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Statistics and Probability for Engineering Applications With Microsoft® Excel by W.J. DeCoursey College of Engineering, University of Saskatchewan Saskatoon Amsterd am Boston London New York Oxford Paris San Diego San Francisco Singapore Sydney Tokyo Newnes is an imprint of Elsevier Science. Copyright © 2003, Elsevier Science (USA). 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, photocopy ing, recording, or otherwise, without the prior written permission of the publisher. Recognizing the importance of preserving what has been written, Elsevier Science prints its books on acid-free paper whenever possible. Library of Congress Cataloging-in-Publication Data ISBN: 0-7506-7618-3 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. The publisher offers special discounts on bulk orders of this book. For information, please contact: Manager of Special Sales Elsevier Science 225 Wildwood Avenue Woburn, MA 01801-2041 Tel: 781-904-2500 Fax: 781-904-2620 For information on all Newnes publications available, contact our World Wide Web home page at: http://www.newnespress.com 10 9 8 7 6 5 4 3 2 1 Printed in the United States of America Contents Preface ................................................................................................ xi What’s on the CD-ROM? ................................................................. xiii List of Symbols .................................................................................. xv 1. Introduction: Probability and Statistics......................................... 1 1.1 Some Important Terms ................................................................... 1 1.2 What does this book contain? ....................................................... 2 2. Basic Probability ............................................................................. 6 2.1 Fundamental Concepts .................................................................. 6 2.2 Basic Rules of Combining Probabilities ......................................... 11 2.2.1 Addition Rule .................................................................... 11 2.2.2 Multiplication Rule ............................................................ 16 2.3 Permutations and Combinations .................................................. 29 2.4 More Complex Problems: Bayes’ Rule .......................................... 34 3. Descriptive Statistics: Summary Numbers ................................... 41 3.1 Central Location .......................................................................... 41 3.2 Variability or Spread of the Data ................................................... 44 3.3 Quartiles, Deciles, Percentiles, and Quantiles ................................ 51 3.4 Using a Computer to Calculate Summary Numbers ...................... 55 4. Grouped Frequencies and Graphical Descriptions ..................... 63 4.1 Stem-and-Leaf Displays ................................................................ 63 4.2 Box Plots ...................................................................................... 65 4.3 Frequency Graphs of Discrete Data .............................................. 66 4.4 Continuous Data: Grouped Frequency ......................................... 66 4.5 Use of Computers ........................................................................ 75 v 5. Probability Distributions of Discrete Variables ........................... 84 5.1 Probability Functions and Distribution Functions .......................... 85 (a) Probability Functions ............................................................... 85 (b) Cumulative Distribution Functions .......................................... 86 5.2 Expectation and Variance ............................................................. 88 (a) Expectation of a Random Variable .......................................... 88 (b) Variance of a Discrete Random Variable. ................................. 89 (c) More Complex Problems. ........................................................ 94 5.3 Binomial Distribution ................................................................. 101 (a) Illustration of the Binomial Distribution ................................. 101 (b) Generalization of Results ...................................................... 102 (c) Application of the Binomial Distribution ............................... 102 (d) Shape of the Binomial Distribution ....................................... 104 (e) Expected Mean and Standard Deviation ................................ 105 (f) Use of Computers ................................................................ 107 (g) Relation of Proportion to the Binomial Distribution. .............. 108 (h) Nested Binomial Distributions ............................................... 110 (i) Extension: Multinomial Distributions. .................................... 111 5.4 Poisson Distribution ................................................................... 117 (a) Calculation of Poisson Probabilities ....................................... 118 (b) Mean and Variance for the Poisson Distribution. ................... 123 (c) Approximation to the Binomial Distribution .......................... 123 (d) Use of Computers ................................................................ 125 5.5 Extension: Other Discrete Distributions ....................................... 131 5.6 Relation Between Probability Distributions and Frequency Distributions ............................................................... 133 (a) Comparisons of a Probability Distribution with Corresponding Simulated Frequency Distributions ................ 133 (b) Fitting a Binomial Distribution. .............................................. 135 (c) Fitting a Poisson Distribution. ................................................ 136 6. Probability Distributions of Continuous Variables ................... 141 6.1 Probability from the Probability Density Function ........................ 141 6.2 Expected Value and Variance ..................................................... 149 6.3 Extension: Useful Continuous Distributions ................................ 155 6.4 Extension: Reliability ................................................................... 156 vi 7. The Normal Distribution............................................................. 157 7.1 Characteristics ............................................................................ 157 7.2 Probability from the Probability Density Function ........................ 158 7.3 Using Tables for the Normal Distribution .................................... 161 7.4 Using the Computer .................................................................. 173 7.5 Fitting the Normal Distribution to Frequency Data ...................... 175 7.6 Normal Approximation to a Binomial Distribution ...................... 178 7.7 Fitting the Normal Distribution to Cumulative Frequency Data .......................................................................... 184 7.8 Transformation of Variables to Give a Normal Distribution .......... 190 8. Sampling and Combination of Variables .................................. 197 8.1 Sampling ................................................................................... 197 8.2 Linear Combination of Independent Variables ............................ 198 8.3 Variance of Sample Means ......................................................... 199 8.4 Shape of Distribution of Sample Means: Central Limit Theorem ................................................................ 205 9. Statistical Inferences for the Mean............................................ 212 9.1 Inferences for the Mean when Variance Is Known ...................... 213 9.1.1 Test of Hypothesis ........................................................... 213 9.1.2 Confidence Interval ......................................................... 221 9.2 Inferences for the Mean when Variance Is Estimated from a Sample ........................................................... 228 9.2.1 Confidence Interval Using the t-distribution .................... 232 9.2.2 Test of Significance: Comparing a Sample Mean to a Population Mean ..................................................... 233 9.2.3 Comparison of Sample Means Using Unpaired Samples .. 234 9.2.4 Comparison of Paired Samples ........................................ 238 10. Statistical Inferences for Variance and Proportion ................. 248 10.1 Inferences for Variance ............................................................... 248 10.1.1 Comparing a Sample Variance with a Population Variance ........................................................ 248 10.1.2 Comparing Two Sample Variances .................................. 252 10.2 Inferences for Proportion ........................................................... 261 10.2.1 Proportion and the Binomial Distribution ........................ 261 vii 10.2.2 Test of Hypothesis for Proportion .................................... 261 10.2.3 Confidence Interval for Proportion .................................. 266 10.2.4 Extension ........................................................................ 269 11. Introduction to Design of Experiments................................... 272 11.1 Experimentation vs. Use of Routine Operating Data ................... 273 11.2 Scale of Experimentation ............................................................ 273 11.3 One-factor-at-a-time vs. Factorial Design .................................... 274 11.4 Replication ................................................................................. 279 11.5 Bias Due to Interfering Factors ................................................... 279 (a) Some Examples of Interfering Factors .................................... 279 (b) Preventing Bias by Randomization ........................................ 280 (c) Obtaining Random Numbers Using Excel .............................. 284 (d) Preventing Bias by Blocking .................................................. 285 11.6 Fractional Factorial Designs ........................................................ 288 12. Introduction to Analysis of Variance ....................................... 294 12.1 One-way Analysis of Variance .................................................... 295 12.2 Two-way Analysis of Variance .................................................... 304 12.3 Analysis of Randomized Block Design ........................................ 316 12.4 Concluding Remarks .................................................................. 320 13. Chi-squared Test for Frequency Distributions ........................ 324 13.1 Calculation of the Chi-squared Function .................................... 324 13.2 Case of Equal Probabilities ......................................................... 326 13.3 Goodness of Fit .......................................................................... 327 13.4 Contingency Tables .................................................................... 331 14. Regression and Correlation ..................................................... 341 14.1 Simple Linear Regression ............................................................ 342 14.2 Assumptions and Graphical Checks ........................................... 348 14.3 Statistical Inferences ................................................................... 352 14.4 Other Forms with Single Input or Regressor ............................... 361 14.5 Correlation ................................................................................ 364 14.6 Extension: Introduction to Multiple Linear Regression ................ 367 viii 15. Sources of Further Information ............................................... 373 15.1 Useful Reference Books ............................................................. 373 15.2 List of Selected References ......................................................... 374 Appendices ...................................................................................... 375 Appendix A: Tables ............................................................................. 376 Appendix B: Some Properties of Excel Useful During the Learning Process ....................................................... 382 Appendix C: Functions Useful Once the Fundamentals Are Understood................................................... 386 Appendix D: Answers to Some of the Problems .................................. 387 Engineering Problem-Solver Index ............................................... 391 Index ................................................................................................ 393 ix Preface This book has been written to meet the needs of two different groups of readers. On one hand, it is suitable for practicing engineers in industry who need a better under­ standing or a practical review of probability and statistics. On the other hand, this book is eminently suitable as a textbook on statistics and probability for engineering students. Areas of practical knowledge based on the fundamentals of probability and statistics are developed using a logical and understandable approach which appeals to the reader’s experience and previous knowledge rather than to rigorous mathematical development. The only prerequisites for this book are a good knowledge of algebra and a first course in calculus. The book includes many solved problems showing applications in all branches of engineering, and the reader should pay close attention to them in each section. The book can be used profitably either for private study or in a class. Some material in earlier chapters is needed when the reader comes to some of the later sections of this book. Chapter 1 is a brief introduction to probability and statistics and their treatment in this work. Sections 2.1 and 2.2 of Chapter 2 on Basic Probability present topics that provide a foundation for later development, and so do sections 3.1 and 3.2 of Chapter 3 on Descriptive Statistics. Section 4.4, which discusses representing data for a continuous variable in the form of grouped fre­ quency tables and their graphical equivalents, is used frequently in later chapters. Mathematical expectation and the variance of a random variable are introduced in section 5.2. The normal distribution is discussed in Chapter 7 and used extensively in later discussions. The standard error of the mean and the Central Limit Theorem of Chapter 8 are important topics for later chapters. Chapter 9 develops the very useful ideas of statistical inference, and these are applied further in the rest of the book. A short statement of prerequisites is given at the beginning of each chapter, and the reader is advised to make sure that he or she is familiar with the prerequisite material. This book contains more than enough material for a one-semester or one-quarter course for engineering students, so an instructor can choose which topics to include. Sections on use of the computer can be left for later individual study or class study if so desired, but readers will find these sections using Excel very useful. In my opinion a course on probability and statistics for undergraduate engineering students should xi include at least the following topics: introduction (Chapter 1), basic probability (sections 2.1 and 2.2), descriptive statistics (sections 3.1 and 3.2), grouped frequency (section 4.4), basics of random variables (sections 5.1 and 5.2), the binomial distribu­ tion (section 5.3) (not absolutely essential), the normal distribution (sections 7.1, 7.2, 7.3), variance of sample means and the Central Limit Theorem (from Chapter 8), statistical inferences for the mean (Chapter 9), and regression and correlation (from Chapter 14). A number of other topics are very desirable, but the instructor or reader can choose among them. It is a pleasure to thank a number of people who have made contributions to this book in one way or another. The book grew out of teaching a section of a general engineering course at the University of Saskatchewan in Saskatoon, and my approach was affected by discussions with the other instructors. Many of the examples and the problems for readers to solve were first suggested by colleagues, including Roy Billinton, Bill Stolte, Richard Burton, Don Norum, Ernie Barber, Madan Gupta, George Sofko, Dennis O’Shaughnessy, Mo Sachdev, Joe Mathews, Victor Pollak, A.B. Bhattacharya, and D.R. Budney. Discussions with Dennis O’Shaughnessy have been helpful in clarifying my ideas concerning the paired t-test and blocking. Example 7.11 is based on measurements done by Richard Evitts. Colleagues were very generous in reading and commenting on drafts of various chapters of the book; these include Bill Stolte, Don Norum, Shehab Sokhansanj, and particularly Richard Burton. Bill Stolte has provided useful comments after using preliminary versions of the book in class. Karen Burlock typed the first version of Chapter 7. I thank all of these for their contributions. Whatever errors remain in the book are, of course, my own responsibility. I am grateful to my editor, Carol S. Lewis, for all her contributions in preparing this book for publication. Thank you, Carol! W.J. DeCoursey Department of Chemical Engineering College of Engineering University of Saskatchewan Saskatoon, SK, Canada S7N 5A9 xii What’s on the CD-ROM? Included on the accompanying CD-ROM: • a fully searchable eBook version of the text in Adobe pdf form • data sets to accompany the examples in the text • in the “Extras” folder, useful statistical software tools developed by the Statistical Engineering Division, National Institute of Science and Technology (NIST). Once again, you are cautioned not to apply any tech­ nique blindly without first understanding its assumptions, limitations, and area of application. Refer to the Read-Me file on the CD-ROM for more detailed information on these files and applications. xiii

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