Design of Experiments with MINITAB Paul G Mathews ASQ Quality Press Milwaukee, Wisconsin American Society for Quality, Quality Press, Milwaukee 53203 © 2005 by ASQ All rights reserved Published 2004 Printed in the United States of America 12 11 10 09 08 07 06 05 04 Library of Congress Cataloging-in-Publication Data Mathews, Paul G., 1960– Design of experiments with MINITAB / Paul G Mathews p cm Includes bibliographical references and index ISBN 0-87389-637-8 (hardcover, case binding : alk paper) Statistical hypothesis testing Experimental design Minitab Science—Statistical methods Engineering—Statistical methods I Title QA277.M377 2004 519.5'7—dc22 2004020013 ISBN 0-87389-637-8 Copyright Protection Notice for the ANSI/ISO 9000 Series Standards: These materials are subject to copyright claims of ISO, ANSI, and ASQ Not for resale No part of this publication may be reproduced in any form, including an electronic retrieval system, without the prior written permission of ASQ All requests pertaining to the ANSI/ISO 9000 Series Standards should be submitted to ASQ No part of this book may be reproduced in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher Publisher: William A Tony Acquisitions Editor: Annemieke Hytinen Project Editor: Paul O’Mara Production Administrator: Randall Benson Special Marketing Representative: David Luth ASQ Mission: The American Society for Quality advances individual, organizational, and community excellence worldwide through learning, quality improvement, and knowledge exchange Attention Bookstores, Wholesalers, Schools and Corporations: ASQ Quality Press books, videotapes, audiotapes, and software are available at quantity discounts with bulk purchases for business, educational, or instructional use For information, please contact ASQ Quality Press at 800-248-1946, or write to ASQ Quality Press, P.O Box 3005, Milwaukee, WI 53201-3005 To place orders or to request a free copy of the ASQ Quality Press Publications Catalog, including ASQ membership information, call 800-248-1946 Visit our Web site at www.asq.org or http://qualitypress.asq.org Printed on acid-free paper Preface WHAT IS DOE? Design of experiments (DOE) is a methodology for studying any response that varies as a function of one or more independent variables or knobs By observing the response under a planned matrix of knob settings, a statistically valid mathematical model for the response can be determined The resulting model can be used for a variety of purposes: to select optimum levels for the knobs; to focus attention on the crucial knobs and eliminate the distractions caused by minor or insignificant knobs; to provide predictions for the response under a variety of knob settings; to identify and reduce the response’s sensitivity to troublesome knobs and interactions between knobs; and so on Clearly, DOE is an essential tool for studying complex systems and it is the only rigorous replacement for the inferior but unfortunately still common practice of studying one variable at a time (OVAT) WHERE DID I LEARN DOE? When I graduated from college and started working at GE Lighting as a physicist/engineer, I quickly found that statistical methods were an integral part of their design, process, and manufacturing operations Although I’d had a mathematical statistics course as an undergraduate physics student, I found that my training in statistics was completely inadequate for survival in the GE organization However, GE knew from experience that this was a major weakness of most if not all of the entry-level engineers coming from any science or engineering program (and still is today), and dealt with the problem by offering a wonderful series of internal statistics courses Among those classes was my first formal training in DOE—a 20-contact-hour course using Hicks, Fundamental Concepts of Design of Experiments To tell the truth, we spent most of our time in that class solving DOE problems with pocket calculators because there was litxiii xiv Preface tle software available at the time Although to some degree the calculations distracted me from the bigger DOE picture, that course made the power and efficiency offered by DOE methods very apparent Furthermore, DOE was part of the GE Lighting culture— if your work plans didn’t incorporate DOE methods they didn’t get approved During my twelve years at GE Lighting I was involved in about one experiment per week Many of the systems that we studied were so complex that there was no other possible way of doing the work While our experiments weren’t always successful, we did learn from our mistakes, and the designs and processes that we developed benefited greatly from our use of DOE methods The proof of our success is shown by the longevity of our findings—many of the designs and processes that we developed years ago are still in use today, even despite recent attempts to modify and improve them Although I learned the basic designs and methods of DOE at GE, I eventually realized that we had restricted ourselves to a relatively small subset of the available experiment designs This only became apparent to me after I started teaching and consulting on DOE to students and corporate clients who had much more diverse requirements I have to credit GE with giving me a strong foundation in DOE, but my students and clients get the credit for really opening my eyes to the true range of possibilities for designed experiments WHY DID I WRITE THIS BOOK? The first DOE courses that I taught were at GE Lighting and Lakeland Community College in Kirtland, Ohio At GE we used RS1 and MINITAB for software while I chose MINITAB for Lakeland The textbooks that I chose for those classes were Montgomery, Design and Analysis of Experiments and Hicks, Fundamental Concepts in the Design of Experiments, however, I felt that both of those books spent too much time describing the calculations that the software took care of for us and not enough time presenting the full capabilities offered by the software Since many students were still struggling to learn DOS while I was trying to teach them to use MINITAB, I supplemented their textbooks with a series of documents that integrated material taken from the textbooks with instructions for using the software As those documents became more comprehensive they evolved into this textbook I still have and occasionally use Montgomery; Box, Hunter, and Hunter, Statistics for Experimenters; Hicks; and other DOE books, but as my own book has become more complete I find that I am using those books less and less often and then only for reference WHAT IS THE SCOPE OF THIS BOOK? I purposely limited the scope of this book to the basic DOE designs and methods that I think are essential for any engineer or scientist to understand This book is limited to the study of quantitative responses using one-way and multi-way classifications, full Preface xv and fractional factorial designs, and basic response-surface designs I’ve left coverage of other experiment designs and analyses, including qualitative and binary responses, Taguchi methods, and mixture designs, to the other books However, students who learn the material in this book and gain experience by running their own experiments will be well prepared to use those other books and address those other topics when it becomes necessary SAMPLE-SIZE CALCULATIONS As a consultant, I’m asked more and more often to make sample-size recommendations for designed experiments Obviously this is an important topic Even if you choose the perfect experiment to study a particular problem, that experiment will waste time and resources if it uses too many runs and it will put you and your organization at risk if it uses too few runs Although the calculations are not difficult, the older textbooks present little or no instruction on how to estimate sample size To a large degree this is not their fault—at the time those books were written the probability functions and tables required to solve sample-size problems were not readily available But now most good statistical and DOE software programs provide that information and at least a rudimentary interface for sample-size calculations This book is unique in that it presents detailed instructions and examples of sample-size calculations for most common DOE problems HOW COULD THIS BOOK BE USED IN A COLLEGE COURSE? This book is appropriate for a one-quarter or one-semester course in DOE Although the book contains a few references to calculus methods, in most cases alternative methods based on simple algebra are also presented Students are expected to have good algebra skills—no calculus is required As prerequisites, students should have completed either: 1) a one-quarter or semester course in statistical methods for quality engineering (such as with Ostle, Turner, Hicks, and McElrath, Engineering Statistics: The Industrial Experience) or 2) a onequarter or semester course in basic statistics (such as with one of Freund’s books) and a one-quarter or semester course in statistical quality control covering SPC and acceptance sampling (such as with Montgomery’s Statistical Quality Control) Students should also have good Microsoft Windows skills and access to a good general statistics package like MINITAB or a dedicated DOE software package Students meeting the prerequisite requirements should be able to successfully complete a course using this textbook in about 40 classroom/ lab hours with 40 to 80 hours of additional time spent reading and solving homework problems Students must have access to software during class/ lab and to solve homework problems xvi Preface WHY MINITAB? Although most DOE textbooks now present and describe the solutions to DOE problems using one or more software packages, I find that they still tend to be superficial and of little real use to readers and students I chose to use MINITAB extensively in this book for many reasons: • The MINITAB program interface is designed to be very simple and easy to use There are many other powerful programs available that don’t get used much because they are so difficult to run • Despite its apparent simplicity, MINITAB also supports many advanced methods • In addition to the tools required to design and analyze experiments, MINITAB supports most of the other statistical analyses and methods that most users need, such as basic descriptive and inferential statistics, SPC, reliability, GR&R studies, process capability, and so on Why buy, learn, and maintain multiple software packages when one will suffice? • MINITAB has a powerful graphics engine with an easy to use interface Most graph attributes are easy to configure and can be edited after a graph is created All but a few of the graphs in this book were originally created in MINITAB • MINITAB has a simple but powerful integrated sample-size calculation interface that can solve the most common sample-size problems This eliminates the need to buy and learn another program that is dedicated to sample-size calculations MINITAB can also be used to solve many more complex samplesize problems that are not included in the standard interface • MINITAB has a very simple integrated system to package a series of instructions to form an executable macro If you can drive a mouse you can write a MINITAB macro MINITAB macros are easy to edit, customize, and maintain and can be made even more powerful with the higher-level MINITAB macro programming language All of the custom analysis macros that are described in this book are provided on the CD-ROM included with the book • MINITAB is relatively free of bugs and errors, and its output is accurate • MINITAB has a very large established user base • MINITAB’s printed documentation, online help, and technical support are all excellent • MINITAB Incorporated is a large company that will be around for many years • Although price should not be a primary factor in selecting statistical or DOE software, MINITAB is priced competitively for both single users and network installations Preface xvii Despite its dedication to MINITAB, I’ve successfully taught DOE from this book to students and clients who use other software packages Generally the user interfaces and outputs of those packages are similar enough to those of MINITAB that most students learn to readily translate from MINITAB into their own program I’ve tried to use the conventions chosen in the MINITAB documentation to present MINITAB references throughout the book MINITAB commands, buttons, text box labels, and pull-down menus are indicated in boldface MINITAB columns like c1, c2, are indicated in typewriter (Courier) font MINITAB file names and extensions are indicated in italics Variable names are capitalized and displayed in the standard font HOW ARE THE BOOK AND SUPPLEMENTARY CD-ROM ORGANIZED? Since many readers and students who would consider this book have rusty statistical skills, a rather detailed review of graphical data presentation methods, descriptive statistics, and inferential statistics is presented in the first three chapters Sample-size calculations for basic confidence intervals and hypothesis tests are also presented in Chapter This is a new topic for many people and this chapter sets the stage for the sample-size calculations that are presented in later chapters Chapter provides a qualitative introduction to the language and concepts of DOE This chapter can be read superficially the first time, but be prepared to return to it frequently as the topics introduced here are addressed in more detail in later chapters Chapters through present experiment designs and analyses for one-way and multi-way classifications Chapter includes superficial treatment of incomplete designs, nested designs, and fixed, random, and mixed models Many readers/students postpone their study of much of Chapter until after they’ve completed the rest of this book or until they have need for that material Chapter provides detailed coverage of linear regression and the use of variable transformations Polynomial and multivariable regression and general linear models are introduced in preparation for the analysis of multivariable designed experiments Chapters 9, 10, and 11 present two-level full factorial, fractional factorial, and response-surface experiment designs, respectively The analysis of data from these experiments using multiple regression methods and the prepackaged MINITAB DOE analyses is presented Although the two-level plus centers designs are not really responsesurface designs, they are included in the beginning of Chapter 11 because of the new concepts and issues that they introduce The supplementary CD-ROM included with the book contains: • Data files from the example problems in the book • Descriptions of simple experiments with toys that could be performed at home or in a DOE class There are experiments involving magic dice, three different kinds of paper helicopters, the strength of rectangular wooden beams, and xviii Preface catapults Paper helicopter templates are provided on graph paper to simplify the construction of helicopters to various specifications • MINITAB macros for analyzing factorial, fractional factorial, and responsesurface designs • MINITAB macros for special functions • A standard set of experiment design files in MINITAB worksheets • Microsoft Excel experiment design files with integrated simulations RUNNING EXPERIMENTS No matter how hard you study this book or how many of the chapter problems or simulations you attempt, you’ll never become a proficient experimenter unless you actually run lots of experiments In many ways, the material in this book is easy and the hard things—the ones no book can capture—are only learned through experience But don’t rush into performing experiments at work where the results could be embarrassing or worse Rather, take the time to perform the simple experiments with toys that are described in the documents on the supplementary CD-ROM If you can, recruit a DOE novice or child to help you perform these experiments Observe your assistant carefully and honestly note the mistakes that you both make because then you’ll be less likely to commit those mistakes again under more important circumstances And always remember that you usually learn more from a failed experiment than one that goes perfectly Table of Contents Preface xiii Acknowledgments xix Chapter Graphical Presentation of Data 1.1 Introduction 1.2 Types of Data 1.3 Bar Charts 1.4 Histograms 1.5 Dotplots 1.6 Stem-and-Leaf Plots 1.7 Box-and-Whisker Plots 1.8 Scatter Plots 1.9 Multi-Vari Charts 1.10 An Introduction to MINITAB 1.10.1 Starting MINITAB 1.10.2 MINITAB Windows 1.10.3 Using the Command Prompt 1.10.4 Customizing MINITAB 1.10.5 Entering Data 1.10.6 Graphing Data 1.10.7 Printing Data and Graphs 1.10.8 Saving and Retrieving Information 1.10.9 MINITAB Macros 1.10.10 Summary of MINITAB Files 1 4 9 11 11 12 13 13 14 15 17 Chapter Descriptive Statistics 2.1 Introduction 2.2 Selection of Samples 2.3 Measures of Location 2.3.1 The Median 19 19 19 20 20 v vi Table of Contents 2.3.2 The Mean 2.4 Measures of Variation 2.4.1 The Range 2.4.2 The Standard Deviation 2.4.3 Degrees of Freedom 2.4.4 The Calculating Form for the Standard Deviation 2.5 The Normal Distribution 2.6 Counting 2.6.1 Multiplication of Choices 2.6.2 Factorials 2.6.3 Permutations 2.6.4 Combinations 2.7 MINITAB Commands to Calculate Descriptive Statistics 21 21 21 22 24 25 26 30 30 31 31 32 34 Chapter Inferential Statistics 3.1 Introduction 3.2 The Distribution of Sample Means (s Known) 3.3 Confidence Interval for the Population Mean (s Known) 3.4 Hypothesis Test for One Sample Mean (s Known) 3.4.1 Hypothesis Test Rationale 3.4.2 Decision Limits Based on Measurement Units 3.4.3 Decision Limits Based on Standard (z) Units 3.4.4 Decision Limits Based on the p Value 3.4.5 Type and Type Errors 3.4.6 One-Tailed Hypothesis Tests 3.5 The Distribution of Sample Means (s Unknown) 3.5.1 Student’s t Distribution 3.5.2 A One-Sample Hypothesis Test for the Population Mean (s Unknown) 3.5.3 A Confidence Interval for the Population Mean (s Unknown) 3.6 Hypothesis Tests for Two Means 3.6.1 Two Independent Samples (s 21 and s 22 Known) 3.6.2 Two Independent Samples (s 21 and s 22 Unknown But Equal) 3.6.3 Two Independent Samples (s 21 and s 22 Unknown and Unequal) 3.6.4 Paired Samples 3.7 Inferences About One Variance (Optional) 3.7.1 The Distribution of Sample Variances 3.7.2 Hypothesis Test for One Sample Variance 3.7.3 Confidence Interval for the Population Variance 3.8 Hypothesis Tests for Two Sample Variances 3.9 Quick Tests for the Two-Sample Location Problem 37 37 38 41 42 42 44 45 46 49 51 52 52 54 55 56 56 56 58 59 61 61 63 64 65 68 Index Terms Links goodness of fit tests 309 with MINITAB 313 Graeco-Latin Square 233 H half-fractional factorial design 406 heteroscedasticity 101 ANOVA residuals 151 regression model residuals 317 histogram creating in MINITAB 13 interpreting 19 homoscedasticity 101 ANOVA residuals 150 regression model residuals 283 two-sample t test 57 unbalanced one-way classification experiments Hsu’s method hyper-Graeco-Latin Square hypothesis tests 160 59 233 37 conclusions, stating 43 decision limits 44 errors 49 for means and variances 74 with MINITAB 79 for normality 75 for one sample mean 42 for one sample variance 63 one-sided 51 for one-way classification by ANOVA 54 144 for paired samples, location 59 procedure 73 This page has been reformatted by Knovel to provide easier navigation Index Terms Links hypothesis tests (Cont.) quick tests for location 68 rationale 42 for regression coefficients 286 for two independent means 56 for two variances 65 Imanishi-Kari, Theresa 119 incomplete factorial designs 231 I inferences 37 inflection points 30 interaction plots 99 204 205 371 383 203 213 355 371 389 interactions 98 376 interquartile range inverse prediction 289 21 K knobs xiii Kruskal-Wallis test 185 L lack of fit 309 with MINITAB 313 lag-one plot 283 Latin Square 232 least significant difference 252 This page has been reformatted by Knovel to provide easier navigation Index Terms Links Levene’s test 151 modified 168 161 linear goodness of fit See lack of fit linear least squares regression line 275 linear regression assumptions 282 coded variables 318 coefficient of determination 293 confidence limits, line 289 confidence limits, regression coefficients 287 correlation 293 design considerations 345 errors in variables 316 general linear models 327 goodness of fit tests 309 hypothesis tests for regression coefficients 285 lack of fit tests 312 with MINITAB 299 multiple regression 320 polynomial models 306 prediction limits 290 rationale 273 regression coefficients 277 regression constant, determining 341 regression slope, determining 337 sample-size calculations 337 transformations to linear form 301 weighted regression 317 location 19 measures logarithmic transform lower quartile lurking variables 20 179 118 172 This page has been reformatted by Knovel to provide easier navigation Index Terms Links M mac files 17 macros (MINITAB) See also specific macro by name exec 15 local 16 makeoneway.mac 176 Mann-Whitney test 56 mean 20 mean deviation 23 mean squares measurement scale median 21 159 20 method of least squares mgf files 275 17 minimum aberration designs 435 MINITAB column identifier 12 column names 12 command prompt 11 commands, modifying 11 customizing 11 data, entering 12 data, graphing 13 data, printing 13 descriptive statistics, calculating 34 file extensions 17 graph gallery 13 graphs, customizing 13 graphs, printing 14 Graphs folder 10 History window 10 macros, creating 15 This page has been reformatted by Knovel to provide easier navigation Index Terms Links MINITAB (Cont.) project file 14 Project Manager 10 Related Documents folder 10 Report Pad 10 saving and retrieving 14 Session window shortcut to MINITAB starting toolbars 10 windows Worksheet MINITAB local mac macros 12 16 coefftnormplot.mac 379 makeoneway.mac 176 mlrk.mac macros 376 randomize.mac 225 unrandomize.mac 225 missing at random 390 missing values in two-level factorial designs 389 mixed model 241 mlrk.mac macros 376 375 378 418 459 378 418 459 models purpose 104 types 100 Mood’s median test 185 mpj files 17 mtb files 17 mtw files 17 multiple comparison tests, after ANOVA 161 Bonferroni’s method 161 Duncan’s multiple range test 164 Dunnett’s test 167 with MINITAB 168 This page has been reformatted by Knovel to provide easier navigation Index Terms Links multiple comparison tests, after ANOVA (Cont.) Sidak’s method 163 Tukey’s multiple comparison test 166 multiple regression 320 with MINITAB 322 multiplication of choices rule multi-vari charts 31 98 multi-way classification ANOVA, with MINITAB 215 multi-way classification designs 213 227 N nested designs 248 analysis with MINITAB 249 ANOVA tables 249 power calculations 261 nested variables 106 noncentral F distribution 253 noncentrality parameter 87 nonlinear problems normal curve standard 248 237 253 301 26 29 normal curve amplitude 35 normal curve graphing 35 normal distribution 26 normality test normal probability plot 75 quantitative tests 78 normal plots See normal probability plots normal probabilities, calculating in MINITAB 35 This page has been reformatted by Knovel to provide easier navigation Index Terms normal probability distribution Links 26 cumulative 28 probability density function 27 normal probability plots 75 with MINITAB 82 no-way classification 192 nuisance variable 114 null hypothesis acceptance interval 43 44 O Occam’s razor 118 one variable at a time xiii one-tailed hypothesis tests one-way classification experiments 143 188 power calculations 252 for random variables 256 sample-size calculations 185 one-way fixed-effects ANOVA 235 one-way random-effects ANOVA 238 193 51 optimization problems 459 orthogonality 114 outliers 152 identifying 98 51 design considerations operating characteristic curves 93 449 284 OVAT See one variable at a time P paired-sample t-test 59 parameters 19 Pareto charts 37 This page has been reformatted by Knovel to provide easier navigation Index Terms Links permutations 31 Plackett-Burman designs 99 sample-size calculation point estimate 432 37 Poisson distribution 182 polynomial regression 306 with MINITAB 307 pooled sample variance 147 pooling 147 population 19 population mean 20 confidence interval population standard deviation 432 21 41 23 post-ANOVA analysis methods 161 post-ANOVA comparisons 161 post-experiment power of the ANOVA 253 power calculations 250 See also sample-size calculations factorial designs with fixed variables 254 263 factorial designs with random variables 256 266 fractional factorial designs 432 nested designs 261 two steps 252 two-level factorial designs 392 two-level factorial designs with centers 467 prediction limits 290 probability density function, normal 27 probability distributions 26 binomial 183 chi-square 61 F 65 normal 26 Poisson 182 Student’s t 52 Weibull 305 This page has been reformatted by Knovel to provide easier navigation Index Terms Links problem statement 124 procedure 11-step DOE 120 process, DOE documenting 123 11-step procedure 120 inputs 94 model 94 outputs 94 variables 105 project binder, designed experiment 136 propagation of error 390 pure error in linear lack of fit test 312 p value 46 importance 48 Q quadratic model See also polynomial model designs for 437 linear goodness of fit test 309 linear lack of fit test 312 qualitative data displaying qualitative predictor 101 qualitative variable 96 redefining as quantitative qualitative variable levels 105 105 quantitative data presenting quantitative predictor 100 quantitative variable 96 uncontrolled 107 quantitative variable levels 103 105 This page has been reformatted by Knovel to provide easier navigation Index Terms quarter-fractional factorial design quick tests for two-sample location Links 406 68 R random samples 20 random sampling 19 random variables in ANOVA 235 238 256 266 randomization, run order 108 109 128 175 randomization by blocking on replicates 114 randomization plan 128 175 210 212 power calculation with MINITAB validation 113 randomized block design 110 randomized complete block design 211 randomize.mac 225 range 21 rank transform 184 regression analysis See linear regression regression coefficients 276 confidence limits 287 hypothesis tests 285 regression line, confidence limits with MINITAB regression line, prediction limits with MINITAB 277 289 289 290 291 regression model 101 repetitions 113 replicates 113 fractional 116 randomization 114 residuals 101 resolution, design 407 This page has been reformatted by Knovel to provide easier navigation 225 Index Terms responses Links 94 types 97 response-surface designs comparisons number of levels of each variable 99 100 437 453 455 number of observations and error degrees of freedom variable levels, safety 454 456 design considerations 474 with MINITAB 458 sample-size calculations 467 variable levels, sensitivity 456 response transformations 177 rotatability 448 run order, randomization 109 128 112 224 with MINITAB runs 301 175 224 108 adding center cells 367 extra 389 missing 118 order 109 randomization 109 389 S sample 19 sample mean 20 21 distribution 38 52 transformation to standard units 45 53 sample median 20 sample selection 19 sample standard deviation 23 calculating form 25 This page has been reformatted by Knovel to provide easier navigation 363 Index Terms Links sample variances, distribution 61 sample-size calculations xv 82 114 127 224 362 See also power calculations Box-Behnken designs 471 central composite designs 473 for confidence intervals for means fractional factorial designs for hypothesis tests for means linear regression for means with MINITAB k 83 432 86 337 89 for designs 470 for 2k designs 467 for two-level factorial designs 392 Satterthwaite method saturated designs scatter plot screening experiments 59 418 99 shape See distribution shape Sidak’s method software, statistical spreadsheet 163 12 square root transform 179 for count data 182 for fraction data 184 standard deviation calculating form 22 26 standard error of the model 101 standard order 108 star points 448 stem-and-leaf plot stepwise regression Student’s t distribution 30 176 459 52 sums of squares, one-way ANOVA calculating forms 159 This page has been reformatted by Knovel to provide easier navigation Index Terms Links sums of squares, one-way ANOVA (Cont.) defining forms sums of squares, two-way ANOVA interaction 155 202 207 T t test statistic 57 t tests Bonferroni’s method 161 one-sample 54 for outliers 284 paired-sample 59 regression coefficients 287 risk of multiple tests 144 two-sample k 57 factorial experiment design 443 total variation 155 transformations 178 choosing 179 to linear form 301 transformed sample mean 45 transforming count data 182 transforming fraction data 183 Tukey’s honest significant difference test 166 Tukey’s multiple range test 166 Tukey’s quick test 53 69 k factorial designs analyzing 370 analyzing with MINITAB 375 center cells, adding 367 confounding 411 creating in MINITAB 372 design considerations 397 This page has been reformatted by Knovel to provide easier navigation Index Terms Links 2k factorial designs (Cont.) observations, extra or missing 389 propagation of error 390 sample size, determining 392 unbalanced experiment 389 k fractional factorial design 399 analyzing with MINITAB 417 confounding 411 creating in MINITAB 415 design considerations 434 generators 406 407 interpretation 421 429 observations, extra or missing 389 propagation of error 390 resolution 407 sample size, determining 432 two-sample Smirnov test 71 two-sample t test 56 unequal variances 58 two-stage nested design 248 plus centers design 441 two-way classification 196 power calculations 257 two-way factorial design 212 Type errors 49 Type errors 49 430 50 U uniform precision 449 unrandomize.mac 224 upper quartile This page has been reformatted by Knovel to provide easier navigation Index Terms Links V variable levels, selection variables 105 94 nested 106 types 96 uncontrolled 107 variables matrix 107 variance 248 23 variance components 239 confidence intervals 240 gage error studies 243 variation 19 between sample, in one-way ANOVA measures 155 21 total, in one-way ANOVA 155 within sample, in one-way ANOVA 155 W Weibull distribution 305 weighted regression 283 Welch method within-sample variation Worksheet Worksheet window 317 59 155 12 Z z value 45 This page has been reformatted by Knovel to provide easier navigation ... MINITAB for software while I chose MINITAB for Lakeland The textbooks that I chose for those classes were Montgomery, Design and Analysis of Experiments and Hicks, Fundamental Concepts in the Design. .. mistakes, and the designs and processes that we developed benefited greatly from our use of DOE methods The proof of our success is shown by the longevity of our findings—many of the designs and processes... Concepts of Design of Experiments To tell the truth, we spent most of our time in that class solving DOE problems with pocket calculators because there was litxiii xiv Preface tle software available