Applied Statistics and Probability for Engineers 5th Ed.

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Applied Statistics and Probability for Engineers Fifth Edition Douglas C. Montgomery Arizona State University George C. Runger Arizona State University John Wiley & Sons, Inc.

To: Meredith, Neil, Colin, and Cheryl Rebecca, Elisa, George, and Taylor EXECUTIVE PUBLISHER Don Fowley ASSOCIATE PUBLISHER Daniel Sayre ACQUISITIONS EDITOR Jennifer Welter PRODUCTION EDITOR Trish McFadden MARKETING MANAGER Christopher Ruel SENIOR DESIGNER Kevin Murphy MEDIA EDITOR Lauren Sapira PHOTO ASSOCIATE Sheena Goldstein EDITORIAL ASSISTANT Alexandra Spicehandler PRODUCTION SERVICES MANAGEMENT Aptara COVER IMAGE Norm Christiansen This book was set in 10/12 pt. TimesNewRomanPS by Aptara and printed and bound by R.R. Donnelley/Willard Division. The cover was printed by Phoenix Color. This book is printed on acid-free paper. ϱ Copyright © 2011 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. To order books or for customer service, please call 1-800-CALL WILEY (225-5945). ISBN–13: 978-0-470-05304-1 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, Fourth Edition by Montgomery, Runger, and Hubele 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, Fifth Edition by Montgomery and Runger Introduction to engineering statistics, with topical coverage appropriate for either a one- or two-semester course. An applied approach to solving real-world engineering problems. Introduction to Statistical Quality Control, Sixth Edition by Douglas C. Montgomery For a first 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, Seventh Edition by Douglas C. Montgomery An introduction to design and analysis of experiments, with the modest prerequisite of a first course in statistical methods. For senior and graduate students and for practitioners, to design and analyze experiments for improving the quality and efficiency of working systems. Introduction to Linear Regression Analysis, Fourth Edition by Montgomery, Peck, and Vining A comprehensive and thoroughly up-to-date look at regression analysis, still the most widely used technique in statistics today. Response Surface Methodology: Process and Product Optimization Using Designed Experiments, Third Edition by Myers, Montgomery, and Anderson-Cook 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 by Myers, Montgomery, and Vining Website: www.wiley.com/college/myers An introductory text or reference on generalized linear 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 time series data, to draw inferences about the data and generate forecasts useful to the decision maker. Minitab and SAS are used to illustrate how the methods are implemented in practice. For advanced undergrad/first-year graduate, with a prerequisite of basic statistical methods. Portions of the book require calculus and matrix algebra.

Preface INTENDED AUDIENCE This is an introductory textbook for a first course in applied statistics and probability for undergraduate students in engineering and the physical or chemical sciences. These individuals play a significant 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 en- gineer 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 first four editions. We have found that engineering students who have completed one or two semesters of calculus should have no difficulty reading almost 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 reorganizing 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 field of statistics and how engineers use statistical methodology as part of the engineering problem-solving process. This chapter also introduces the reader to some engineer- ing applications of statistics, including building empirical models, designing engineering experiments, and monitoring manufacturing processes. These topics are discussed in more depth in subsequent chapters. Chapters 2, 3, 4, and 5 cover the basic concepts of probability, discrete and continuous random vari- ables, 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 summary and de- scription techniques, including stem-and-leaf plots, histograms, box plots, and probability plotting; and several types of time series plots. Chapter 7 discusses sampling distributions, the central limit theorem, and point estimation of parameters. This chapter also introduces some of the important properties of esti- mators, the method of maximum likelihood, the method of moments, and Bayesian estimation. Chapter 8 discusses interval estimation for a single sample. Topics included are confidence intervals for means, variances or standard deviations, proportions, prediction intervals, and tolerance intervals. Chapter 9 discusses hypothesis tests for a single sample. Chapter 10 presents tests and confidence intervals for two samples. This material has been extensively rewritten and reorganized. There is detailed information and examples of methods for determining appropriate 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 vi

PREFACE vii the concepts behind them. We give a logical, heuristic development of the procedures rather than a formal, mathematical 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 adequacy 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 undergrad- uate 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 ran- domization, blocking, factorial designs, interactions, graphical data analysis, and fractional factorials are emphasized. Chapter 15 introduces statistical quality control, emphasizing the control chart and the fun- damentals of statistical process control. WHAT’S NEW IN THIS EDITION? We received much feedback from users of the fourth edition of the book, and in response we have made substantial changes in this new edition. • The most obvious change is that the chapter on nonparametric methods is gone. We have inte- grated most of this material into Chapter 9 and 10 on statistical hypothesis testing, where we think it is a much better fit if instructors elect to cover these techniques. • Another substantial change is the increased emphasis on the use of P-value in hypothesis test- ing. Many sections of several chapters were rewritten to reflect this. • We have also rewritten and modified many portions of the book to improve the explanations and try to make the concepts easier to understand. • We have added brief comments at the end of examples to emphasize the practical interpretations of the results. • We have also added approximately 200 new homework exercises. FEATURED IN THIS BOOK 4-4 MEAN AND VARIANCE OF A CONTINUOUS RANDOM VARIABLE Definitions, Key Concepts, and Equations The mean and variance can also be defined for a continuous random variable. Integration Throughout the text, definitions and key con- replaces summation in the discrete definitions. If a probability density function is viewed as a cepts and equations are highlighted by a box loading on a beam as in Fig. 4-1, the mean is the balance point. to emphasize their importance. Mean Suppose X is a continuous random variable with probability density function f(x). and The mean or expected value of X, denoted as ␮ or E(X), is Variance Learning Objectives ϱ (4-4) Learning Objectives at the start Ύ␮ ϭ E1X 2 ϭ xf 1x2 dx of each chapter guide the students in what they are Ϫϱ expected to take away from this chapter and serve as a study reference. The variance of X, denoted as V(X) or ␴2, is ϱϱ Ύ Ύ␴2 ϭ V1X 2 ϭ 1x Ϫ ␮22f 1x2 dx ϭ x2f 1x2 dx Ϫ ␮2 Ϫϱ Ϫϱ The standard deviation of X is ␴ ϭ 2␴2. LEARNING OBJECTIVES The equivalence of the two formulas for variance can be derived from the same approach used for discrete random variables. After careful study of this chapter you should be able to do the following: 1. Determine probabilities from probability density functions EXAMPLE 4-6 Electric Current 2. Determine probabilities from cumulative distribution functions and cumulatiFvoer dthisetrciobpupteiorncufrurnenc-t measurement in Example 4-1, the mean The variance of X is tions from probability density functions, and the reverse of X is 20 3. Calculate means and variances for continuous random variables 20 Ύ 20 Ύ4. Understand the assumptions for some common continuous probability distributions E1X 2 ϭ 20 V1X 2 ϭ 1x Ϫ 1022f 1x2 dx ϭ 0.051x Ϫ 1023ր3 ` ϭ 33.33 5. Select an appropriate continuous probability distribution to calculate probabilities in specific applications 0 xf 1x2 dx ϭ 0.05x2ր2 ` ϭ 10 0 0 0 6. Calculate probabilities, determine means and variances for some common continuous probability distributions 7. Standardize normal random variables The expected value of a function h(X ) of a continuous random variable is also defined in a 8. Use the table for the cumulative distribution function of a standard normal distribution to calcu- straightforward manner. late probabilities 9. Approximate probabilities for some binomial and Poisson distributions Expected Value of a If X is a continuous random variable with probability density function f(x), (4-5) Function of a ϱ Continuous Random ΎE3h1X 2 4 ϭ h1x2 f 1x2 dx Variable Ϫϱ

viii PREFACE Seven-Step Procedure for Hypothesis Testing 9-1.6 General Procedure for Hypothesis Tests The text introduces a sequence of seven steps in This chapter develops hypothesis-testing procedures for many practical problems. Use of the applying hypothesis-testing methodology and following sequence of steps in applying hypothesis-testing methodology is recommended. explicitly exhibits this procedure in examples. 1. Parameter of interest: From the problem context, identify the parameter of interest. 2. Null hypothesis, H0: State the null hypothesis, H0. 3. Alternative hypothesis, H1: Specify an appropriate alternative hypothesis, H1. 4. Test statistic: Determine an appropriate test statistic. 5. Reject H0 if: State the rejection criteria for the null hypothesis. 6. Computations: Compute any necessary sample quantities, substitute these into the equation for the test statistic, and compute that value. 7. Draw conclusions: Decide whether or not H0 should be rejected and report that in the problem context. Steps 1–4 should be completed prior to examination of the sample data. This sequence of steps will be illustrated in subsequent sections. Figures Table 11-1 Oxygen and Hydrocarbon Levels Numerous figures throughout the text illustrate statistical concepts in multiple Observation Hydrocarbon Level Purity 100 formats. Number x (%) y (%) 98 y 1 0.99 90.01 (Oxygen 2 1.02 89.05 96 3 1.15 91.43 purity) 4 1.29 93.74 Purity ( y) 94 β 0 + β 1 (1.25) 5 1.46 96.73 β 0 + β 1 (1.00) 6 1.36 94.45 92 7 0.87 87.59 8 1.23 91.77 90 9 1.55 99.42 10 1.40 93.65 88 11 1.19 93.54 12 1.15 92.52 86 0.95 1.05 1.15 1.25 1.35 1.45 1.55 13 0.98 90.56 0.85 Hydrocarbon level (x) 14 1.01 89.54 15 1.11 89.85 Figure 11-1 Scatter diagram of oxygen purity versus hydrocarbon 16 1.20 90.39 level from Table 11-1. 17 True reg1r.e2s6sion line 93.25 18 μY⎜x =1.β302+ β1x 93.41 19 = 75 + 15x 94.98 20 1.43 87.33 0.95 x = 1.00 x = 1.25 x (Hydrocarbon level) Figure 11-2 The distribution of Y for a given value of x for the oxygen purity–hydrocarbon data. Minitab Output Character Stem-and-Leaf Display Throughout the book, we have Stem-and-leaf of Strength presented output from Minitab as typical examples of what can be done N = 80 Leaf Unit = 1.0 with modern statistical software. 176 287 397 5 10 1 5 8 11 0 5 8 11 12 0 1 3 17 13 1 3 3 4 5 5 25 14 1 2 3 5 6 8 9 9 37 15 0 0 1 3 4 4 6 7 8 8 8 8 (10) 16 0003357789 33 17 0 1 1 2 4 4 5 6 6 8 23 18 0 0 1 1 3 4 6 16 19 0 3 4 6 9 9 10 20 0 1 7 8 6 21 8 Figure 6-6 A stem- 5 22 1 8 9 and-leaf diagram from Minitab. 2 23 7 1 24 5

PREFACE ix Example Problems EXAMPLE 10-1 Paint Drying Time 4. Test statistic: The test statistic is A product developer is interested in reducing the drying time A set of example problems provides the stu- of a primer paint. Two formulations of the paint are tested; for- z0˛ ϭ x1 Ϫ x2 Ϫ 0 dent with detailed solutions and comments for mulation 1 is the standard chemistry, and formulation 2 has a interesting, real-world situations. Brief practi- new drying ingredient that should reduce the drying time. ␴21 ϩ ␴22 cal interpretations have been added in this From experience, it is known that the standard deviation of B n1 n2 edition. drying time is 8 minutes, and this inherent variability should be unaffected by the addition of the new ingredient. Ten spec- where ␴12 ϭ ␴22 ϭ 1822 ϭ 64 and n1 ϭ n2 ϭ 10. imens are painted with formulation 1, and another 10 speci- mens are painted with formulation 2; the 20 specimens are 5. Reject H0 if: Reject H0: ␮1 ϭ ␮2 if the P-value is less painted in random order. The two sample average drying times than 0.05. are x1˛ ϭ 121 minutes and x2˛ ϭ 112 minutes, respectively. What conclusions can the product developer draw about the 6. Computations: Since x1 ϭ 121 minutes and x2 ϭ 112 effectiveness of the new ingredient, using ␣ ϭ 0.05? minutes, the test statistic is We apply the seven-step procedure to this problem as z0 ϭ 121 Ϫ 112 ϭ 2.52 follows: 1822 1822 B 10 ϩ 10 1. Parameter of interest: The quantity of interest is the dif- ference in mean drying times, ␮1 Ϫ ␮2, and ⌬0 ϭ 0. 7. Conclusion: Since z0 ϭ 2.52, the P-value is P ϭ 1 Ϫ ⌽12.522 ϭ 0.0059, so we reject H0 at the ␣ ϭ 0.05 level 2. Non hypothesis: H˛0: ␮1 Ϫ ␮2 ϭ 0, or H :0˛ ˛ ␮1 ϭ ␮2. Practical Interpretation: We conclude that adding the 3. Alternative hypothesis: H˛1: ␮1 Ͼ ␮2. We want to reject new ingredient to the paint significantly reduces the drying H0 if the new ingredient reduces mean drying time. time. This is a strong conclusion. Exercises EXERCISES FOR SECTION 5-5 Each chapter has an extensive collection 5-67. Suppose that X is a random variable with probability 5-73. Suppose that X has the probability distribution of exercises, including end-of-section exercises that emphasize the material in distribution that section, supplemental exercises at fX 1x2 ϭ 1ր4, x ϭ 1, 2, 3, 4 fX 1x2 ϭ 1, 1 Յ x Յ 2 the end of the chapter that cover the 8 Find the probability distribution of the random variable scope of chapter topics and require the Find the prob(aab)ilFitiyndditshtreibpurtoiobnaboifliYtyϭdis2tXribϩut1io.n of the random 5-68. Let X beYaϭbi2nXomϩia1l0r.andom variable with p ϭ 0.25 vYaϭriaebXle. The random variabSleuXpphlaesmtheentparol bEaxbielritcyisdeisstri- student to make a decision about the avnardianbleϭY3ϭ. F(5Xbi-n)72d.0F.tihnedStpuhrpeopbeoxasbpeieltcihttyeadtdXvisahtlrauisbeuaotfuionYni.foorfmthperorbaanbdiolimty 5-74. approach they will use to solve the bution 5-75. Show that the following function satisfies the proper- problem, and mind-expanding exer- distribution cises that often require the student to ties of a joint probability mass function: 5-69. Suppose that X is a continuous random variable with x extend the text material somewhat or to fX 1x2 ϭ 8, 0ՅxՅ4 probability distribution apply it in a novel situation. fX 1x2 ϭ 1, 0 Յ x Յ 1 xy f (x, y) Answers are provided to most 1txh2atϭth1ex8p,robab0ilՅityxdՅistr6ibution ShowfX of the random Find the probability distri0bution of Y ϭ 0(X Ϫ 2)2. 1͞4 odd-numbered exercises in Appendix C variable 1͞8 in the text, and the WileyPLUS online Y ϭ Ϫ2 ln X is chi-squared with two degrees of freedom. 01 learning environment includes for stu- 5-71. A random variable X has the following probability 10 1͞8 dents complete detailed solutions to distribution: 11 1͞4 selected exercises. fX 1x2 ϭ eϪx, x Ն 0 22 1͞4 (a) Find the probability ddiissttrrMiibbuuIttNiioonDn ff-ooErr XYY PϭϭAXXN21.ր2D. ING EXERCD(aeI)tSePErm1XSinϽe the following: (b) P1X Յ 12 (b) Find the probability 0.5, Y Ͻ 1.52 (c) Find the probability distribution for Y ϭ ln X. (c) P1X Ͻ 1.52 (d) P1X Ͼ 0.5, Y Ͻ 1.52 5-96.5-7S2h.owTthheatveifloXci1t,y Xo2f,appa,rtXicpleairne aingdaespisenadreanntd,om xavraaerniiadnbdhlee(pye)n, dise((efna))t.fDuMneactertigrominnianolenplEyr(oXobf)a,ybE.il(SiYthy)o,wdVi(stXthrai)bt, uaXntidaonnVd(oYYf).the random vari- continVuowuisthrapnrdoobmabivliatryiadbilsetsri,bPu(tXio1nʦ A1, X2 ʦ A2, p , Xp ʦ Ap) ϭ P(X1 ʦ A1)P(X2 ʦ A2) p P(Xp ʦ Ap) for any 5-100. This exearbclieseXextends the hypergeometric dis- regions A1, A2, p , ApfVin1vt2hϭe raavn2geeϪbovf X1, vXϾ2, p0 , Xp tribution to m(gu)ltiCpolendviatiroianbalepsr.oCbaobnislidtyerdiastrpiobpuutiloantionf Y given that X ϭ 1 respectively. with N items(ohf) kEd1Yiff0 eXreϭnt 1ty2 pes. Assume there are N1 5-97.whSerheowb isthaatcoifnsXta1n, tXt2h,apt d,epXepndasreonintdheepteenmdpeenrtaturieteomfsthoef type (1i,) NA2rieteXmasnodf Ytypined2e,p.e..n,dNeknitt?eWmshyofotrywpehyk not? randomgasvarnidabthleesmanadssYofϭthce1pXa1rtϩiclec.2 X2 ϩ p ϩ cp Xp, so that N1 ϩ N( j2)ϩCpalcϩulapte, tNhek ϭcorNre. lSautipopnobsetwtheaetnaXraann-d Y. V((ba1))Y F2Tiϭhnedckt12ihVnee1Xvtiac1l2ueϩneeorcfg22ytVho1eXfc2to2hneϩsptaapnrttiϩcal.eci2psVW1Xpϭ2 mV 2ր2 . dom sample o5f-7si6z.e nTishesepleecrcteedn,tawgiethoofutpreeopplalecegmiveennt, an antirheumatoid fFrionmd tthhee popumlaetdioicna. tLioent Xw1h, oX2s,u.f..f,eXr ksedveenroet,emthoedneuramteb,eorr minor side effects You may apssruombaebtihliattythdeisrtarinbduotmionvaorfiaWbl.es are continuous. of items of eaacrhety1p0e, i2n0t,haensdam70p%le,soretshpaetcXti1v,eXly2., Aϩspsuϩme that people react p, Xk ϭ n. Show that for feasible values of the parame- 5-98. Suppose that the joint probability function of ters n, x1, x2, p, xk, N1, N2, p, Nk, the probability is P (X1 ϭ the continuous random variables X and Y is constant on axN11b aNx22b axNnkb the rectangle 0 Ͻ x Ͻ a, 0 Ͻ y Ͻ b. Show that X and Y N p are independent. x1, X2 ϭ x2, p, Xk ϭ xk) ϭ 5-99. Suppose that the range of the continuous variables X and Y is 0 Ͻ x Ͻ a and 0 Ͻ y Ͻ b. Also an b suppose that the joint probability density function fXY (x, y) ϭ g (x)h( y), where g (x) is a function only of Important Terms and Concepts IMPORTANT TERMS AND CONCEPTS At the end of each chapter is a list of Bivariate distribution Conditional variance Joint probability density Multinomial distribution important terms and concepts for an Bivariate normal Contour plots function Reproductive property of easy self-check and study tool. Correlation distribution Covariance Joint probability mass the normal Conditional mean function distribution STUDENT RESOURCES • Data Sets Data sets for all examples and exercises in the text. Visit the student section of the book Web site at www.wiley.com/college/montgomery to access these materials. • Student Solutions Manual Detailed solutions for selected problems in the book. The Student Solutions Manual may be purchased from the Web site at www.wiley.com/college/montgomery.