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Applied Statistics

and Probability

for Engineers

Third Edition

Douglas C. Montgomery

Arizona State University

George C. Runger

Arizona State University

John Wiley & Sons, Inc.

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Library of Congress Cataloging-in-Publication Data

Montgomery, Douglas C. Applied statistics and probability for engineers / Douglas C. Montgomery, George C. Runger.—3rd ed. p. cm. Includes bibliographical references and index. ISBN 0-471-20454-4 (acid-free paper)

1. Statistics. 2. Probabilities. I. Runger, George C. II. Title.

QA276.12.M645 2002 519.5—dc 2002016765

Printed in the United States of America.

10 9 8 7 6 5 4 3 2 1

v

Preface

This is an introductory textbook for a first course in applied statistics and probability for un- dergraduate 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 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 ex- amples 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 expe- riences. 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 mate- rial for two courses in the hope that more students will see the important applications of sta- tistics in their everyday work and elect a second course. We believe that this book will also serve as a useful reference.

ORGANIZATION OF THE BOOK

We have retained the relatively modest mathematical level of the first two 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 understand- ing 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. 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 va- riety 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. After struggling with these issues, we decided to divide the text into two components; a set of core topics, many of which are most

PREFACE vii

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 mind-expanding exercises that often require the student to extend the text material somewhat or to apply it in a novel situation. As noted above, answers are provided to most odd-numbered exercises and the e-Text contains com- plete solutions to selected exercises.

USING THE BOOK

This is a very flexible textbook because instructors’ ideas about what should be in a first 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 first course in statistics for engineers should be primarily an applied sta- tistics 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 though 10 on confi- dence 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 16 (four lectures). This leaves about three to four pe- riods 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 e-Text 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 concepts. 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 output from Minitab as typical examples of what can be done with modern sta- tistical software. In teaching, we have used other software packages, including Statgraphics, JMP, and Statisticia. We did not clutter up the book with examples from many different packages because how the instructor integrates the software into the class is ultimately more important than which package is used. All text data is available in electronic form on the e-Text CD. 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 on the PC local area net- works. We have found that this greatly improves the pace of the course and student under- standing of the material.

viii PREFACE

USING THE WEB

Additional resources for students and instructors can be found at www.wiley.com/college/ montgomery/.

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 first two editions provided excellent suggestions that we have tried to incorporate in this revision. We also thank Professors Manuel D. Rossetti (University of Arkansas), Bruce Schmeiser (Purdue University), Michael G. Akritas (Penn State University), and Arunkumar Pennathur (University of Texas at El Paso) for their insightful reviews of the manuscript of the third edition. 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. John Wiley and Sons, Prentice Hall, the Institute of Mathematical Statistics, and the editors of Biometrics allowed us to use copyrighted material, for which we are grateful. Thanks are also due to Dr. Lora Zimmer, Dr. Connie Borror, and Dr. Alejandro Heredia-Langner for their outstanding work on the solutions to exercises.

Douglas C. Montgomery George C. Runger

x CONTENTS

CHAPTER 5 Joint Probability

Distributions 141

5-1 Two Discrete Random Variables 142 5-1.1 Joint Probability Distributions 142 5-1.2 Marginal Probability Distributions 144 5-1.3 Conditional Probability Distributions 146 5-1.4 Independence 148 5-2 Multiple Discrete Random Variables 151 5-2.1 Joint Probability Distributions 151 5-2.2 Multinomial Probability Distribution 154 5-3 Two Continuous Random Variables 157 5-3.1 Joint Probability Distributions 157 5-3.2 Marginal Probability Distributions 159 5-3.3 Conditional Probability Distributions 162 5-3.4 Independence 164 5-4 Multiple Continuous Random Variables 167 5-5 Covariance and Correlation 171 5-6 Bivariate Normal Distribution 177 5-7 Linear Combinations of Random Variables 180 5-8 Functions of Random Variables (CD Only) 185 5-9 Moment Generating Functions (CD Only) 185 5-10 Chebyshev’s Inequality (CD Only) 185

CHAPTER 6 Random Sampling

and Data Description 189

6-1 Data Summary and Display 190 6-2 Random Sampling 195 6-3 Stem-and-Leaf Diagrams 197 6-4 Frequency Distributions and Histograms 203 6-5 Box Plots 207 6-6 Time Sequence Plots 209 6-7 Probability Plots 212 6-8 More About Probability Plotting (CD Only) 216

CHAPTER 7 Point Estimation of

Parameters 220

7-1 Introduction 221 7-2 General Concepts of Point Estimation 222 7-2.1 Unbiased Estimators 222 7-2.2 Proof that S is a Biased Estimator of  (CD Only) 224 7-2.3 Variance of a Point Estimator 224 7-2.4 Standard Error: Reporting a Point Estimator 225 7-2.5 Bootstrap Estimate of the Standard Error (CD Only) 226 7-2.6 Mean Square Error of an Estimator 226 7-3 Methods of Point Estimation 229 7-3.1 Method of Moments 229 7-3.2 Method of Maximum Likelihood 230 7-3.3 Bayesian Estimation of Parameters (CD Only) 237 7-4 Sampling Distributions 238 7-5 Sampling Distribution of Means 239

CHAPTER 8 Statistical Intervals

for a Single Sample 247

8-1 Introduction 248 8-2 Confidence Interval on the Mean of a Normal Distribution, Variance Known 249 8-2.1 Development of the Confidence Interval and Its Basic Properties 249 8-2.2 Choice of Sample Size 252 8-2.3 One-sided Confidence Bounds 253 8-2.4 General method to Derive a Confidence Interval 253 8-2.5 A Large-Sample Confidence Interval for  254 8-2.6 Bootstrap Confidence Intervals (CD Only) 256 8-3 Confidence Interval on the Mean of a Normal Distribution, Variance Unknown 257 8-3.1 The t Distribution 258 8-3.2 Development of the t Distribution (CD Only) 259 8-3.3 The t Confidence Interval on  259

CONTENTS xi

8-4 Confidence Interval on the Variance and Standard Deviation of a Normal Distribution 261 8-5 A Large-Sample Confidence Interval for a Population Proportion 265 8-6 A Prediction Interval for a Future Observation 268 8-7 Tolerance Intervals for a Normal Distribution 270

CHAPTER 9 Tests of Hypotheses

for a Single Sample 277

9-1 Hypothesis Testing 278 9-1.1 Statistical Hypotheses 278 9-1.2 Tests of Statistical Hypotheses 280 9-1.3 One-Sided and Two-Sided Hypotheses 286 9-1.4 General Procedure for Hypothesis Testing 287 9-2 Tests on the Mean of a Normal Distribution, Variance Known 289 9-2.1 Hypothesis Tests on the Mean 289 9-2.2 P -Values in Hypothesis Tests 292 9-2.3 Connection Between Hypothesis Tests and Confidence Intervals 293 9-2.4 Type II Error and Choice of Sample Size 293 9-2.5 Large Sample Test 297 9-2.6 Some Practical Comments on Hypothesis Tests 298 9-3 Tests on the Mean of a Normal Distribution, Variance Unknown 300 9-3.1 Hypothesis Tests on the Mean 300 9-3.2 P -Value for a t -Test 303 9-3.3 Choice of Sample Size 304 9-3.4 Likelihood Ratio Approach to Development of Test Procedures (CD Only) 305 9-4 Tests on the Variance and Standard Deviation of a Normal Distribution 307 9-4.1 The Hypothesis Testing Procedures 307 9-4.2 -Error and Choice of Sample Size 309

9-5 Tests on a Population Proportion 310 9-5.1 Large-Sample Tests on a Proportion 310 9-5.2 Small-Sample Tests on a Proportion (CD Only) 312 9-5.3 Type II Error and Choice of Sample Size 312 9-6 Summary of Inference Procedures for a Single Sample 315 9-7 Testing for Goodness of Fit 315 9-8 Contingency Table Tests 320

CHAPTER 10 Statistical Inference

for Two Samples 327

10-1 Introduction 328 10-2 Inference For a Difference in Means of Two Normal Distributions, Variances Known 328 10-2.1 Hypothesis Tests for a Difference in Means, Variances Known 329 10-2.2 Choice of Sample Size 331 10-2.3 Identifying Cause and Effect 333 10-2.4 Confidence Interval on a Difference in Means, Variances Known 334 10-3 Inference For a Difference in Means of Two Normal Distributions, Variances Unknown 337 10-3.1 Hypothesis Tests for a Difference in Means, Variances Unknown 337 10-3.2 More About the Equal Variance Assumption (CD Only) 344 10-3.3 Choice of Sample Size 344 10-3.4 Confidence Interval on a Difference in Means, Variances Unknown 345 10-4 Paired t -Test 349 10-5 Inference on the Variances of Two Normal Distributions 355 10-5.1 The F Distribution 355 10-5.2 Development of the F Distribution (CD Only) 357 10-5.3 Hypothesis Tests on the Ratio of Two Variances 357 10-5.4 -Error and Choice of Sample Size 359 10-5.5 Confidence Interval on the Ratio of Two Variances 359

CONTENTS xiii

13-2.4 More About Multiple Comparisons (CD Only) 481 13-2.5 Residual Analysis and Model Checking 481 13-2.6 Determining Sample Size 482 13-2.7 Technical Details about the Analysis of Variance (CD Only) 485 13-3 The Random Effects Model 487 13-3.1 Fixed Versus Random Factors 487 13-3.2 ANOVA and Variance Components 487 13-3.3 Determining Sample Size in the Random Model (CD Only) 490 13-4 Randomized Complete Block Design 491 13-4.1 Design and Statistical Analysis 491 13-4.2 Multiple Comparisons 497 13-4.3 Residual Analysis and Model Checking 498 13-4.4 Randomized Complete Block Design with Random Factors (CD Only) 498

CHAPTER 14 Design of

Experiments with Several

Factors 505

14-1 Introduction 506 14-2 Some Applications of Designed Experiments (CD Only) 506 14-3 Factorial Experiments 506 14-4 Two-Factor Factorial Experiments 510 14-4.1 Statistical Analysis of the Fixed- Effects Model 511 14-4.2 Model Adequacy Checking 517 14-4.3 One Observation Per Cell 517 14-4.4 Factorial Experiments with Random Factors: Overview 518 14-5 General Factorial Experiments 520 14-6 Factorial Experiments with Random Factors (CD Only) 523 14-7 2 k^ Factorial Designs 523 14-7.1 2^2 Design 524 14-7.2 2 k^ Design for k  3 Factors 529 14-7.3 Single Replicate of the 2 k Design 537 14-7.4 Addition of Center Points to a 2 k Design (CD Only) 541

14-8 Blocking and Confounding in the 2 k Design 543 14-9 Fractional Replication of the 2 k Design 549 14-9.1 One Half Fraction of the 2 k^ Design 549 14-9.2 Smaller Fractions: The 2 k  p Fractional Factorial 555 14-10 Response Surface Methods and Designs (CD Only) 564

CHAPTER 15 Nonparametric

Statistics 571

15-1 Introduction 572 15-2 Sign Test 572 15-2.1 Description of the Test 572 15-2.2 Sign Test for Paired Samples 576 15-2.3 Type II Error for the Sign Test 578 15-2.4 Comparison to the t -Test 579 15-3 Wilcoxon Signed-Rank Test 581 15-3.1 Description of the Test 581 15-3.2 Large-Sample Approximation 583 15-3.3 Paired Observations 583 15-3.4 Comparison to the t -Test 584 15-4 Wilcoxon Rank-Sum Test 585 15-4.1 Description of the Test 585 15-4.2 Large-Sample Approximation 587 15-4.3 Comparison to the t -Test 588 15-5 Nonparametric Methods in the Analysis of Variance 589 15-5.1 Kruskal-Wallis Test 589 15-5.2 Rank Transformation 591

CHAPTER 16 Statistical Quality

Control 595

16-1 Quality Improvement and Statistics 596 16-2 Statistical Quality Control 597 16-3 Statistical Process Control 597 16-4 Introduction to Control Charts 598 16-4.1 Basic Principles 598 16-4.2 Design of a Control Chart 602 16-4.3 Rational Subgroups 603 16-4.4 Analysis of Patterns on Control Charts 604 16-5 and R or S Control Chart 607 16-6 Control Charts for Individual Measurements 615

X

xiv CONTENTS

16-7 Process Capability 619 16-8 Attribute Control Charts 625 16-8.1 P Chart (Control Chart for Proportion) 625 16-8.2 U Chart (Control Chart for Defects per Unit) 627 16-9 Control Chart Performance 630 16-10 Cumulative Sum Control Chart 632 16-11 Other SPC Problem-Solving Tools 639 16-12 Implementing SPC 641

APPENDICES 649

APPENDIX A: Statistical Tables

and Charts 651

Table I Summary of Common Probability Distributions 652 Table II Cumulative Standard Normal Distribution 653 Table III Percentage Points ^2 , of the Chi- Squared Distribution 655 Table IV Percentage Points t (^) , of the t -distribution 656

Table V Percentage Points f (^) ,v 1 ,v 2 of the F -distribution 657 Chart VI Operating Characteristic Curves 662 Table VII Critical Values for the Sign Test 671 Table VIII Critical Values for the Wilcoxon Signed-Rank Test 671 Table IX Critical Values for the Wilcoxon Rank-Sum Test 672 Table X Factors for Constructing Variables Control Charts 673 Table XI Factors for Tolerance Intervals 674

APPENDIX B: Bibliography 677

APPENDIX C: Answers to

Selected Exercises

GLOSSARY 689

INDEX 703

PROBLEM SOLUTIONS

2 CHAPTER 1 THE ROLE OF STATISTICS IN ENGINEERING

available for some of the text sections that appear on CD only. These exercises may be found within the e-Text immediately following the section they accompany.

1-1 THE ENGINEERING METHOD AND STATISTICAL THINKING

An engineer is someone who solves problems of interest to society by the efficient application of scientific principles. Engineers accomplish this by either refining an existing product or process or by designing a new product or process that meets customers’ needs. The engineering, or scientific, 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 scientific 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. Refine 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 confirm that the proposed solution to the prob- lem is both effective and efficient. 8. Draw conclusions or make recommendations based on the problem solution. The steps in the engineering method are shown in Fig. 1-1. Notice that the engineering method features a strong interplay between the problem, the factors that may influence 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 final solution. Consequently, engineers must know how to efficiently plan experiments, collect data, analyze and interpret the data, and understand how the observed data are related to the model they have proposed for the problem under study. The field of statistics deals with the collection, presentation, analysis, and use of data to make decisions, solve problems, and design products and processes. Because many aspects of engineering practice involve working with data, obviously some knowledge of statistics is important to any engineer. Specifically, statistical techniques can be a powerful aid in design- ing new products and systems, improving existing designs, and designing, developing, and improving production processes.

Figure 1-1 The engineering method.

Develop a clear description

Identify the important factors

Propose or refine a model

Conduct experiments

Manipulate the model

Confirm the solution

Conclusions and recommendations

1-1 THE ENGINEERING METHOD AND STATISTICAL THINKING 3

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, consider 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 condition of the vehicle over time (which could include factors such as tire inflation, engine compression, 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 gives us a framework for describing this variability 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, sup- pose that an engineer is designing a nylon connector to be used in an automotive engine application. The engineer is considering establishing the design specification on wall thick- ness at 332 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

(1-1)

where  is a constant and  is a random disturbance. The constant remains the same with every measurement, but small changes in the environment, test equipment, differences in the indi- vidual parts themselves, and so forth change the value of . If there were no disturbances,  would always equal zero and X would always be equal to the constant . 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 see eas- ily 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 difficult to identify any specific patterns in the vari- ability, although the dot diagram is a convenient way to see any unusual data features. 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

X    

12 13 14 15 Pull-off force Figure 1-2 Dot diagram of the pull-off force data when wall thickness is 3/32 inch.

12 13 14 15 Pull-off force

3 32 inch inch

= 1 = 8

Figure 1-3 Dot diagram of pull-off force for two wall thicknesses.

1-2 COLLECTING ENGINEERING DATA 5

The wafers-from-lots example is called an enumerative study. A sample is used to make an inference to the population from which the sample is selected. The connector example is called an analytic study. A sample is used to make an inference to a conceptual (future) population. The statistical analyses are usually the same in both cases, but an analytic study clearly requires an assumption of stability. See Fig. 1-5, on page 4.

1-2 COLLECTING ENGINEERING DATA

1-2.1 Basic Principles

In the previous section, we illustrated some simple methods for summarizing data. In the en- gineering environment, the data is almost always a sample that has been selected from some population. Three basic methods of collecting data are A retrospective study using historical data An observational study A designed experiment 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 ex- amples of these data collection methods.

1-2.2 Retrospective Study

Montgomery, Peck, and Vining (2001) describe an acetone-butyl alcohol distillation column for which concentration of acetone in the distillate or output product stream is an important variable. Factors that may affect the distillate are the reboil temperature, the con- densate temperature, and the reflux rate. Production personnel obtain and archive the following records: The concentration of acetone in an hourly test sample of output product The reboil temperature log, which is a plot of the reboil temperature over time The condenser temperature controller log The nominal reflux rate each hour The reflux rate should be held constant for this process. Consequently, production personnel change this very infrequently. A retrospective study would use either all or a sample of the historical process data archived over some period of time. The study objective might be to discover the relationships among the two temperatures and the reflux 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 reflux rate and acetone con- centration, because the reflux rate didn’t change much over the historical period. 2. The archived data on the two temperatures (which are recorded almost continu- ously) do not correspond perfectly to the acetone concentration measurements (which are made hourly). It may not be obvious how to construct an approximate correspondence.

6 CHAPTER 1 THE ROLE OF STATISTICS IN ENGINEERING

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 difficult to assess their real impact on acetone concentration. 4. Within the narrow ranges that they do vary, the condensate temperature tends to in- crease with the reboil temperature. Consequently, the effects of these two process variables on acetone concentration may be difficult to separate. As you can see, a retrospective study may involve a lot of data, but that 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 outliers (or unusual values), or data on other important factors may not have been collected and archived. In the distillation column, for example, the specific concentrations of butyl alco- hol and acetone in the input feed stream are a very important factor, 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 identify interesting phenomena, but solid and reliable explanations of these phenomena are often difficult to obtain.

1-2.3 Observational Study

In an observational study, the engineer observes the process or population, disturbing it as lit- tle as possible, and records the quantities of interest. Because these studies are usually con- ducted 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 reflux 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 fac- tor could be studied. Generally, an observational study tends to solve problems 1 and 2 above 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 control- lable 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 ex- periment; that is, a deliberate change was made in the wall thickness of the connector with the objective of discovering whether or not a greater pull-off force could be obtained. Designed experiments play a very important role in engineering design and development and in the improvement of manufacturing processes. Generally, when products and processes are designed and developed with designed experiments, they enjoy better performance, higher reliability, and lower overall costs. Designed experiments also play a crucial role in reducing the lead time for engineering design and development activities. For example, consider the problem involving the choice of wall thickness for the nylon connector. 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