Applied Statistics and Probability for Engineers (6th Edition) PDF

Index of Applications in Examples and Exercises BIOLOGICAL Visual accommodation Exercises 6-11, 6-16, 6-75 Weight of swine or guinea pigs...

Index of Applications in Examples and Exercises BIOLOGICAL Visual accommodation Exercises 6-11, 6-16, 6-75 Weight of swine or guinea pigs Exercises 9-142, 13-48 Amino acid composition of soybean meal Exercise 8-52 Wheat grain drying Exercises 13-47, 15-41 Anaerobic respiration Exercise 2-144 Blood CHEMICAL Cholesterol level Exercise 15-10 Acid-base titration Exercises 2-60, 2-132, 3-12, Glucose level Exercises 13-25, 14-37 5-48 Hypertension Exercises 4-143, 8-31, 11-8, Alloys Examples 6-4, 8-5 11-30, 11-46 Exercises 10-21, 10-44, 10-59, Body mass index (BMI) Exercise 11-35 13-38, 15-17 Body temperature Exercise 9-59 Contamination Exercise 2-128, 4-113 Cellular replication Exercises 2-193, 3-100 Cooking oil Exercise 2-79 Circumference of orange trees Exercise 10-46 Etching Exercises 10-19, 10-65, 10-34 Deceased beetles under Infrared focal plane arrays Exercise 9-146 autolysis and putrefaction Exercise 2-92 Melting point of a binder Exercise 9-42 Diet and weight loss Exercises 10-43, 10-77, 15-35 Metallic material transition Examples 8-1, 8-2 Disease in plants Exercise 14-76 Moisture content in raw material Exercise 3-6 Dugongs (sea cows) length Exercise 11-15 Mole fraction solubility Exercises 12-75, 12-91 Fatty acid in margarine Exercises 8-36, 8-66, 8-76, Mole ratio of sebacic acid Exercise 11-91 9-147, 9-113 Pitch carbon analysis Exercises 12-10, 12-36, 12-50, Gene expression Exercises 6-65, 13-50, 15-42 12-60, 12-68 Gene occurrence Exercises 2-195, 3-11 Plasma etching Examples 14-5, 14-8 Gene sequences Exercises 2-25, 2-192, 3-13, Exercise 7-32 3-147 Polymers Exercises 7-15, 10-8, 13-12, Grain quality Exercise 8-21 13-24 Height of plants Exercises 4-170, 4-171 Propellant Height or weight of people Exercises 4-44, 4-66, 5-64, Bond shear strength Examples 15-1, 15-2, 15-4 6-30, 6-37, 6-46, 6-63, 6-73, Exercises 11-11, 11-31, 9-68 11-49, 15-32 Insect fragments in chocolate bars Exercises 3-134, 4-101 Burning rate Examples 9-1, 9-2, 9-3, 9-4, 9-5 IQ for monozygotic twins Exercise 10-45 Exercise 10-6 Leaf transmutation Exercises 2-88, 3-123 Purity Exercise 15-42 Leg strength Exercises 8-30, 9-64 Thermal barrier coatings Exercise 10-75 Light-dependent photosynthesis Exercise 2-24 Nisin recovery Exercises 12-14, 12-32, 12-50, CHEMICAL ENGINEERING 12-64, 12-84, 14-83 Pesticides and grape infestation Exercise 10-94 Aluminum smelting Exercise 10-92 Potato spoilage Exercise 13-14 Automobile basecoat Exercises 14-56, 14-68 Protein Blow molding Exercise 16-59 in Livestock feed Exercise 14-75 Catalyst usage Exercise 10-17 in Milk Exercises 13-13, 13-25, 13-33 Concentration Examples 16-2, 16-6 from Peanut milk Exercise 9-143 Exercises 5-46, 6-68, 6-84, Protopectin content in tomatoes Exercises 13-40, 15-40 10-9, 10-54, 15-64 Rat muscle Exercise 6-15 Conversion Exercise 12-3 Rat tumors Exercise 8-50 Cooling system in a nuclear submarine Exercise 9-130 Rat weight Exercise 8-57 Copper content of a plating bath Exercises 15-8, 15-34, 15-58 Rejuvenated mitochondria Exercises 2-96, 3-88 Dispensed syrup in soda machine Exercises 8-29, 8-63, 8-75 Root vole population Exercise 14-16 Dry ash value of paper pulp Exercise 14-57 Sodium content of cornlakes Exercise 9-61 Fill volume and capability Examples 5-35, 8-6, 9-8, 9-9 Soil Exercises 3-24, 12-1, 12-2, Exercises 2-180, 3-146, 3-151, 12-23, 12-24, 12-41, 12-42 4-62, 4-63, 5-62, 9-100, 10-4, Splitting cell Exercise 4-155 10-85, 10-90, 14-43, 15-38 St John’s Wort Example 10-14 Filtration rate Exercise 14-44 Stork sightings Exercises 4-100, 11-96 Fish preparation Exercise 13-46 Sugar content Exercises 8-46, 9-83, 9-114 Flow metering devices Examples 15-3, 15-5 Synapses in the granule cell layer Exercise 9-145 Exercises 9-126, 9-127 Tar content in tobacco Exercise 8-95 Foam expanding agents Exercises 10-16, 10-56, 10-88 Taste evaluation Exercises 14-13, 14-31, 14-34, Green liquor Exercise 12-100 14-50, 14-54 Hardwood concentration Example 13-2 Tissues from an ivy plant Exercise 2-130 Exercise 14-11 Impurity level in chemical product Exercises 15-3, 15-15 Wearing seat belts Exercises 10-82, 10-83 Injection molding Example 14-9 Exercises 2-15, 2-137, 10-70 COMMUNICATIONS, COMPUTERS, AND NETWORKS Laboratory analysis of chemical Cell phone signal bars Examples 5-1, 5-3 process samples Exercise 2-43 Cellular neural network speed Exercise 8-39 Maximum heat of a hot tub Exercise 10-33 Code for a wireless garage door Exercise 2-34 Na2S concentration Exercises 11-7, 11-29, 11-41, Computer clock cycles Exercise 3-8 11-62 Computer networks Example 4-21 NbOCl3 Exercise 6-36 Exercises 2-10, 2-64, 2-164, Oxygen purity Examples 11-1, 11-2, 11-3, 3-148, 3-175, 4-65, 4-94 11-4, 11-5, 11-6, 11-7 Corporate Web site errors Exercise 4-84 pH Digital channel Examples 2-3, 3-4, 3-6, 3-9, and Catalyst concentration Exercise 14-61 3-12, 3-16, 3-24, 4-15, 5-7, of Plating bath Exercises 15-1, 15-13 5-9, 5-10 of a Solution Exercise 6-17 Electronic messages Exercises 3-158, 4-98, 4-115 of a Water sample Exercise 2-11 Email routes Exercise 2-184 Product color Exercise 14-45 Encryption-decryption system Exercise 2-181 Product solution strength Errors in a communications channel Examples 3-22, 4-17, 4-20 in recirculation unit Exercise 14-38 Exercises 2-2, 2-4, 2-46, 3-40, Pulp brightness Exercise 13-31 4-116, 5-5, 5-12, 6-94, 9-135 Reaction Time Example 4-5 Passwords Exercises 2-81, 2-97, 2-194, Exercises 2-13, 2-33, 4-56 3-91, 3-108 Redox reaction experiments Exercise 2-65 Programming design languages Exercise 10-40 Shampoo foam height Exercises 8-91, 9-15, 9-16, Response time in computer 9-17, 9-18, 9-19, 9-128 operation system Exercise 8-82 Stack loss of ammonia Exercises 12-16, 12-34, 12-52, Software development cost Exercise 13-49 12-66, 12-85 Telecommunication preixes Exercise 2-45 Temperature Telecommunications Examples 3-1, 3-14 Firing Exercise 13-15 Exercises 2-17, 3-2, 3-85, Furnace Exercises 6-55, 6-109 3-105, 3-132, 3-155, 4-95, of Hall cell solution Exercise 11-92 4-105, 4-111, 4-117, 4-160, Vapor deposition Exercises 13-28, 13-32 5-78, 9-98, 15-9 Vapor phase oxidation of naphthalene Exercise 6-54 Transaction processing performance Viscosity Exercises 6-66, 6-88, 6-90, and OLTP benchmark Exercises 2-68, 2-175, 5-10, 6-96, 12-73, 12-103, 14-64, 5-34, 10-7 15-20, 15-36, 15-86 Viruses Exercise 3-75 Water temperature from power Web browsing Examples 3-25, 5-12, 5-13 plant cooling tower Exercise 9-40 Exercises 2-32, 2-191, 3-159, Water vapor pressure Exercise 11-78 4-87, 4-140, 5-6 Wine Examples 12-14, 12-15 Exercises 6-35, 6-51 ELECTRONICS CIVIL ENGINEERING Automobile engine controller Examples 9-10, 9-11 Bipolar transistor current Exercise 14-7 Cement and Concrete Calculator circuit response Exercises 13-6, 13-18 Hydration Example 10-8 Circuits Examples 2-35, 7-3 Mixture heat Exercises 9-10, 9-11, 9-12, Exercises 2-135, 2-136, 2-170, 9-13, 9-14 2-177, 2-190 Mortar briquettes Exercise 15-79 Conductivity Exercise 12-105 Strength Exercises 4-57, 15-24 Current Examples 4-1, 4-5, 4-8, 4-9, Tensile strength Exercise 15-25 4-12, 16-3 Compressive strength Exercises 13-3, 13-9, 13-19, Exercises 10-31, 15-30 14-14, 14-24, 14-48, 7-7, 7-8, Drain and leakage current Exercises 13-41, 11-85 8-13, 8-18, 8-37, 8-69, 8-80, Electromagnetic energy absorption Exercise 10-26 8-87, 8-90, 15-5 Error recovery procedures Exercises 2-18, 2-166 Intrinsic permeability Exercises 11-1, 11-23, 11-39, Inverter transient point Exercises 12-98, 12-99, 12-102 11-52 Magnetic tape Exercises 2-189, 3-125 Highway pavement cracks Exercise 3-138, 4-102 Nickel charge Exercises 2-61, 3-48 Pavement delection Exercises 11-2, 11-16, 11-24, Parallel circuits Example 2-34 11-40 Power consumption Exercises 6-89, 11-79, 12-6, Retained strength of asphalt Exercises 13-11,13-23 12-26, 12-44, 12-58, 12-80 Speed limits Exercises 8-59, 10-60 Power supply Example 9-13 Trafic Exercises 3-87, 3-149, 3-153, Exercises 2-3, 9-20, 9-21, 9-190 9-22, 9-23, 9-24, 9-28 Printed circuit cards Example 2-10 Temperature in Phoenix, AZ Exercise 8-49 Exercises 2-42, 3-122 Temperature of sewage discharge Exercises 6-92, 6-97 Redundant disk array Exercise 2-127 Voters and air pollution Exercises 9-27, 9-94 Resistors Example 7-1 Waste water treatment tank Exercise 2-37 Exercise 6-86 Water demand and quality Exercises 4-68, 9-137 Solder connections Exercises 3-1, 15-43, 15-45 Watershed yield Exercise 11-70 Strands of copper wire Exercise 2-77 Surface charge Exercise 14-15 MATERIALS Surface mount technology (SMT) Example 16-5 Baked density of carbon anodes Exercise 14-4 Transistor life Exercise 7-51 Ceramic substrate Example 16-4 Voltage measurement errors Exercise 4-48N Coating temperature Exercises 10-24, 10-60 ENERGY Coating weight and surface roughness Exercise 2-90 Compressive strength Exercises 7-56, 11-60 Consumption in Asia Exercises 6-29, 6-45, 6-59 Flow rate on silicon wafers Exercises 13-2, 13-16, 15-28 Enrichment percentage Insulation ability Exercise 14-5 of reactor fuel rods Exercises 8-41, 8-71, 8-88 Insulation luid breakdown time Exercises 6-8, 6-74 Fuel octane ratings Exercises 6-22, 6-26, 6-38, Izod impact test Exercises 8-28, 8-62, 8-74, 6-42, 6-58, 6-78, 10-7 9-66, 9-80 Gasoline cost by month Exercise 15-98 Luminescent ink Exercise 5-28 Gasoline mileage Exercises 10-89, 11-6, 11-17, Paint drying time Examples 10-1, 10-2, 10-3 11-28, 11-44, 11-56, 12-27, Exercises 14-2, 14-19, 15-8, 12-55, 12-57, 12-77, 12-89, 15-16 15-37 Particle size Exercises 4-33, 16-17 Heating rate index Exercise 14-46 Photoresist thickness Exercise 5-63 Petroleum imports Exercise 6-72 Plastic breaking strength Exercises 10-5, 10-20, 10-55 Released from cells Exercise 2-168 Polycarbonate plastic Example 2-8 Renewable energy consumption Exercise 15-78 Exercises 2-66, 2-76 Steam usage Exercises 11-5, 11-27, 11-43, Rockwell hardness Exercises 10-91, 9-115, 15-17 11-55 Temperature of concrete Exercise 9-58 Wind power Exercises 4-132, 11-9 Tensile strength of ENVIRONMENTAL Aluminum Example 10-4 Fiber Exercises 7-3, 7-4, 13-3, 13-17 Arsenic Example 10-6 Steel Example 10-9 Exercises 12-12, 12-30, 12-48, Exercise 9-44 12-62, 12-76, 12-88, 13-39 Paper Example 13-1 Asbestos Exercises 4-85, 4-169 Exercises 4-154, 11-86 Biochemical oxygen demand (BOD) Exercises 11-13, 11-33, 11-51 Titanium content Exercises 8-47, 9-79, 15-2, Calcium concentration in lake water Exercise 8-9 15-12 Carbon dioxide in the atmosphere Exercise 3-58 Tube brightness in TV sets Exercises 7-12, 8-35, 8-67, Chloride in surface streams Exercises 11-10, 11-32, 11-48, 8-79, 9-148, 9-67, 14-1 11-59 Cloud seeding Exercise 9-60 MECHANICAL Earthquakes Exercises 6-63, 9-102, 11-15, 15-46 Aircraft manufacturing Examples 6-6, 12-12, 14-1, Emissions and luoride emissions Exercises 2-28, 15-34 15-6, 16-1 Global temperature Exercises 6-83, 11-74 Exercises 6-8, 8-97, 10-42, Hydrophobic organic substances Exercise 10-93 15-31, 15-13, 15-74 Mercury contamination Example 8-4 Artillery shells Exercise 9-106 Ocean wave height Exercise 4-181 Beam delamination Exercises 8-32, 8-64 Organic pollution Example 3-18 Bearings Examples 8-7, 8-8 Oxygen concentration Exercises 8-94, 9-63, 9-140 Exercise 9-95 Ozone levels Exercises 2-9, 11-90 Diameter Exercises 4-181, 9-42, 15-6, Radon release Exercises 13-8, 13-20 15-14 Rainfall in Australia Exercises 8-33, 8-65, 8-77 Wear Example 4-25 Suspended solids in lake water Exercises 6-32, 6-48, 6-60, Exercises 5-22, 4-127, 12-19, 6-80, 9-70 12-39, 12-45, 12-67 (Text continued at the back of book.)  Applied Statistics and Probability for Engineers Sixth Edition Douglas C. Montgomery Arizona State University George C. Runger Arizona State University  To: Meredith, Neil, Colin, and Cheryl Rebecca, Elisa, George and Taylor VICE PRESIDENT AND PUBLISHER Don Fowley AQUISITIONS EDITOR Linda Ratts OPERATIONS MANAGER Melissa Edwards CONTENT EDITOR Wendy Ashenberg SENIOR CONTENT MANAGER Kevin Holm SENIOR PRODUCTION EDITOR Jill Spikereit EXECUTIVE MARKETING MANAGER Christopher Ruel PRODUCT DESIGNER Jennifer Welter DESIGN DIRECTOR Harry Nolan DESIGNER Kristine Carney PHOTO EDITOR Felicia Ruocco PRODUCTION MANAGEMENT SERVICES Laserwords COVER DESIGN Wendy Lai Cover Photo © PaulFleet/iStockphoto Chapter Opener Photos: © babyblueut/iStockphoto, Erik Isakson/Getty Images, Thinkstock Images/Comstock Images/Getty Images, © amriphoto/ iStockphoto, © Ralph125/iStockphoto, © Caro/Alamy, © EasyBuy4u/iStockphoto, © jkullander/iStockphoto, © Alain Nogues/Sygma/Corbis, © HKPNC/iStockphoto, © EdStock/iStockphoto, © skynesher/iStockphoto, © Vasko/iStockphoto, © RedHelga/iStockphoto, © doc-stock/©Corbis, Courtesy of Mike Johnson, www.redbead.com This book was set by Laserwords Private Limited and printed and bound by RR Donnelley. The cover was printed by RR Donnelley. This book is printed on acid free paper. Copyright © 2014, 2011, 2008, 2005, John Wiley & Sons, Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc. 222 Rosewood Drive, Danvers, MA 01923, website www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030-5774, (201)748-6011, fax (201)748-6008, website http://www.wiley.com/go/permissions. Evaluation copies are provided to qualiied academics and professionals for review purposes only, for use in their courses during the next academic year. These copies are licensed and may not be sold or transferred to a third party. Upon completion of the review period, please return the evalu- ation copy to Wiley. Return instructions and a free of charge return shipping label are available at www.wiley.com/go/returnlabel. Outside of the United States, please contact your local representative. Library of Congress Cataloging-in-Publication Data ISBN-13 9781118539712 ISBN (BRV)-9781118645062 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 Wiley Books by These Authors Website: www.wiley.com/college/montgomery Engineering Statistics, Fifth Edition by Montgomery and Runger Introduction to engineering statistics, with topical coverage appropriate for a one-semester course. A modest mathematical level, and an applied approach. Applied Statistics and Probability for Engineers, Sixth Edition by Montgomery and Runger Introduction to engineering statistics, with topical coverage appropriate for eityher a one or two- semester course. An applied approach to solving real-world engineering problems. Introduction to Statistical Quality Control, Seventh Edition by Douglas C. Montgomery For a irst course in statistical quality control. A comprehensive treatment of statistical methodology for quality control and improvement. Some aspects of quality management are also included such as the six-sigma approach. Design and Analysis of Experiments, Eighth Edition by Douglas C. Montgomery An introduction to design and analysis of expieriments, with the modes prerequisite of a irst course in statistical methods. For senior and graduate students for a practitioners, to design and analyze experiments for improving the quality and eficiency of working systems. Introduction to Linear Regression Analysis, Fifth Edition by Mongomery, Peck and Vining A comprehensive and thoroughly up to date look at regression analysis still most widely used technique in statistics today. Response Surface Methodology: Process and Product Optimization Using Designed Experiments, Third Edition Website: www.wiley.com/college/myers The exploration and optimization of response surfaces, for graduate courses in experimental design and for applied statisticians, engineers and chemical and physical scientists. Generalized Linear Models: with Applications in Engineering and the Sciences, Second Edition by Myers, Montgomery and Vining Website: www.wiley.com/college/myers An introductory text or reference on generalized linewar models (GLMs). The range of theoretical topics and applications appeals both to students and practicing professionals. Introduction to Time Series Analysis and Forecasting by Montgomery, Jennings and Kulahci Methods for modeling and analyzing times series data, to draw inferences about the datat and generate forecasts useful to he decision maker. Minitab and SAS are used to illustrate how the methods are implemented in practice. For Advanced undergrad/irst-year graduate, with a prereq- uisite of basic statistical methods. Portions of the book require calculus and matrix algebra. Preface INTENDED AUDIENCE This is an introductory textbook for a irst course in applied statistics and probability for undergraduate students in engineering and the physical or chemical sciences. These indi- viduals play a signiicant role in designing and developing new products and manufacturing systems and processes, and they also improve existing systems. Statistical methods are an important tool in these activities because they provide the engineer with both descriptive and analytical methods for dealing with the variability in observed data. Although many of the methods we present are fundamental to statistical analysis in other disciplines, such as business and management, the life sciences, and the social sciences, we have elected to focus on an engineering-oriented audience. We believe that this approach will best serve students in engineering and the chemical/physical sciences and will allow them to concentrate on the many applications of statistics in these disciplines. We have worked hard to ensure that our examples and exercises are engineering- and science-based, and in almost all cases we have used examples of real data—either taken from a published source or based on our consulting experiences. We believe that engineers in all disciplines should take at least one course in statistics. Unfortunately, because of other requirements, most engineers will only take one statistics course. This book can be used for a single course, although we have provided enough material for two courses in the hope that more students will see the important applications of statistics in their everyday work and elect a second course. We believe that this book will also serve as a useful reference. We have retained the relatively modest mathematical level of the irst ive editions. We have found that engineering students who have completed one or two semesters of calculus and have some knowledge of matrix algebra should have no dificulty reading all of the text. It is our intent to give the reader an understanding of the methodology and how to apply it, not the mathematical theory. We have made many enhancements in this edition, including reorganiz- ing and rewriting major portions of the book and adding a number of new exercises. ORGANIZATION OF THE BOOK Perhaps the most common criticism of engineering statistics texts is that they are too long. Both instructors and students complain that it is impossible to cover all of the topics in the book in one or even two terms. For authors, this is a serious issue because there is great variety in both the content and level of these courses, and the decisions about what material to delete without limiting the value of the text are not easy. Decisions about which topics to include in this edition were made based on a survey of instructors. Chapter 1 is an introduction to the ield of statistics and how engineers use statistical meth- odology as part of the engineering problem-solving process. This chapter also introduces the reader to some engineering applications of statistics, including building empirical models, designing engineering experiments, and monitoring manufacturing processes. These topics are discussed in more depth in subsequent chapters.  Preface vii Chapters 2, 3, 4, and 5 cover the basic concepts of probability, discrete and continuous random variables, probability distributions, expected values, joint probability distributions, and independence. We have given a reasonably complete treatment of these topics but have avoided many of the mathematical or more theoretical details. Chapter 6 begins the treatment of statistical methods with random sampling; data sum- mary and description techniques, including stem-and-leaf plots, histograms, box plots, and probability plotting; and several types of time series plots. Chapter 7 discusses sampling dis- tributions, the central limit theorem, and point estimation of parameters. This chapter also introduces some of the important properties of estimators, the method of maximum likeli- hood, the method of moments, and Bayesian estimation. Chapter 8 discusses interval estimation for a single sample. Topics included are conidence intervals for means, variances or standard deviations, proportions, prediction intervals, and tol- erance intervals. Chapter 9 discusses hypothesis tests for a single sample. Chapter 10 presents tests and conidence intervals for two samples. This material has been extensively rewritten and reorganized. There is detailed information and examples of methods for determining appropri- ate sample sizes. We want the student to become familiar with how these techniques are used to solve real-world engineering problems and to get some understanding of the concepts behind them. We give a logical, heuristic development of the procedures rather than a formal, mathe- matical one. We have also included some material on nonparametric methods in these chapters. Chapters 11 and 12 present simple and multiple linear regression including model ade- quacy checking and regression model diagnostics and an introduction to logistic regression. We use matrix algebra throughout the multiple regression material (Chapter 12) because it is the only easy way to understand the concepts presented. Scalar arithmetic presentations of multiple regression are awkward at best, and we have found that undergraduate engineers are exposed to enough matrix algebra to understand the presentation of this material. Chapters 13 and 14 deal with single- and multifactor experiments, respectively. The notions of randomization, blocking, factorial designs, interactions, graphical data analysis, and frac- tional factorials are emphasized. Chapter 15 introduces statistical quality control, emphasiz- ing the control chart and the fundamentals of statistical process control. WHAT’S NEW IN THIS EDITION We received much feedback from users of the ifth edition of the book, and in response we have made substantial changes in this new edition. r
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BEEFE viii Preface FEATURED IN THIS BOOK Deinitions, Key Concepts, and Equations Throughout the text, deinitions and key concepts and equations are highlighted by a box to emphasize their importance. Learning Objectives Learning Objectives at the start of each chapter guide the students in what they are expected to take away from this chapter and serve as a study reference. Seven-Step Procedure for Hypothesis Testing The text introduces a sequence of seven steps in applying hypothesis-testing methodology and explicitly exhibits this procedure in examples.  Preface ix Figures Numerous igures throughout the text illustrate statistical concepts in multiple formats. Computer Output Example throughout the book, use computer output to illustrate the role of modern statistical software. Example Problems A set of example problems provides the student with detailed solutions and comments for interesting, real-world situations. Brief practical interpretations have been added in this edition. x Preface Exercises Each chapter has an extensive collection of exercises, including end-of-section exercises that emphasize the material in that section, supplemental exercises at the end of the chapter that cover the scope of chapter topics and require the student to make a decision about the approach they will use to solve the problem, and mind-expanding exercises that often require the student to extend the text material somewhat or to apply it in a novel situation. Answers are provided to most odd- numbered exercises in Appendix C in the text, and the WileyPLUS online learning environment includes for students complete detailed solutions to selected exercises. Important Terms and Concepts At the end of each chapter is a list of important terms and concepts for an easy self-check and study tool. STUDENT RESOURCES r
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Data Sets Data sets for all examples and exercises in the text. r
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Section on Logistic Regression  Preface xi These instructor-only resources are password-protected. Visit the instructor section of the book Web site at www.wiley.com/college/montgomery to register for a password to access these materials. COMPUTER SOFTWARE We have used several different packages, including Excel, to demonstrate computer usage. Minitab can be used for most exercises. A student version of Minitab is available as an option to purchase in a set with this text. Student versions of software often do not have all the functionality that full versions do. Consequently, student versions may not support all the concepts presented in this text. If you would like to adopt for your course the set of this text with the student version of Minitab, please contact your local Wiley representative at www.wiley.com/college/rep. Alternatively, students may ind information about how to purchase the professional version of the software for academic use at www.minitab.com. WileyPLUS This online teaching and learning environment integrates the entire digital textbook with the most effective instructor and student resources to it every learning style. With WileyPLUS: r
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course on statistics for engineers vary widely, as do the abilities of different groups of stu- dents. Therefore, we hesitate to give too much advice, but will explain how we use the book. We believe that a irst course in statistics for engineers should be primarily an applied statistics course, not a probability course. In our one-semester course we cover all of Chapter 1 (in one or two lectures); overview the material on probability, putting most of the emphasis on the normal distribution (six to eight lectures); discuss most of Chapters 6 through 10 on conidence intervals and tests (twelve to fourteen lectures); introduce regression models in Chapter 11 (four lectures); give an introduction to the design of experiments from Chapters 13 and 14 (six lectures); and present the basic concepts of statistical process control, including the Shewhart control chart from Chapter 15 (four lectures). This leaves about three to four periods for exams and review. Let us emphasize that the purpose of this course is to introduce engineers to how statistics can be used to solve real-world engineering problems, not to weed out the less mathematically gifted students. This course is not the “baby math-stat” course that is all too often given to engineers. If a second semester is available, it is possible to cover the entire book, including much of the supplemental material, if appropriate for the audience. It would also be possible to assign and work many of the homework problems in class to reinforce the understanding of the con- cepts. Obviously, multiple regression and more design of experiments would be major topics in a second course. USING THE COMPUTER In practice, engineers use computers to apply statistical methods to solve problems. Therefore, we strongly recommend that the computer be integrated into the class. Throughout the book we have presented typical example of the output that can be obtained with modern statistical software. In teaching, we have used a variety of software packages, including Minitab, Stat- graphics, JMP, and Statistica. We did not clutter up the book with operational details of these different packages because how the instructor integrates the software into the class is ultimate- ly more important than which package is used. All text data are available in electronic form on the textbook Web site. In some chapters, there are problems that we feel should be worked using computer software. We have marked these problems with a special icon in the margin. In our own classrooms, we use the computer in almost every lecture and demonstrate how the technique is implemented in software as soon as it is discussed in the lecture. Student versions of many statistical software packages are available at low cost, and students can either purchase their own copy or use the products available through the institution. We have found that this greatly improves the pace of the course and student understanding of the material. Users should be aware that inal answers may differ slightly due to different numerical preci- sion and rounding protocols among softwares.  Preface xiii ACKNOWLEDGMENTS We would like to express our grateful appreciation to the many organizations and individuals who have contributed to this book. Many instructors who used the previous editions provided excellent suggestions that we have tried to incorporate in this revision. We would like to thank the following who assisted in contributing to and/or reviewing material for the WileyPLUS course: Michael DeVasher, Rose-Hulman Institute of Technology Craig Downing, Rose-Hulman Institute of Technology Julie Fortune, University of Alabama in Huntsville Rubin Wei, Texas A&M University We would also like to thank the following for their assistance in checking the accuracy and completeness of the exercises and the solutions to exercises. Dr. Abdelaziz Berrado Dr. Connie Borror Aysegul Demirtas Kerem Demirtas Patrick Egbunonu, Sindhura Gangu James C. Ford Dr. Alejandro Heredia-Langner Dr. Jing Hu Dr. Busaba Laungrungrong Dr. Fang Li Dr. Nuttha Lurponglukana 4BSBI
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5SB Dr. Lora Zimmer We are also indebted to Dr. Smiley Cheng for permission to adapt many of the statistical tables from his excellent book (with Dr. James Fu), Statistical Tables for Classroom and Exam Room. Wiley, Prentice Hall, the Institute of Mathematical Statistics, and the editors of Biomet- rics allowed us to use copyrighted material, for which we are grateful. Douglas C. Montgomery George C. Runger Contents Inside Front cover Index of Applications Chapter 4 Continuous Random Variables Examples and Exercises and Probability Distributions 107 4-1 Continuous Random Variables 108 Chapter 1 The Role of Statistics in Engineering 1 4-2 Probability Distributions and Probability Density Functions 108 1-1 The Engineering Method and Statistical 4-3 Cumulative Distribution Functions 112 Thinking 2 4-4 Mean and Variance of a Continuous 1-2 Collecting Engineering Data 4 Random Variable 114 1-2.1 Basic Principles 4 4-5 Continuous Uniform Distribution 116 1-2.2 Retrospective Study 5 4-6 Normal Distribution 119 1-2.3 Observational Study 5 4-7 Normal Approximation to the Binomial and 1-2.4 Designed Experiments 6 Poisson Distributions 128 1-2.5 Observing Processes Over Time 8 4-8 Exponential Distribution 133 1-3 Mechanistic and Empirical Models 11 4-9 Erlang and Gamma Distributions 139 1-4 Probability and Probability Models 12 4-10 Weibull Distribution 143 Chapter 2 Probability 15 4-11 Lognormal Distribution 145 2-1 Sample Spaces and Events 16 4-12 Beta Distribution 148 2-1.1 Random Experiments 16 Chapter 5 Joint Probability Distributions 155 2-1.2 Sample Spaces 17 2-1.3 Events 20 5-1 Two or More Random Variables 156 2-1.4 Counting Techniques 22 5-1.1 Joint Probability Distributions 156 2-2 Interpretations and Axioms of Probability 30 5-1.2 Marginal Probability Distributions 159 2-3 Addition Rules 35 5-1.3 Conditional Probability Distributions 161 2-4 Conditional Probability 40 5-1.4 Independence 164 2-5 Multiplication and Total Probability 5-1.5 More Than Two Random Variables 167 Rules 45 5-2 Covariance and Correlation 174 2-6 Independence 49 5-3 Common Joint Distributions 179 
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 5-3.1 Multinomial Probability Distribution 179 2-8 Random Variables 57 5-3.2 Bivariate Normal Distribution 181 5-4 Linear Functions of Random Variables 184 Chapter 3 Discrete Random Variables and 5-5 General Functions of Random Variables 188 Probability Distributions 65 5-6 Moment-Generating Functions 191 3-1 Discrete Random Variables 66 Chapter 6 Descriptive Statistics 199 3-2 Probability Distributions and Probability Mass Functions 67 6-1 Numerical Summaries of Data 200 3-3 Cumulative Distribution Functions 71 6-2 Stem-and-Leaf Diagrams 206 3-4 Mean and Variance of a Discrete Random 6-3 Frequency Distributions and Histograms 213 Variable 74 6-4 Box Plots 217 3-5 Discrete Uniform Distribution 78 6-5 Time Sequence Plots 219 3-6 Binomial Distribution 80 6-6 Scatter Diagrams 225 3-7 Geometric and Negative Binomial 6-7 Probability Plots 230 Distributions 86 Chapter 7 Point Estimation of Parameters and 3-7.1 Geometric Distribution 86 Sampling Distributions 239 3-8 Hypergeometric Distribution 93 3-9 Poisson Distribution 98 7-1 Point Estimation 240  Contents xv 7-2 Sampling Distributions 9-1.3 One-Sided and Two-Sided and the Central Limit Theorem 241 Hypotheses 313 7-3 General Concepts of Point Estimation 249 9-1.4 P-Values in Hypothesis Tests 314 7-3.1 Unbiased Estimators 249 9-1.5 Connection Between Hypothesis Tests 7-3.2 Variance of a Point Estimator 251 and Conidence Intervals 316 7-3.3 Standard Error: Reporting a Point 9-1.6 General Procedure for Hypothesis Estimate 251 Tests 318 7.3.4 Bootstrap Standard Error 252 9-2 Tests on the Mean of a Normal Distribution, 7-3.5 Mean Squared Error of an Estimator 254 Variance Known 322 7-4 Methods of Point Estimation 256 9-2.1 Hypothesis Tests on the Mean 322 7-4.1 Method of Moments 256 9-2.2 Type II Error and Choice of Sample 7-4.2 Method of Maximum Likelihood 258 Size 325 7-4.3 Bayesian Estimation of 9-2.3 Large-Sample Test 329 Parameters 264 9-3 Tests on the Mean of a Normal Distribution, Variance Unknown 331 Chapter 8 Statistical Intervals for a 9-3.1 Hypothesis Tests on the Mean 331 Single Sample 271 9-3.2 Type II Error and Choice of Sample 8-1 Conidence Interval on the Mean of a Normal Size 336 Distribution, Variance Known 273 9-4 Tests on the Variance and Standard 8-1.1 Development of the Conidence Interval Deviation of a Normal Distribution 340 and Its Basic Properties 273 9-4.1 Hypothesis Tests on the Variance 341 8-1.2 Choice of Sample Size 276 9-4.2 Type II Error and Choice of Sample 8-1.3 One-Sided Conidence Bounds 277 Size 343 8-1.4 General Method to Derive a Conidence 9-5 Tests on a Population Proportion 344 Interval 277 9-5.1 Large-Sample Tests on a Proportion 344 8-1.5 Large-Sample Conidence Interval 9-5.2 Type II Error and Choice of Sample for μ 279 Size 347 8-2 Conidence Interval on the Mean of a Normal 9-6 Summary Table of Inference Procedures Distribution, Variance Unknown 282 for a Single Sample 350 8-2.1 t Distribution 283 9-7 Testing for Goodness of Fit 350 8-2.2 t Conidence Interval on μ 284 9-8 Contingency Table Tests 354 8-3 Conidence Interval on the Variance and 9-9 Nonparametric Procedures 357 Standard Deviation of a Normal 9-9.1 The Sign Test 358 Distribution 287 9-9.2 The Wilcoxon Signed-Rank Test 362 8-4 Large-Sample Conidence Interval 9-9.3 Comparison to the t-Test 364 for a Population Proportion 291 9-10 Equivalence Testing 365 8-5 Guidelines for Constructing Conidence 9-11 Combining P-Values 367 Intervals 296 Chapter 10 Statistical Inference for 8.6 Bootstrap Conidence Interval 296 Two Samples 373 8-7 Tolerance and Prediction Intervals 297 8-7.1 Prediction Interval for a Future 10-1 Inference on the Difference in Means of Two Observation 297 Normal Distributions, Variances Known 374 8-7.2 Tolerance Interval for a Normal 10-1.1 Hypothesis Tests on the Difference in Distribution 298 Means, Variances Known 376 10-1.2 Type II Error and Choice of Sample Chapter 9 Tests of Hypotheses for a Size 377 Single Sample 305 10-1.3 Conidence Interval on the Difference in Means, Variances Known 379 9-1 Hypothesis Testing 306 10-2 Inference on the Difference in Means of two 9-1.1 statistical hypotheses 306 Normal Distributions, Variances Unknown 383 9-1.2 Tests of Statistical Hypotheses 308 xvi Contents 10-2.1 Hypotheses Tests on the Difference in 11-7.1 Residual Analysis 453 Means, Variances Unknown 383 11-7.2 Coeficient of Determination 10-2.2 Type II Error and Choice of Sample (R2) 454 Size 389 11-8 Correlation 457 10-2.3 Conidence Interval on the Difference in 11-9 Regression on Transformed Variables 463 Means, Variances Unknown 390 11-10 Logistic Regression 467 10-3 A Nonparametric Test for the Difference in Two Chapter 12 Multiple Linear Regression 477 Means 396 10-3.1 Description of the Wilcoxon Rank-Sum 12-1 Multiple Linear Regression Model 478 Test 397 12-1.1 Introduction 478 10-3.2 Large-Sample Approximation 398 12-1.2 Least Squares Estimation of the 10-3.3 Comparison to the t-Test 399 Parameters 481 10-4 Paired t-Test 400 12-1.3 Matrix Approach to Multiple Linear 10-5 Inference on the Variances of Two Normal Regression 483 Distributions 407 12-1.4 Properties of the Least Squares 10-5.1 F Distribution 407 Estimators 488 10-5.2 Hypothesis Tests on the Ratio of Two 12-2 Hypothesis Tests In Multiple Linear Variances 409 Regression 497 10-5.3 Type II Error and Choice of Sample 12-2.1 Test for Signiicance Size 411 of Regression 497 10-5.4 Conidence Interval on the Ratio of Two 12-2.2 Tests on Individual Regression Variances 412 Coeficients and Subsets of 10-6 Inference on Two Population Coeficients 500 Proportions 414 12-3 Conidence Intervals In Multiple Linear 10-6.1 Large-Sample Tests on the Difference in Regression 506 Population Proportions 414 12-3.1 Conidence Intervals on Individual 10-6.2 Type II Error and Choice of Sample Regression Coeficients 506 Size 416 12-3.2 Conidence Interval on the Mean 10-6.3 Conidence Interval on the Difference in Response 507 Population Proportions 417 12-4 Prediction of New Observations 508 10-7 Summary Table and Road Map for Inference 12-5 Model Adequacy Checking 511 Procedures for Two Samples 420 12-5.1 Residual Analysis 511 12-5.2 Inluential Observations 514 Chapter 11 Simple Linear Regression 12-6 Aspects of Multiple Regression and Correlation 427 Modeling 517 11-1 Empirical Models 428 12-6.1 Polynomial Regression Models 517 11-2 Simple Linear Regression 431 12-6.2 Categorical Regressors and Indicator 11-3 Properties of the Least Squares Variables 519 Estimators 440 12-6.3 Selection of Variables and Model 11-4 Hypothesis Tests in Simple Linear Building 522 Regression 441 12-6.4 Multicollinearity 529 11-4.1 Use of t-Tests 441 Chapter 13 Design and Analysis of Single-Factor 11-4.2 Analysis of Variance Approach to Test Experiments:The Analysis of Variance 539 Signiicance of Regression 443 11-5 Conidence Intervals 447 13-1 Designing Engineering Experiments 540 11-5.1 Conidence Intervals on the Slope and 13-2 Completely Randomized Single-Factor Intercept 447 Experiment 541 11-5.2 Conidence Interval on the Mean 13-2.1 Example: Tensile Strength 541 Response 448 13-2.2 Analysis of Variance 542 11-6 Prediction of New Observations 449 13-2.3 Multiple Comparisons Following the 11-7 Adequacy of the Regression Model 452 ANOVA 549  Contents xvii 13-2.4 Residual Analysis and Model 15-5 Process Capability 692 Checking 551 15-6 Attribute Control Charts 697 13-2.5 Determining Sample Size 553 15-6.1 P Chart (Control Chart for 13-3 The Random-Effects Model 559 Proportions) 697 13-3.1 Fixed Versus Random Factors 559 15-6.2 U Chart (Control Chart for Defects per 13-3.2 ANOVA and Variance Components 560 Unit) 699 13-4 Randomized Complete Block Design 565 15-7 Control Chart Performance 704 13-4.1 Design and Statistical Analysis 565 15-8 Time-Weighted Charts 708 13-4.2 Multiple Comparisons 570 15-8.1 Cumulative Sum Control Chart 709 13-4.3 Residual Analysis and Model 15-8.2 Exponentially Weighted Moving- Checking 571 Average Control Chart 714 15-9 Other SPC Problem-Solving Tools 722 Chapter 14 Design of Experiments with Several 15-10 Decision Theory 723 Factors 575 15-10.1 Decision Models 723 14-1 Introduction 576 15-10.2 Decision Criteria 724 14-2 Factorial Experiments 578 15-11 Implementing SPC 726 14-3 Two-Factor Factorial Experiments 582 Appendix A. Statistical Tables and Charts 737 14-3.1 Statistical Analysis of the Fixed-Effects Model 582 Table I Summary of Common Probability 14-3.2 Model Adequacy Checking 587 Distributions 738 14-3.3 One Observation per Cell 588 Table II Cumulative Binomial Probabilities 14-4 General Factorial Experiments 591 P ( X ≤ x ) 739 14-5 2k Factorial Designs 594 Table III Cumulative Standard Normal 14-5.1 22 Design 594 Distribution 742 14-5.2 2k Design for k≥3 Factors 600 Table IV Percentage Points χ2α,v of the Chi-Squared 14-5.3 Single Replicate of the 2k Design 607 Distribution 744 14-5.4 Addition of Center Points to Table V Percentage Points tα,v of the t a 2k Design 611 Distribution 745 14-6 Blocking and Confounding in the 2k Table VI Percentage Points fα, v1 , v2 of the F Design 619 Distribution 746 14-7 Fractional Replication of the 2k Design 626 Chart VII Operating Characteristic Curves 751 14-7.1 One-Half Fraction of the Table VIII Critical Values for the Sign Test 760 2k Design 626 Table IX Critical Values for the Wilcoxon Signed-Rank 14-7.2 Smaller Fractions: The 2k–p Fractional Test 760 Factorial 632 Table X Critical Values for the Wilcoxon Rank-Sum 14-8 Response Surface Methods and Designs 643 Test 761 Table XI Factors for Constructing Variables Control Chapter 15 Statistical Quality Control 663 Charts 762 15-1 Quality Improvement and Statistics 664 Table XII Factors for Tolerance Intervals 762 15-1.1 Statistical Quality Control 665 Appendix B: Bibliography 765 15-1.2 Statistical Process Control 666 15-2 Introduction to Control Charts 666 Appendix C: Answers to Selected Exercises 769 15-2.1 Basic Principles 666 15-2.2 Design of a Control Chart 670 Glossary 787 15-2.3 Rational Subgroups 671 15-2.4 Analysis of Patterns on Control Charts Index 803 672 – 15-3 X and R or S Control Charts 674 Index of applications in examples and 15-4 Control Charts for Individual exercises, continued 809 Measurements 684  1 The Role of Statistics in Engineering Statistics is a science that helps us make decisions and draw Chapter Outline conclusions in the presence of variability. For example, civil engineers working in the transportation ield are concerned 1-1 The Engineering Method and Statistical about the capacity of regional highway systems. A typical Thinking problem related to transportation would involve data regarding this speciic system’s number of nonwork, home-based trips, 1-2 Collecting Engineering Data the number of persons per household, and the number of vehi- 1-2.1 Basic Principles cles per household. The objective would be to produce a trip- 1-2.2 Retrospective Study generation model relating trips to the number of persons per 1-2.3 Observational Study household and the number of vehicles per household. A statis- 1-2.4 Designed Experiments tical technique called regression analysis can be used to con- 1-2.5 Observing Processes Over Time struct this model. The trip-generation model is an important tool for transportation systems planning. Regression methods 1-3 Mechanistic and Empirical Models are among the most widely used statistical techniques in engi- 1-4 Probability and Probability Models neering. They are presented in Chapters 11 and 12. The hospital emergency department (ED) is an important part of the healthcare delivery system. The process by which patients arrive at the ED is highly variable and can depend on the hour of the day and the day of the week, as well as on longer-term cyclical variations. The service process is also highly variable, depending on the types of services that the patients require, the number of patients in the ED, and how the ED is staffed and organized. An ED’s capacity is also limited; consequently, some patients experience long waiting times. How long do patients wait, on average? This is an important question for healthcare providers. If waiting times become excessive, some patients will leave without receiving treatment LWOT. Patients who LWOT are a serious problem, because they do not have their medical concerns addressed and are at risk for further problems and complications. Therefore, another 1 2 Chapter 1/The Role of Statistics in Engineering important question is: What proportion of patients LWOT from the ED? These questions can be solved by employing probability models to describe the ED, and from these models very precise estimates of waiting times and the number of patients who LWOT can be obtained. Probability models that can be used to solve these types of problems are discussed in Chapters 2 through 5. The concepts of probability and statistics are powerful ones and contribute extensively to the solutions of many types of engineering problems. You will encounter many examples of these applications in this book. Learning Objectives After careful study of this chapter, you should be able to do the following: 1. Identify the role that statistics can play in the engineering problem-solving process 2. Discuss how variability affects the data collected and used for making engineering decisions 3. Explain the difference between enumerative and analytical studies 4. Discuss the different methods that engineers use to collect data 5. Identify the advantages that designed experiments have in comparison to other methods of collecting engineering data 6. Explain the differences between mechanistic models and empirical models 7. Discuss how probability and probability models are used in engineering and science 1-1 The Engineering Method and Statistical Thinking An engineer is someone who solves problems of interest to society by the eficient application of scientiic principles. Engineers accomplish this by either reining an existing product or process or by designing a new product or process that meets customers’ needs. The engineering, or scientiic, method is the approach to formulating and solving these problems. The steps in the engineering method are as follows: 1. Develop a clear and concise description of the problem. 2. Identify, at least tentatively, the important factors that affect this problem or that may play a role in its solution. 3. Propose a model for the problem, using scientiic or engineering knowledge of the phenomenon being studied. State any limitations or assumptions of the model. 4. Conduct appropriate experiments and collect data to test or validate the tentative model or conclusions made in steps 2 and 3. 5. Reine the model on the basis of the observed data. 6. Manipulate the model to assist in developing a solution to the problem. 7. Conduct an appropriate experiment to conirm that the proposed solution to the problem is both effective and eficient. 8. Draw conclusions or make recommendations based on the problem solution. The steps in the engineering method are shown in Fig. 1-1. Many engineering sciences employ the engineering method: the mechanical sciences (statics, dynamics), luid science, thermal science, electrical science, and the science of materials. Notice that the engineer- ing method features a strong interplay among the problem, the factors that may inluence its solution, a model of the phenomenon, and experimentation to verify the adequacy of the model and the proposed solution to the problem. Steps 2–4 in Fig. 1-1 are enclosed in a box, indicating that several cycles or iterations of these steps may be required to obtain the inal solution. Consequently, engineers must know how to eficiently plan experiments, collect data, analyze and interpret the data, and understand how the observed data relate to the model they have proposed for the problem under study.  Section 1-1/The Engineering Method and Statistical Thinking 3 Develop a Identify the Propose or Manipulate Confirm Conclusions clear important refine a the the and description factors model model solution recommendations Conduct experiments FIGURE 1-1 The engineering method. The Science of Data The ield of statistics deals with the collection, presentation, analysis, and use of data to make decisions, solve problems, and design products and processes. In simple terms, statistics is the sci- ence of data. Because many aspects of engineering practice involve working with data, obviously knowledge of statistics is just as important to an engineer as are the other engineering sciences. Speciically, statistical techniques can be powerful aids in designing new products and systems, improving existing designs, and designing, developing, and improving production processes. Variability Statistical methods are used to help us describe and understand variability. By variability, we mean that successive observations of a system or phenomenon do not produce exactly the same result. We all encounter variability in our everyday lives, and statistical thinking can give us a useful way to incorporate this variability into our decision-making processes. For example, con- sider the gasoline mileage performance of your car. Do you always get exactly the same mileage performance on every tank of fuel? Of course not — in fact, sometimes the mileage performance varies considerably. This observed variability in gasoline mileage depends on many factors, such as the type of driving that has occurred most recently (city versus highway), the changes in the vehicle’s condition over time (which could include factors such as tire inlation, engine com- pression, or valve wear), the brand and/or octane number of the gasoline used, or possibly even the weather conditions that have been recently experienced. These factors represent potential sources of variability in the system. Statistics provides a framework for describing this vari- ability and for learning about which potential sources of variability are the most important or which have the greatest impact on the gasoline mileage performance. We also encounter variability in dealing with engineering problems. For example, suppose that an engineer is designing a nylon connector to be used in an automotive engine application. The engineer is considering establishing the design speciication on wall thickness at 3 32 inch but is somewhat uncertain about the effect of this decision on the connector pull-off force. If the pull-off force is too low, the connector may fail when it is installed in an engine. Eight prototype units are produced and their pull-off forces measured, resulting in the following data (in pounds): 12.6, 12.9, 13.4, 12.3, 13.6, 13.5, 12.6, 13.1. As we anticipated, not all of the prototypes have the same pull-off force. We say that there is variability in the pull-off force measurements. Because the pull-off force measurements exhibit variability, we consider the pull-off force to be a random variable. A convenient way to think of a random variable, say X, that represents a measurement is by using the model X 5m 1e (1-1) where m is a constant and e is a random disturbance. The constant remains the same with every measurement, but small changes in the environment, variance in test equipment, differences in the individual parts themselves, and so forth change the value of e. If there were no distur- bances, e would always equal zero and X would always be equal to the constant m. However, this never happens in the real world, so the actual measurements X exhibit variability. We often need to describe, quantify, and ultimately reduce variability. Figure 1-2 presents a dot diagram of these data. The dot diagram is a very useful plot for displaying a small body of data—say, up to about 20 observations. This plot allows us to easily see two features of the data: the location, or the middle, and the scatter or variability. When the number of observations is small, it is usually dificult to identify any speciic patterns in the variability, although the dot diagram is a convenient way to see any unusual data features. 4 Chapter 1/The Role of Statistics in Engineering = 3 inch 32 12 13 14 15 12 13 14 15 = 1 inch 8 Pull-off force Pull-off force FIGURE 1-2 Dot diagram of the pull-off force FIGURE 1-3 Dot diagram of pull-off force for two wall data when wall thickness is 3 32 inch. thicknesses. The need for statistical thinking arises often in the solution of engineering problems. Consider the engineer designing the connector. From testing the prototypes, he knows that the average pull- off force is 13.0 pounds. However, he thinks that this may be too low for the intended application, so he decides to consider an alternative design with a thicker wall, 1 8 inch in thickness. Eight pro- totypes of this design are built, and the observed pull-off force measurements are 12.9, 13.7, 12.8, 13.9, 14.2, 13.2, 13.5, and 13.1. The average is 13.4. Results for both samples are plotted as dot diagrams in Fig. 1-3. This display gives the impression that increasing the wall thickness has led to an increase in pull-off force. However, there are some obvious questions to ask. For instance, how do we know that another sample of prototypes will not give different results? Is a sample of eight prototypes adequate to give reliable results? If we use the test results obtained so far to conclude that increasing the wall thickness increases the strength, what risks are associated with this deci- sion? For example, is it possible that the apparent increase in pull-off force observed in the thicker prototypes is due only to the inherent variability in the system and that increasing the thickness of the part (and its cost) really has no effect on the pull-off force? Population and Often, physical laws (such as Ohm’s law and the ideal gas law) are applied to help design prod- Samples ucts and processes. We are familiar with this reasoning from general laws to speciic cases. But it is also important to reason from a speciic set of measurements to more general cases to answer the previous questions. This reasoning comes from a sample (such as the eight connectors) to a population (such as the connectors that will be in the products that are sold to customers). The reasoning is referred to as statistical inference. See Fig. 1-4. Historically, measurements were obtained from a sample of people and generalized to a population, and the terminology has remained. Clearly, reasoning based on measurements from some objects to measurements on all objects can result in errors (called sampling errors). However, if the sample is selected properly, these risks can be quantiied and an appropriate sample size can be determined. 1-2 Collecting Engineering Data 1-2.1 BASIC PRINCIPLES In the previous subsection, we illustrated some simple methods for summarizing data. Some- times the data are all of the observations in the population. This results in a census. However, in the engineering environment, the data are almost always a sample that has been selected from the population. Three basic methods of collecting data are r
A retrospective study using historical data r
An observational study r
A designed experiment Physical Population laws Statistical inference Types of reasoning FIGURE 1-4 Statistical Product Sample inference is one designs type of reasoning.  Section 1-2/Collecting Engineering Data 5 An effective data-collection procedure can greatly simplify the analysis and lead to improved understanding of the population or process that is being studied. We now consider some examples of these data-collection methods. 1-2.2 RETROSPECTIVE STUDY Montgomery, Peck, and Vining (2012) describe an acetone-butyl alcohol distillation column for which concentration of acetone in the distillate (the output product stream) is an important variable. Factors that may affect the distillate are the reboil temperature, the condensate temperature, and the relux rate. Production personnel obtain and archive the following records: r
The concentration of acetone in an hourly test sample of output product r
The reboil temperature log, which is a record of the reboil temperature over time r
The condenser temperature controller log r
The nominal relux rate each hour The relux rate should be held constant for this process. Consequently, production personnel change this very infrequently. Hazards of Using A retrospective study would use either all or a sample of the historical process data archived Historical Data over some period of time. The study objective might be to discover the relationships among the two temperatures and the relux rate on the acetone concentration in the output product stream. However, this type of study presents some problems: 1. We may not be able to see the relationship between the relux rate and acetone concentration because the relux rate did not change much over the historical period. 2. The archived data on the two temperatures (which are recorded almost continuously) do not correspond perfectly to the acetone concentration measurements (which are made hourly). It may not be obvious how to construct an approximate correspondence. 3. Production maintains the two temperatures as closely as possible to desired targets or set points. Because the temperatures change so little, it may be dificult to assess their real impact on acetone concentration. 4. In the narrow ranges within which they do vary, the condensate temperature tends to increase with the reboil temperature. Consequently, the effects of these two process vari- ables on acetone concentration may be dificult to separate. As you can see, a retrospective study may involve a signiicant amount of data, but those data may contain relatively little useful information about the problem. Furthermore, some of the relevant data may be missing, there may be transcription or recording errors resulting in outli- ers (or unusual values), or data on other important factors may not have been collected and archived. In the distillation column, for example, the speciic concentrations of butyl alcohol and acetone in the input feed stream are very important factors, but they are not archived because the concentrations are too hard to obtain on a routine basis. As a result of these types of issues, statistical analysis of historical data sometimes identiies interesting phenomena, but solid and reliable explanations of these phenomena are often dificult to obtain. 1-2.3 OBSERVATIONAL STUDY In an observational study, the engineer observes the process or population, disturbing it as little as possible, and records the quantities of interest. Because these studies are usually conducted for a relatively short time period, sometimes variables that are not routinely measured can be included. In the distillation column, the engineer would design a form to record the two temperatures and the relux rate when acetone concentration measurements are made. It may even be possible to measure the input feed stream concentrations so that the impact of this factor could be studied. 6 Chapter 1/The Role of Statistics in Engineering Generally, an observational study tends to solve problems 1 and 2 and goes a long way toward obtaining accurate and reliable data. However, observational studies may not help resolve problems 3 and 4. 1-2.4 DESIGNED EXPERIMENTS In a designed experiment, the engineer makes deliberate or purposeful changes in the controlla- ble variables of the system or process, observes the resulting system output data, and then makes an inference or decision about which variables are responsible for the observed changes in output performance. The nylon connector example in Section 1-1 illustrates a designed experiment; that is, a deliberate change was made in the connector’s wall thickness with the objective of dis- covering whether or not a stronger pull-off force could be obtained. Experiments designed with basic principles such as randomization are needed to establish cause-and-effect relationships. Much of what we know in the engineering and physical-chemical sciences is developed through testing or experimentation. Often engineers work in problem areas in which no scien- tiic or engineering theory is directly or completely applicable, so experimentation and obser- vation of the resulting data constitute the only way that the problem can be solved. Even when there is a good underlying scientiic theory that we may rely on to explain the phenomena of interest, it is almost always necessary to conduct tests or experiments to conirm that the the- ory is indeed operative in the situation or environment in which it is being applied. Statistical thinking and statistical methods play an important role in planning, conducting, and analyzing the data from engineering experiments. Designed experiments play a very important role in engineering design and development and in the improvement of manufacturing processes. For example, consider the problem involving the choice of wall thickness for the nylon connec- tor. This is a simple illustration of a designed experiment. The engineer chose two wall thicknesses for the connector and performed a series of tests to obtain pull-off force measurements at each wall thickness. In this simple comparative experiment, the engineer is interested in determining whether there is any difference between the 3 32- and 1 8-inch designs. An approach that could be used in analyzing the data from this experiment is to compare the mean pull-off force for the 3 32 -inch design to the mean pull-off force for the 1 8-inch design using statistical hypothesis testing, which is discussed in detail in Chapters 9 and 10. Generally, a hypothesis is a statement about some aspect of the system in which we are interested. For example, the engineer might want to know if the mean pull-off force of a 3 32-inch design exceeds the typical maximum load expected to be encountered in this application, say, 12.75 pounds. Thus, we would be interested in testing the hypothesis that the mean strength exceeds 12.75 pounds. This is called a single-sample hypothesis- testing problem. Chapter 9 presents techniques for this type of problem. Alternatively, the engineer might be interested in testing the hypothesis that increasing the wall thickness from 3 32 to 1 8 inch results in an increase in mean pull-off force. It is an example of a two-sample hypothesis-testing problem. Two-sample hypothesis-testing problems are discussed in Chapter 10. Designed experiments offer a very powerful approach to studying complex systems, such as the distillation column. This process has three factors—the two temperatures and the relux rate—and we want to investigate the effect of these three factors on output acetone concentra- tion. A good experimental design for this problem must ensure that we can separate the effects of all three factors on the acetone concentration. The speciied values of the three factors used in the experiment are called factor levels. Typically, we use a small number of levels such as two or three for each factor. For the distillation column problem, suppose that we use two lev- els, “high’’ and “low’’ (denoted +1 and -1, respectively), for each of the three factors. A very reasonable experiment design strategy uses every possible combination of the factor levels to form a basic experiment with eight different settings for the process. This type of experiment is called a factorial experiment. See Table 1-1 for this experimental design. Figure 1-5 illustrates that this design forms a cube in terms of these high and low levels. With eac