output in both the Baton Rouge plant and the Tucson plant is 50 Based on www.freebookslides.com 70 CHAPTER Baton Rouge 48 49 50 _ X 51 52 Tucson 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 _ X Hourly Production CHART 3–6 Hourly Production of Computer Monitors at the Baton Rouge and Tucson Plants the two means, you might conclude that the distributions of the hourly outputs are identical Production records for hours at the two plants, however, reveal that this conclusion is not correct (see Chart 3–6) Baton Rouge production varies from 48 to 52 assemblies per hour Production at the Tucson plant is more erratic, ranging from 40 to 60 per hour Therefore, the hourly output for Baton Rouge is clustered near the mean of 50; the hourly output for Tucson is more dispersed We will consider several measures of dispersion The range is based on the maximum and minimum values in the data set; that is, only two values are considered The variance and the standard deviation use all the values in a data set and are based on deviations from the arithmetic mean Range The simplest measure of dispersion is the range It is the difference between the maximum and minimum values in a data set In the form of an equation: RANGE Range = Maximum value − Minimum value (3–6) The range is widely used in production management and control applications because it is very easy to calculate and understand EXAMPLE Refer to Chart 3–6 above Find the range in the number of computer monitors produced per hour for the Baton Rouge and the Tucson plants Interpret the two ranges SOLUTION The range of the hourly production of computer monitors at the Baton Rouge plant is 4, found by the difference between the maximum hourly production of 52 and www.freebookslides.com 71 DESCRIBING DATA: NUMERICAL MEASURES the minimum of 48 The range in the hourly production for the Tucson plant is 20 computer monitors, found by 60 − 40 We therefore conclude that (1) there is less dispersion in the hourly production in the Baton Rouge plant than in the Tucson plant because the range of computer monitors is less than a range of 20 computer monitors and (2) the production is clustered more closely around the mean of 50 at the Baton Rouge plant than at the Tucson plant (because a range of is less than a range of 20) Thus, the mean production in the Baton Rouge plant (50 computer monitors) is a more representative measure of location than the mean of 50 computer monitors for the Tucson plant Variance A limitation of the range is that it is based on only two values, the maximum and the minimum; it does not take into consideration all of the values The variance does It measures the mean amount by which the values in a population, or sample, vary from their mean In terms of a definition: VARIANCE The arithmetic mean of the squared deviations from the mean The following example illustrates how the variance is used to measure dispersion EXAMPLE The chart below shows the number of cappuccinos sold at the Starbucks in the Orange County airport and the Ontario, California, airport between 4 and p.m for a sample of days last month © Sorbis/Shutterstock.com Determine the mean, median, range, and variance for each location Comment on the similarities and differences in these measures SOLUTION The mean, median, and range for each of the airport locations are reported as part of an Excel spreadsheet www.freebookslides.com 72 CHAPTER Notice that all three of the measures are exactly the same Does this indicate that there is no difference in the two sets of data? We get a clearer picture if we calculate the variance First, for Orange County: Variance = Σ(x − μ) (−302 ) + (−102 ) + 02 + 102 + 302 2,000 = = = 400 N 5 The variance is 400 That is, the average squared deviation from the mean is 400 The following shows the detail of determining the variance for the number of cappuccinos sold at the Ontario Airport Variance = Σ(x − μ) (−302 ) + (−52 ) + 02 + 52 + 302 1,850 = = = 370 N 5 So the mean, median, and range of the cappuccinos sold are the same at the two airports, but the variances are different The variance at Orange County is 400, but it is 370 at Ontario Let’s interpret and compare the results of our measures for the two Starbucks airport locations The mean and median of the two locations are exactly the same, 50 cappuccinos sold These measures of location suggest the two distributions are the same The range for both locations is also the same, 60 However, recall that www.freebookslides.com DESCRIBING DATA: NUMERICAL MEASURES 73 the range provides limited information about the dispersion because it is based on only two values, the minimum and maximum. The variances are not the same for the two airports The variance is based on the differences between each observation and the arithmetic mean It shows the closeness or clustering of the data relative to the mean or center of the distribution Compare the variance for Orange County of 400 to the variance for Ontario of 370 Based on the variance, we conclude that the dispersion for the sales distribution of the Ontario Starbucks is more concentrated—that is, nearer the mean of 50—than for the Orange County location The variance has an important advantage over the range It uses all the values in the computation Recall that the range uses only the highest and the lowest values SELF-REVIEW 3–6 The weights of containers being shipped to Ireland are (in thousands of pounds): 95 103 105 110 104 105 112 90 (a) What is the range of the weights? (b) Compute the arithmetic mean weight. (c) Compute the variance of the weights. EXERCISES For Exercises 35–38, calculate the (a) range, (b) arithmetic mean, (c) variance, and (d) interpret the statistics 35 36 37 38 39 During last weekend’s sale, there were five customer service representatives on duty at the Electronic Super Store The numbers of HDTVs these representatives sold were 5, 8, 4, 10, and 3. The Department of Statistics at Western State University offers eight sections of basic statistics Following are the numbers of students enrolled in these sections: 34, 46, 52, 29, 41, 38, 36, and 28 Dave’s Automatic Door installs automatic garage door openers The following list indicates the number of minutes needed to install 10 door openers: 28, 32, 24, 46, 44, 40, 54, 38, 32, and 42. All eight companies in the aerospace industry were surveyed as to their return on investment last year The results are: 10.6%, 12.6%, 14.8%, 18.2%, 12.0%, 14.8%, 12.2%, and 15.6% Ten young adults living in California rated the taste of a newly developed sushi pizza topped with tuna, rice, and kelp on a scale of to 50, with indicating they did not like the taste and 50 that they did The ratings were: 34 39 40 46 33 31 34 14 15 45 In a parallel study, 10 young adults in Iowa rated the taste of the same pizza The ratings were: 28 25 35 16 25 29 24 26 17 20 As a market researcher, compare the potential for sushi pizza in the two markets 40 The personnel files of all eight employees at the Pawnee location of Acme Carpet Cleaners Inc revealed that during the last 6-month period they lost the following number of days due to illness: 2 0 6 3 10 4 1 2 ... 2 015 2 014 2,562,840 2,434,707 2 ,17 8,587 2,065, 612 2,0 71, 446 1, 975,368 1, 814 ,268 1, 687, 313 1, 320, 217 1, 2 81, 777 1, 238,535 1, 166,389 638 ,19 5 607,539 526,024 489, 711 480,3 31 418 ,497 294,602 3 01, 187... 2009 19 .3 2 010 30.5 10 2 011 41. 1 11 2 012 44.9 12 2 013 32.6 13 2 014 32.5 20 10 2 014 30 2 013 26.5 2 012 2007 H 40 2 011 G 2 010 26.2 2009 2006 2008 24.3 F ExxonMobile Annual Earnings 50 2007 16 .7... Terrence St Julian 19 95 To better understand customers, Mr Selig 883 3 ,14 0 299 2 ,19 7 17 5 15 9the1 ,10 5 434 615 14 9 asked 11 .65 1, 116 6 81 1,294 12 754 1, 206 1, 448 870 90.8 944 1, 255 and is run today