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(BQ) Part 1 book Statistical techniques in business & economics has contents: What is statistics, describing data - numerical measures, describing data - displaying and exploring data, a survey of probability concepts, discrete probability distributions, sampling methods and the central limit theorem
www.downloadslide.com Statistical Techniques in Business & Economics Seventeenth Edition LIND MARCHAL WATHEN www.downloadslide.com Statistical Techniques in BUSINESS & ECONOMICS www.downloadslide.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? A Primer on SAP Implementation First Edition Schroeder, Goldstein, and Rungtusanatham Operations Management in the Supply Chain: Decisions and Cases Sixth Edition Stevenson Operations Management Eleventh Edition BUSINESS MATH Slater and Wittry Practical Business Math Procedures Eleventh Edition Slater and Wittry Practical Business Math Procedures, Brief Edition Eleventh Edition BUSINESS STATISTICS Bowerman, O’Connell, and Murphree Business Statistics in Practice Seventh Edition Bowerman, O’Connell, Murphree, and Orris Essentials of Business Statistics Fourth Edition Doane and Seward Applied Statistics in Business and Economics Fourth Edition Lind, Marchal, and Wathen Basic Statistics for Business and Economics Eighth Edition Lind, Marchal, and Wathen Statistical Techniques in Business and Economics Seventeenth Edition Jaggia and Kelly Business Statistics: Communicating with Numbers First Edition Jaggia and Kelly Essentials of Business Statistics: Communicating with Numbers First Edition www.downloadslide.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.downloadslide.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 This book is printed on acid-free paper LWI 21 20 19 18 17 16 ISBN 978-1-259-66636-0 MHID 1-259-66636-0 Chief Product Officer, SVP Products & Markets: G Scott Virkler Vice President, General Manager, Products & Markets: Marty Lange Vice President, Content Design & Delivery: Betsy Whalen Managing Director: Tim Vertovec Senior Brand Manager: Charles Synovec Director, Product Development: Rose Koos Product Developers: Michele Janicek / Ryan McAndrews Senior Director, Digital Content Development: Douglas Ruby Marketing Manager: Trina Maurer Director, Content Design & Delivery: Linda Avenarius Program Manager: Mark Christianson Content Project Managers: Harvey Yep (Core) / Bruce Gin (Assessment) Buyer: Susan K Culbertson Design: Matt Backhaus Cover Image: © Corbis / Glow Images Content Licensing Specialists: Melissa Homer (Image) / Beth Thole (Text) Typeface: 9.5/11 Proxima Nova Compositor: Aptara®, Inc Printer: LSC Communications All credits appearing on page or at the end of the book are 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.downloadslide.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.downloadslide.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.downloadslide.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.downloadslide.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 swordtails (44 female and 36 male) 60 are orange swordtails (36 female and www.downloadslide.com TheandGeneral 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.downloadslide.com 338 CHAPTER 10 know if the mean time in the lot is more than 15 minutes A sample of 12 recent customers showed they were in the lot the following lengths of time, in minutes 30 24 28 22 14 2 39 23 23 28 12 31 At the 05 significance level, is it reasonable to conclude that the mean time in the lot is more than 15 minutes? SOLUTION We begin by stating the null hypothesis and the alternate hypothesis In this case, the question is whether the population mean could be more than 15 minutes So this is a one-tailed test We state the two hypotheses as follows: H0: μ ≤ 15 H1: μ > 15 There are 11 degrees of freedom, found by n − = 12 − = 11 The critical t value is 1.796, found by referring to Appendix B.5 for a one-tailed test, using α = 05 with 11 degrees of freedom The decision rule is: Reject the null hypothesis if the computed t is greater than 1.796 This information is summarized in Chart 10–7 Rejection region a = 05 1.796 Scale of t CHART 10–7 Rejection Region, One-Tailed Test, Student’s t Distribution, α = 05 We calculate the sample mean using formula (3–2) and the sample standard deviation using formula (3–8) The sample mean is 23 minutes, and the sample standard deviation is 9.835 minutes The details of the calculations are shown in Table 10–2 TABLE 10–2 Calculations of Sample Mean and Standard Deviation Parking Times Customer x, Minutes (x − x )2 Chmura Will Crompton Craver Cao Nowlin Esposito Colvard Hoefle Lawler Trask Grullon Total 30 24 28 22 14 39 23 23 28 12 31 276 49 25 81 441 256 0 25 121 64 1064 x= Σx 276 = = 23 n 12 s=√ Σ(x − x ) 1064 =√ = 9.835 n−1 12 − www.downloadslide.com 339 ONE-SAMPLE TESTS OF HYPOTHESIS Now we are ready to compute the value of t, using formula (10–2) t= x−μ s∕√n = 23 − 15 = 2.818 9.835∕√12 The null hypothesis that the population mean is less than or equal to 15 minutes is rejected because the computed t value of 2.818 lies in the area to the right of 1.796 We conclude that the time customers spend in the lot is more than 15 minutes This result indicates that the airport may need to add more parking places SELF-REVIEW 10–3 The mean life of a battery used in a digital clock is 305 days The lives of the batteries follow the normal distribution The battery was recently modified to last longer A sample of 20 of the modified batteries had a mean life of 311 days with a standard deviation of 12 days Did the modification increase the mean life of the battery? (a) State the null hypothesis and the alternate hypothesis (b) Show the decision rule graphically Use the 05 significance level (c) Compute the value of t What is your decision regarding the null hypothesis? Briefly summarize your results EXERCISES 9 Given the following hypotheses: H0: μ ≤ 10 H1: μ > 10 A random sample of 10 observations is selected from a normal population The sample mean was 12 and the sample standard deviation Using the 05 significance level: a State the decision rule b Compute the value of the test statistic c What is your decision regarding the null hypothesis? 10 Given the following hypotheses: H0: μ = 400 H1: μ ≠ 400 A random sample of 12 observations is selected from a normal population The sample mean was 407 and the sample standard deviation Using the 01 significance level: a State the decision rule b Compute the value of the test statistic c What is your decision regarding the null hypothesis? 11 The Rocky Mountain district sales manager of Rath Publishing Inc., a college textbook publishing company, claims that the sales representatives make an average of 40 sales calls per week on professors Several reps say that this estimate is too low To investigate, a random sample of 28 sales representatives reveals that the mean number of calls made last week was 42 The standard deviation of the sample is 2.1 calls Using the 05 significance level, can we conclude that the mean number of calls per salesperson per week is more than 40? 12 The management of White Industries is considering a new method of assembling its golf cart The present method requires a mean time of 42.3 minutes to assemble a cart The mean assembly time for a random sample of 24 carts, using the new method, was 40.6 minutes, and the standard deviation of the sample was 2.7 minutes Using the 10 level of significance, can we conclude that the assembly time using the new method is faster? www.downloadslide.com 340 CHAPTER 10 13 The mean income per person in the United States is $50,000, and the distribution of incomes follows a normal distribution A random sample of 10 residents of Wilmington, Delaware, had a mean of $60,000 with a standard deviation of $10,000 At the 05 level of significance, is that enough evidence to conclude that residents of Wilmington, Delaware, have more income than the national average? 14 Most air travelers now use e-tickets Electronic ticketing allows passengers to not worry about a paper ticket, and it costs the airline companies less to handle than paper ticketing However, in recent times the airlines have received complaints from passengers regarding their e-tickets, particularly when connecting flights and a change of airlines were involved To investigate the problem, an independent watchdog agency contacted a random s ample of 20 airports and collected information on the number of complaints the airport had with e-tickets for the month of March The information is reported below 14 14 16 12 12 14 13 16 15 14 12 15 15 14 13 13 12 13 10 13 At the 05 significance level, can the watchdog agency conclude the mean number of complaints per airport is less than 15 per month? a What assumption is necessary before conducting a test of hypothesis? b Plot the number of complaints per airport in a frequency distribution or a dot plot Is it reasonable to conclude that the population follows a normal distribution? c Conduct a test of hypothesis and interpret the results A Statistical Software Solution The Minitab statistical software system, used in earlier chapters and the previous section, provides an efficient method for conducting a one-sample test of hypothesis for a population mean The steps to generate the following output are shown in A ppendix C An additional feature of most statistical software packages is to report the p-value, which gives additional information on the null hypothesis The p-value is the probability of a t value as extreme or more extreme than the computed t value, given that the null hypothesis is true Using the Minitab analysis from the previous cell phone parking lot example, the p-value of 008 is the likelihood of a t value of 2.82 or larger, given a population mean of 15 Thus, comparing the p-value to the significance level tells us whether the null hypothesis was close to being rejected, barely rejected, and so on www.downloadslide.com 341 ONE-SAMPLE TESTS OF HYPOTHESIS To explain further, refer to the diagram below The p-value of 008 is the brown shaded area and the significance level is the total amber and brown shaded area Because the p-value of 008 is less than the significance level of 05, the null hypothesis is rejected Had the p-value been larger—say, 06, 19, or 57—than the significance level, the null hypothesis would not be rejected Rejection region a = 05 Scale of t 1.796 2.818 In the preceding example, the alternate hypothesis was one-sided, and the upper (right) tail of the t distribution contained the rejection region The p-value is the area to the right of 2.818 for a t distribution with 11 degrees of freedom What if we were conducting a two-sided test, so that the rejection region is in both the upper and the lower tails? That is, in the cell phone parking lot example, if H1 were stated as μ ≠ 15, we would have reported the p-value as the area to the right of 2.818 plus the value to the left of −2.818 Both of these values are 008, so the p-value is 008 + 008 = 016 How can we estimate a p-value without a computer? To illustrate, recall that, in the example/solution regarding the length of time at the cell phone parking lot, we rejected the null hypothesis that μ ≤ 15 and accepted the alternate hypothesis that μ > 15 The significance level was 05, so logically the p-value is less than 05. To estimate the p-value more accurately, go to Appendix B.5 and find the row with 11 degrees of freedom The computed t value of 2.818 is between 2.718 and 3.106 (A portion of Appendix B.5 is reproduced as Table 10–3.) The one-tailed significance level corresponding to 2.718 is 01, and for 3.106 it is 005 Therefore, the p-value is between 005 and 01 The usual practice is to report that the p-value is less than the larger of the two significance levels So we would report “the p-value is less than 01.” TABLE 10–3 A Portion of Student’s t Distribution Confidence Intervals 80% 90% 95% 98% 99% 99.9% Level of Significance for One-Tailed Test, α df 0.10 0.05 0025 0.01 0.005 0.0005 Level of Significance for Two-Tailed Test, α 0.20 0.10 0.05 0.02 0.01 0.001 ⋮ 10 ⋮ 1.383 1.372 ⋮ 1.833 1.812 ⋮ 2.262 2.228 ⋮ 2.821 2.764 ⋮ 3.250 3.169 ⋮ 4.781 4.587 11 12 13 14 15 1.363 1.356 1.350 1.345 1.341 1.796 1.782 1.771 1.761 1.753 2.201 2.179 2.160 2.145 2.131 2.718 2.681 2.650 2.624 2.602 3.106 3.055 3.012 2.977 2.947 4.437 4.318 4.221 4.140 4.073 www.downloadslide.com 342 SELF-REVIEW CHAPTER 10 10–4 A machine is set to fill a small bottle with 9.0 grams of medicine A sample of eight bottles revealed the following amounts (grams) in each bottle 9.2 8.7 8.9 8.6 8.8 8.5 8.7 9.0 At the 01 significance level, can we conclude that the mean weight is less than 9.0 grams? (a) State the null hypothesis and the alternate hypothesis (b) How many degrees of freedom are there? (c) Give the decision rule (d) Compute the value of t What is your decision regarding the null hypothesis? (e) Estimate the p-value EXERCISES 15 Given the following hypotheses: H0: μ ≥ 20 H1: μ > 20 A random sample of five resulted in the following values: 18, 15, 12, 19, and 21 Assume a normal population Using the 01 significance level, can we conclude the population mean is less than 20? a State the decision rule b Compute the value of the test statistic c What is your decision regarding the null hypothesis? d Estimate the p-value 16 Given the following hypotheses: H0: μ = 100 H1: μ ≠ 100 A random sample of six resulted in the following values: 118, 105, 112, 119, 105, and 111 Assume a normal population Using the 05 significance level, can we conclude the mean is different from 100? a State the decision rule b Compute the value of the test statistic c What is your decision regarding the null hypothesis? d Estimate the p-value 17 The amount of water consumed each day by a healthy adult follows a normal distribution with a mean of 1.4 liters A health campaign promotes the consumption of at least 2.0 liters per day A sample of 10 adults after the campaign shows the following consumption in liters: 1.5 1.6 1.5 1.4 1.9 1.4 1.3 1.9 1.8 1.7 At the 01 significance level, can we conclude that water consumption has increased? Calculate and interpret the p-value The liquid chlorine added to swimming pools to combat algae has a relatively 18 short shelf life before it loses its effectiveness Records indicate that the mean shelf life of a 5-gallon jug of chlorine is 2,160 hours (90 days) As an experiment, Holdlonger was added to the chlorine to find whether it would increase the shelf life A sample of nine jugs of chlorine had these shelf lives (in hours): 2,159 2,170 2,180 2,179 2,160 2,167 2,171 2,181 2,185 At the 025 level, has Holdlonger increased the shelf life of the chlorine? Estimate the p-value www.downloadslide.com ONE-SAMPLE TESTS OF HYPOTHESIS 19 343 A Washington, D.C., “think tank” announces the typical teenager sent 67 text messages per day in 2017 To update that estimate, you phone a sample of 12 teenagers and ask them how many text messages they sent the previous day Their responses were: 51 175 47 49 44 54 145 203 21 59 42 100 At the 05 level, can you conclude that the mean number is greater than 67? Estimate the p-value and describe what it tells you 20 Hugger Polls contends that an agent conducts a mean of 53 in-depth home surveys every week A streamlined survey form has been introduced, and Hugger wants to evaluate its effectiveness The number of in-depth surveys conducted during a week by a random sample of 15 agents are: 53 57 50 55 58 54 60 52 59 62 60 60 51 59 56 LO10-7 Compute the probability of a Type II error At the 05 level of significance, can we conclude that the mean number of interviews conducted by the agents is more than 53 per week? Estimate the p-value TYPE II ERROR Recall that the level of significance, identified by the symbol α, is the probability that the null hypothesis is rejected when it is true This is called a Type I error The most common levels of significance are 05 and 01 and are set by the researcher at the outset of the test In a hypothesis-testing situation there is also the possibility that a null hypothesis is not rejected when it is actually false This is called a Type II error The probability of a Type II error is identified by the Greek letter beta (β) In contrast to selecting a value for α in the hypothesis testing procedure, the value of β is calculated after the hypothesis testing procedure is finished The following example illustrates the details of determining the value of β EXAMPLE Western Wire Products purchases steel bars to make cotter pins Past experience indicates that the mean tensile strength of all incoming shipments is 10,000 psi and that the standard deviation, σ, is 400 psi To monitor the quality of the cotter pins, samples of 100 pins are randomly selected and tested for their strength In our hypothesis testing procedure the hypotheses are: H0: μ = 10,000 H1: μ ≠ 10,000 To determine if a shipment of steel bars meets the quality standard, Western Wire Products set up a rule for the quality-control inspector to follow: “Take a sample of 100 steel bars Test each of the bars for tensile strength Using a 05 significance level, accept the shipment if the sample mean ( x ) strength falls between 9,922 psi and 10,078 psi.” These values are the critical values for the hypothesis test If the sample mean is more than 10,078 or less than 9,922, the hypothesis is rejected and we conclude that the shipment does not meet the quality standard Refer to Chart 10–8, Graph A Given that the population mean is 10,000 psi, designated μ0, with a standard deviation of 400, the distribution shows the regions where the hypothesis is rejected and where it is not rejected, that is, whether the shipment meets the quality standard for tensile strength www.downloadslide.com 344 CHAPTER 10 Suppose that the result of testing 100 bars results in a sample mean of 9,900 psi Clearly, based on Graph A, the shipment does not meet the quality standard and is rejected with a 05 probability of a Type I error, a small chance of rejecting the shipment in error To calculate the Type II error, we assume that the sample mean of 9,900 psi is the true population mean Compare Graph A and Graph B Graph A represents the company’s distribution of tensile strength centered on 10,000 psi Graph B, based on the sample data, suggests that the distribution is centered on 9,900 psi Now, let’s use Graph B to determine the probability of a Type II error, β From the quality standards we know that 9,922 psi is used to reject the null hypothesis Any sample mean greater than 9,922 and less than 10,078 is accepted If the distribution is really centered on 9,900 psi, it is possible to find sample means more than 9,922, and we would fail to reject the null hypothesis, μ = 10,000 This is the area in Graph B labeled “Probability of β” where the null hypothesis would not be rejected The graph indicates that the probability of a Type II error is 2912 We can also calculate the power of the test as (1 − β) The power is the probability of not making a Type II error, rejecting the null hypothesis correctly Here, the power of the test is 7088 Graph A Reject lot Reject lot –1.96 9,922 x– 10,000 1.96 x– 10,078 psi Graph B 5000 2088 9,900 2912 9,922 xc psi CHART 10–8 Charts Showing Type I and Type II Errors SOLUTION The probability of committing a Type II error, is represented by the shaded area in Chart 10–8, Graph B It is computed by determining the area under the normal curve that lies above 9,922 pounds The calculation of the areas under the normal curve was discussed in Chapter Reviewing briefly, first determine the probability of the sample mean falling between 9,900 and 9,922 Then this probability is subtracted from 5000 (which represents all the area to the right of the mean of 9,900) to arrive at the probability of making a Type II error www.downloadslide.com 345 ONE-SAMPLE TESTS OF HYPOTHESIS The number of standard errors (z value) between the mean of the incoming lot (9,900), designated by μ1, and xC , representing the critical value for 9,922, is computed by: z= TYPE II ERROR xc − μ1 σ∕√n (10–3) With n = 100 and σ = 400, the value of z is 0.55: xc − μ1 9,922 − 9,900 22 = = = 0.55 40 σ∕√n 400∕√100 The area under the curve between 9,900 and 9,922 (a z value of 0.55) is 2088 The area under the curve beyond 9,922 pounds is 5000 − 2088, or 2912; this is the probability of making a Type II error—that is, accepting an incoming lot of steel bars, based on the company’s standard of 10,000 psi, when the sample suggests that the population mean is 9,900 psi and the shipment should be rejected Let’s use another example, illustrated in Chart 10–9 Suppose the testing of a sample of 100 bars results in a mean of 10,120 psi Based on the company’s standards, represented in Graph A, the shipment should be rejected The sample mean of 10,120 is more than the critical value of 10,078 Based on the sample mean, Graph C shows the distribution of sample means based on the distribution centered on the sample mean, 10,120 If the distribution is centered on the sample mean of 10,120, there is a probability that a sample mean could be less than the upper critical value of 10,078 In Graph C, this is labeled, β To find the probability, we calculate the z value of 10,078 in Graph C: z= z= xc − μ1 σ∕√n = 10,078 − 10,120 = −1.05 400∕√100 Region A Rejection region a = 025 Rejection region a = 025 –1.96 s x– 9,922 m0 10,000 1.96 s x– 10,078 psi Region C Probability of making a Type II error Probability of not making a Type II error b 0.1469 0.3531 10,078 xc CHART 10–9 Type I and Type II Errors (Another Example) m1 10,120 2b 0.5000 psi www.downloadslide.com 346 CHAPTER 10 The probability that z is less than −1.05 is 1469, found by 5000 − 3531 Therefore, β, or the probability of a Type II error, is 1469 The power of the test is 1.000 − 1469 or 8531 Caution: If the difference between μ0 and μ1 is relatively small, the probability of a Type II error may occur in both tails This eventuality is not considered here Using the methods illustrated by Charts 10–8 Region B and 10–9 Region C, the probability of a Type II error can be determined for any sample mean used as an estimate of μ1 To review, in hypothesis testing, we select the probability of making a Type I error, α Naturally, we pick small probabilities, typically less than 0.10 The probability of a Type II error depends on the sample result The larger the difference between the hypothesized mean and the sample mean, the smaller the probability of a Type II error, failing to reject the null hypothesis when it should have been rejected For this example/solution, Table 10–4 shows the probabilities of a Type II error and the power of the test for selected values of μ1 The right column gives the probability of not making a Type II error, which is also known as the power of a test TABLE 10–4 Probabilities of a Type II Error for μ0 = 10,000 Pounds and Selected Alternative Means, 05 Level of Significance Selected Mean (μ1) 9,820 9,880 9,900 9,940 10,000 10,060 10,100 10,120 10,180 Probability of Type II Error (β) 0054 1469 2912 6736 — * 6736 2912 1469 0054 Power of a Test (1 − β) 9946 8531 7088 3264 — 3264 7088 8531 9946 *It is not possible to make a Type II error when μ1 = μ0 SELF-REVIEW 10–5 Refer to the previous example Suppose the true mean of an incoming lot of steel bars is 10,180 psi What is the probability that the quality control inspector will accept the bars as having a mean of 10,000 psi? (It sounds implausible that steel bars will be rejected if the tensile strength is higher than specified However, it may be that the cotter pin has a dual function in an outboard motor It may be designed not to shear off if the motor hits a small object, but to shear off if it hits a rock Therefore, the steel should not be too strong.) The light area in Chart 10–9, Region C, represents the probability of falsely accepting the hypothesis that the mean tensile strength of the incoming steel is 10,000 psi What is the probability of committing a Type II error? EXERCISES 21 Refer to Table 10–4 and the example just completed With n = 100, σ = 400, xc = 9,922 and μ1 = 9,880, verify that the probability of a Type II error is 1469 22 Refer to Table 10–4 and the example just completed With n = 100, σ = 400, xc = 10,078, and μ1 = 10,100, verify that the probability of a Type II error is 2912 23 The management of KSmall Industries is considering a new method of assembling a computer The current assembling method requires a mean time of 60 minutes www.downloadslide.com 347 ONE-SAMPLE TESTS OF HYPOTHESIS with a standard deviation of 2.7 minutes Using the new method, the mean assembly time for a random sample of 24 computers was 58 minutes Using the 10 level of significance, can we conclude that the assembly time using the new method is faster? What is the probability of a Type II error? 24 A recent national survey found that parents read an average (mean) of 10 books per month to their children under five years old The population standard deviation is The distribution of books read per month follows the normal distribution A random sample of 25 households revealed that the mean number of books read last month was 12 At the 01 significance level, can we conclude that parents read more than the average number of books to their children? What is the probability of a Type II error? CHAPTER SUMMARY I The objective of hypothesis testing is to verify the validity of a statement about a population parameter II The steps to conduct a test of hypothesis are: A State the null hypothesis (H0) and the alternate hypothesis (H1) B Select the level of significance The level of significance is the likelihood or probability of rejecting a true null hypothesis The most frequently used probabilities used as significance levels are 01, 05, and 10 As a probability, any value between and 1.00 is possible, but we prefer small probabilities of making a Type I error C Select the test statistic A test statistic is a value calculated from sample information used to determine whether to reject the null hypothesis Two test statistics were considered in this chapter a The standard normal distribution (the z distribution) is used when the population follows the normal distribution and the population standard deviation is known b The t distribution is used when the population follows the normal distribution and the population standard deviation is unknown D State the decision rule The decision rule indicates the condition or conditions when the null hypothesis is rejected In a two-tailed test, the rejection region is evenly split between the upper and lower tails In a one-tailed test, all of the rejection region is in either the upper or the lower tail E Select a sample, compute the value of the test statistic, and make a decision regarding the null hypothesis F Interpret the results of your decision III A p-value is the probability that the value of the test statistic is as extreme as the value computed, when the null hypothesis is true IV When testing a hypothesis about a population mean: A If the population standard deviation, σ, is known, the test statistic is the standard normal distribution and is determined from: z= x−μ σ∕√n (10–1) B If the population standard deviation is not known, s is substituted for σ The test statistic is the t distribution, and its value is determined from: t= x−μ s∕√n (10–2) www.downloadslide.com 348 CHAPTER 10 The major characteristics of the t distribution are: It is a continuous distribution It is mound-shaped and symmetrical It is flatter, or more spread out, than the standard normal distribution There is a family of t distributions, depending on the number of degrees of freedom V There are two types of errors that can occur in a test of hypothesis A A Type I error occurs when a true null hypothesis is rejected The probability of making a Type I error is equal to the level of significance This probability is designated by the Greek letter α B A Type II error occurs when a false null hypothesis is not rejected The probability of making a Type II error is designated by the Greek letter β The likelihood of a Type II error must be calculated comparing the hypothesized distribution to an alternate distribution based on sample results PRONUNCIATION KEY SYMBOL MEANING PRONUNCIATION H0 H1 α/2 xc Null hypothesis Alternate hypothesis Two-tailed significance level Limit of the sample mean H sub zero H sub one Alpha divided by x bar sub c μ0 Assumed population mean mu sub zero CHAPTER EXERCISES 25 According to the local union president, the mean gross income of plumbers in the Salt Lake City area follows the normal probability distribution with a mean of $45,000 and a standard deviation of $3,000 A recent investigative reporter for KYAK TV found, for a sample of 120 plumbers, the mean gross income was $45,500 At the 10 significance level, is it reasonable to conclude that the mean income is not equal to $45,000? Determine the p-value 26 Rutter Nursery Company packages its pine bark mulch in 50-pound bags From a long history, the production department reports that the distribution of the bag weights follows the normal distribution and the standard deviation of the packaging process is 3 pounds per bag At the end of each day, Jeff Rutter, the production manager, weighs 10 bags and computes the mean weight of the sample Below are the weights of 10 bags from today’s production 45.6 47.7 47.6 46.3 46.2 47.4 49.2 55.8 47.5 48.5 a Can Mr Rutter conclude that the mean weight of the bags is less than 50 pounds? Use the 01 significance level b In a brief report, tell why Mr Rutter can use the z distribution as the test statistic c Compute the p-value 27 A new weight-watching company, Weight Reducers International, advertises that those who join will lose an average of 10 pounds after the first two weeks The standard deviation is 2.8 pounds A random sample of 50 people who joined the weight reduction program revealed a mean loss of pounds At the 05 level of significance, can we conclude that those joining Weight Reducers will lose less than 10 pounds? Determine the p-value 28 Dole Pineapple Inc is concerned that the 16-ounce can of sliced pineapple is being overfilled Assume the standard deviation of the process is 03 ounce The quality-control department took a random sample of 50 cans and found that the arithmetic mean weight was 16.05 ounces At the 5% level of significance, can we conclude that the mean weight is greater than 16 ounces? Determine the p-value www.downloadslide.com 349 ONE-SAMPLE TESTS OF HYPOTHESIS 29 According to a recent survey, Americans get a mean of hours of sleep per night A random sample of 50 students at West Virginia University revealed the mean length of time slept last night was hours and 48 minutes (6.8 hours) The standard deviation of the sample was 0.9 hour At the 5% level of significance, is it reasonable to conclude that students at West Virginia sleep less than the typical American? Compute the p-value 30 A statewide real estate sales agency, Farm Associates, specializes in selling farm property in the state of Nebraska Its records indicate that the mean selling time of farm property is 90 days Because of recent drought conditions, the agency believes that the mean selling time is now greater than 90 days A statewide survey of 100 recently sold farms revealed a mean selling time of 94 days, with a standard deviation of 22 days At the 10 significance level, has there been an increase in selling time? 31 According to the Census Bureau, 3.13 people reside in the typical American household A sample of 25 households in Arizona retirement communities showed the mean number of residents per household was 2.86 residents The standard deviation of this sample was 1.20 residents At the 05 significance level, is it reasonable to conclude the mean number of residents in the retirement community household is less than 3.13 persons? 32 A recent article in Vitality magazine reported that the mean amount of leisure time per week for American men is 40.0 hours You believe this figure is too large and decide to conduct your own test In a random sample of 60 men, you find that the mean is 37.8 hours of leisure per week and that the standard deviation of the sample is 12.2 hours Can you conclude that the information in the article is untrue? Use the 05 significance level Determine the p-value and explain its meaning 33 A recent survey by nerdwallet.com indicated Americans paid a mean of $6,658 interest on credit card debt in 2017 A sample of 12 households with children revealed the follow amounts At the 05 significance level is it reasonable to conclude that these households paid more interest? 7077 34 5744 6753 7381 7625 6636 7164 7348 8060 5848 9275 7052 A recent article in The Wall Street Journal reported that the home equity loan rate is now less than 4% A sample of eight small banks in the Midwest revealed the following home equity loan rates (in percent): 3.6 4.1 5.3 3.6 4.9 4.6 5.0 4.4 At the 01 significance level, can we conclude that the home equity loan rate for small banks is less than 4%? Estimate the p-value 35 A recent study revealed the typical American coffee drinker consumes an average of 3.1 cups per day A sample of 12 senior citizens revealed they consumed the following amounts of coffee, reported in cups, yesterday 3.1 3.3 3.5 2.6 2.6 4.3 4.4 3.8 3.1 4.1 3.1 3.2 At the 05 significance level, these sample data suggest there is a difference between the national average and the sample mean from senior citizens? 36 The postanesthesia care area (recovery room) at St Luke’s Hospital in Maumee, Ohio, was recently enlarged The hope was that the change would increase the mean number of patients served per day to more than 25 A random sample of 15 days revealed the following numbers of patients 25 27 25 26 25 28 28 27 24 26 25 29 25 27 24 At the 01 significance level, can we conclude that the mean number of patients per day is more than 25? Estimate the p-value and interpret it www.downloadslide.com 350 CHAPTER 10 37 www.golfsmith.com receives an average of 6.5 returns per day from online shoppers For a sample of 12 days, it received the following numbers of returns 0 4 3 4 9 4 5 9 1 6 7 10 At the 01 significance level, can we conclude the mean number of returns is less than 6.5? 38 During recent seasons, Major League Baseball has been criticized for the length of the games A report indicated that the average game lasts hours and 30 minutes A sample of 17 games revealed the following times to completion (Note that the minutes have been changed to fractions of hours, so that a game that lasted hours and 24 minutes is reported at 2.40 hours.) 2.98 2.40 2.70 2.25 3.23 3.17 2.93 3.18 2.80 2.38 3.75 3.20 3.27 2.52 2.58 4.45 2.45 Can we conclude that the mean time for a game is less than 3.50 hours? Use the 05 39 significance level Watch Corporation of Switzerland claims that its watches on average will neither gain nor lose time during a week A sample of 18 watches provided the following gains (+) or losses (−) in seconds per week −0.38 −0.20 −0.38 −0.32 +0.32 −0.23 +0.30 +0.25 −0.10 −0.37 −0.61 −0.48 −0.47 −0.64 −0.04 −0.20 −0.68 +0.05 Is it reasonable to conclude that the mean gain or loss in time for the watches is 0? Use the 05 significance level Estimate the p-value 40 Listed below is the annual rate of return (reported in percent) for a sample of 12 taxable mutual funds 4.63 4.15 4.76 4.70 4.65 4.52 4.70 5.06 4.42 4.51 4.24 4.52 Using the 05 significance level, is it reasonable to conclude that the mean rate of return is more than 4.50%? 41 Many grocery stores and large retailers such as Kroger and Walmart have installed self-checkout systems so shoppers can scan their own items and cash out themselves How customers like this service and how often they use it? Listed below is the number of customers using the service for a sample of 15 days at a Walmart location 120 108 120 114 118 91 118 92 104 104 112 97 118 108 117 Is it reasonable to conclude that the mean number of customers using the self-checkout system is more than 100 per day? Use the 05 significance level 42 For a recent year, the mean fare to fly from Charlotte, North Carolina, to Chicago, Illinois, on a discount ticket was $267 A random sample of 13 round-trip discount fares on this route last month shows: $321 $286 $290 $330 $310 $250 $270 $280 $299 $265 $291 $275 $281 At the 01 significance level, can we conclude that the mean fare has increased? What is the p-value? 43 The publisher of Celebrity Living claims that the mean sales for personality magazines that feature people such as Megan Fox or Jennifer Lawrence are 1.5 million copies per week A sample of 10 comparable titles shows a mean weekly sales last week of 1.3 million copies with a standard deviation of 0.9 million copies Do these data contradict the publisher’s claim? Use the 0.01 significance level www.downloadslide.com 351 ONE-SAMPLE TESTS OF HYPOTHESIS 44 A United Nations report shows the mean family income for Mexican migrants to the United States is $27,000 per year A FLOC (Farm Labor Organizing Committee) evaluation of 25 Mexican family units reveals a mean to be $30,000 with a sample standard deviation of $10,000 Does this information disagree with the United Nations report? Apply the 0.01 significance level 45 The number of “destination weddings” has skyrocketed in recent years For example, many couples are opting to have their weddings in the Caribbean A Caribbean vacation resort recently advertised in Bride Magazine that the cost of a Caribbean wedding was less than $30,000 Listed below is a total cost in $000 for a sample of Caribbean weddings 29.7 29.4 31.7 29.0 29.1 30.5 29.1 29.8 At the 05 significance level, is it reasonable to conclude the mean wedding cost is less than $30,000 as advertised? 46 The American Water Works Association reports that the per capita water use in a single-family home is 69 gallons per day Legacy Ranch is a relatively new housing development The builders installed more efficient water fixtures, such as low-flush toilets, and subsequently conducted a survey of the residences Thirty-six owners responded, and the sample mean water use per day was 64 gallons with a standard deviation of 8.8 gallons per day At the 10 level of significance, is that enough evidence to conclude that residents of Legacy Ranch use less water on average? 47 A cola-dispensing machine is set to dispense 9.00 ounces of cola per cup, with a standard deviation of 1.00 ounce The manufacturer of the machine would like to set the control limit in such a way that, for samples of 36, 5% of the sample means will be greater than the upper control limit, and 5% of the sample means will be less than the lower control limit a At what value should the control limit be set? b If the population mean shifts to 8.6, what is the probability of detecting the change? c If the population mean shifts to 9.6, what is the probability of detecting the change? 48 The owners of the Westfield Mall wished to study customer shopping habits From earlier studies, the owners were under the impression that a typical shopper spends 0.75 hour at the mall, with a standard deviation of 0.10 hour Recently the mall owners added some specialty restaurants designed to keep shoppers in the mall longer The consulting firm, Brunner and Swanson Marketing Enterprises, was hired to evaluate the effects of the restaurants A sample of 45 shoppers by Brunner and Swanson revealed that the mean time spent in the mall had increased to 0.80 hour a Develop a hypothesis test to determine if the mean time spent in the mall changed Use the 10 significance level b Suppose the mean shopping time actually increased from 0.75 hour to 0.79 hours What is the probability of making a Type II error? c When Brunner and Swanson reported the information in part (b) to the mall owners, the owners believed that the probability of making a Type II error was too high How could this probability be reduced? 49 The following null and alternate hypotheses are given H0: μ ≤ 50 H1: μ > 50 Suppose the population standard deviation is 10 The probability of a Type I error is set at 01 and the probability of a Type II error at 30 Assume that the population mean shifts from 50 to 55 How large a sample is necessary to meet these requirements? 50 An insurance company, based on past experience, estimates the mean damage for a natural disaster in its area is $5,000 After introducing several plans to prevent loss, it randomly samples 200 policyholders and finds the mean amount per claim was $4,800 with a standard deviation of $1,300 Does it appear the prevention plans were effective in reducing the mean amount of a claim? Use the 05 significance level 51 A national grocer’s magazine reports the typical shopper spends minutes in line waiting to check out A sample of 24 shoppers at the local Farmer Jack’s showed a mean of 7.5 minutes with a standard deviation of 3.2 minutes Is the waiting time at the local Farmer Jack’s less than that reported in the national magazine? Use the 05 significance level www.downloadslide.com 352 CHAPTER 10 D A T A A N A LY T I C S 52 The North Valley Real Estate data reports information on the homes sold last year a Adam Marty recently joined North Valley Real Estate and was assigned twenty homes to market and show When he was hired, North Valley assured him that the twenty homes would be fairly assigned to him When he reviewed the selling prices of his assigned homes, he thought that the prices were much below the average of $357,000 Adam was able to find the data of how the other agents in the firm were assigned to the homes Use statistical inference to analyze the “fairness” that homes were assigned to the agents 53 Refer to the Baseball 2016 data, which report information on the 30 Major League Baseball teams for the 2016 season a Conduct a test of hypothesis to determine whether the mean salary of the teams was different from $100.0 million Use the 05 significance level b Using a 5% significance level, conduct a test of hypothesis to determine whether the mean attendance was more than 2,000,000 per team 54 Refer to the Lincolnville School District bus data a Select the variable for the number of miles traveled last month Conduct a hypothesis test to determine whether the mean miles traveled last month equals 10,000 Use the 01 significance level Find the p-value and explain what it means b A study of school bus fleets reports that the average per bus maintenance cost is $4,000 per year Using the maintenance cost variable, conduct a hypothesis test to determine whether the mean maintenance cost for Lincolnville’s bus fleet is more than $4,000 at the 05 significance level Determine the p-value and report the results ... 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