r

Modern Engineering Statistics

LAWRENCE L LAPIN

San Jose State University

An Alexander Kugushev Book

Duxbury Press An Imprint of Wadsworth Publishing Company

l(T)P® An International Thomson Publishing Company Belmont, CA • Albany, NY • Boston • Cincinnati • Detroit • Johannesburg

London • Madrid • Melbourne • Mexico City • New York Paris • Singapore • Tokyo • Toronto • Washington

PREFACE xiii

CHAPTER I

I N T R O D U C T I O N I

l-l The Meaning and Role of Statistics I A Working Definition of Statistics 2 The Role of Modern Statistics 2 Types of Statistics: Descriptive, Inferential,

and Exploratory 2

I -2 Statistical Data 4 Classifications for Data and Variables 4 Types of Quantitative Data 5

I -3 The Population and the Sample 6 Distinguishing among the Data Set,

Population, and Sample 7 Deductive and Inductive Statistics 8 Statistical Error 10

1-4 The Need for Samples 11 Economic Advantages of Samples 11 Further Reasons for Sampling 13 Accuracy and Sampling 14

I -5 Selecting the Sample 15 Sample Selection Using Random

Numbers 15 Presumed Randomness and Computer-

Generated Random Numbers 17

I -6 Engineering Applications of Statistics 18 Statistical Process Control 28 Quality Assessment 19 Model Building and Predicting 19 Communicating with and Acting on

Experimental Results 19 Assessing Design Reliability 20 Experimental Design 20

v

CHAPTER 2

DESCRIBING.DISPLAYING, A N D EXPLORING STATISTICAL DATA 22

2-1 The Frequency Distribution 22 The Histogram and Frequency Curve 23 Stem-and-Leaf Plots 25 Computer-Generated Displays 26 Frequency Distributions for Qualitative

Variables 27 Relative and Cumulative Frequency

Distributions 28 Multidimensional Data Displays 30 Common Forms of the Frequency

Distribution 30

2-2 Summary Statistical Measures: Location 38 Statistics and Parameters 39 The Arithmetic Mean 39 The Median 40 The Mode 41 Finding the Median and Mode with a Stem-

and-Leaf Plot 43 Frequency Distribution Forms and Summary

Measures 43 Percentiles, Fractiles, and Quartiles 44

2-3 Summary Statistical Measures: Variability 50 The Importance of Variability 50 The Range 51 Interquartile Range and Box Plots 52 The Variance and Standard Deviation 53 The Meaning of the Standard Deviation 55 Empirical Rule 55 Composite Summary Measures 57

( vi ) Contents

2-4 Summary Statistical Measures: The Proportion 61

CHAPTER 3

STATISTICAL PROCESS C O N T R O L 66

3-1 The Control Chart 66 Statistical Preliminaries 67 Concept of Statistical Control 68 The Control Chart 70

3-2 Control Charts for Quantitative Data 73 Computing Control Limits for the Mean

Using Specifications 73 Control Limits for the Range and Standard

Deviation Using Specifications 74 Using Control Charts to Uncover Process

Instability 76 Control Charts for the Mean When No

Specifications Are Given 77 Control Limits for the Range and Standard

Deviation When No Specifications Are Given 79

Implementing Statistical Control 81

3-3 Control Charts for Qualitative Data Using the Proportion 83 Control Limits for the Proportion with

Given Specifications 84 Control Limits for the Proportion When

No Specifications are Given 85 Using the Control Chart 86

3-4 Further Issues in Statistical Quality Control 89 Acceptance Sampling 89 Optimization of Quality Loss Function:

Taguchi Method 89

CHAPTER 4

MAKING PREDICTIONS: REGRESSION ANALYSIS 90

4-1 Linear Regression Using Least Squares 90 The Method of Least Squares 91 Rationale and Meaning of Least Squares 93 Measuring the Variability of Results 94

Computer-Assisted Regression Analysis 97

4-2 Correlation and Regression Analysis 102 The Correlation Coefficient 102 Computing the Correlation Coefficient 102

4-3 Multiple Regression Analysis 107 Regression in Three Dimensions 107 Advantages of Multiple Regression 111 Residuais and the Standard Error of the

Estimate 113 Regression with Many Variables 114

CHAPTER 5

STATISTICAL ANALYSIS IN MODEL BUILDING 129

5-1 Nonlinear Regression 129 Using a Linear Surrogate 130 Transforming Variables to Get a Linear

Relationship 130 Finding a Polynomial and Other Multiple

Regression Procedures 132

5-2 Curvilinear Regression 133 Logarithmic Transformations 133 Reciprocal Transformations 136

5-3 Polynomial Regression 138 Parabolic Regression 139 Regression with Higher-Power

Polynomials 140 Polynomial Multiple Regression 141

5-4 Multiple Regression with Indicator Variables 143 The Basic Multiple Regression Model 144 Advantages of Using Indicator

Variables 145 Interactive Multiple Regression with

Indicator Variables 146

CHAPTER 6

PROBABILITY 156

6-1 Fundamental Concepts of Probability 156 Elementary Events and the Sample

Space 156

Event Sets 157 Basic Definitions of Probability 157 Certain and Impossible Events 158 Finding Probabilities from

Experimentation 159 Logically Deducted Probabilities 159

6-2 Probabilities for Compound Events 162 Applying the Basic Definition 162 The Addition Law 163 Application to Complementary Events 164 General Addition Law 164 Statistical Independence 165 The Multiplication Law for Independent

Events 166

6-3 Conditional Probability 170 Establishing Independence by Comparing

Probabilities 171

6-4 The Multiplication Law, Probability Trees, and Sampling 172 The General Multiplication Law 173 Multiplication Law for Several Events 175 The Probability Tree Diagram 175 Probability and Sampling 176

6-5 Predicting the Reliability of Systems 181 Systems with Series Components 181 Systems with Parallel Components 182 Increasing System Reliability 183 Complex Modular Systems 184

CHAPTER 7

RANDOM VARIABLES A N D PROBABILITY DISTRIBUTIONS 191

7-1 Random Variables and Probability Distributions 191 The Random Variable as a Function 192 The Discrete Probability Distribution 193 Continuous Random Variables 194 The Probability Density Function 195

7-2 ExpectedValue andVariance 198 Expected Value 199 Variance and Standard Deviation of a

Random Variable 199

Contents f vii 1

Some Important Properties of Expected Value and Variance 200

Expected Value and Variance of a Continuous Random Variable 201

7-3 The Binomial Distribution 204 The Bernoulli Process 204 Binomial Probabilities 205 Counting Paths of Like Type: Factorials and

Combinations 207 Number of Combinations 208 The Probability Mass Function 208 Expected Value and Variance 210 Cumulative Probability and the Binomial

Probability Table 211 Binomial Distribution and Sampling 214

7-4 The Normal Distribution 216 The Normal Distribution and the Population

Frequency Curve 217 The Normally Distributed Random

Variable 219 The Standard Normal Distribution 220 Probabilities for Any Normal Random

Variable 221 Expected Value, Variance, and

Percentiles 224 Practical Limitations of the Normal

Distribution 225

CHAPTER 8

IMPORTANT PROBABILITY DISTRIBUTIONS IN ENGINEERING 228

8-1 The Poisson Distribution 228 The Poisson Process 229 Poisson Probabilities 229 Parameter Levels and Poisson

Probabilities 230 Importance of Poisson Assumptions 231 Poisson Distribution Function and Probability

Table 232 Expected Value and Variance 233

8-2 The Exponential Distribution 235 Finding Exponential Probabilities 235 Expected Value, Variance, and

Percentiles 237

f viii ) Contents

Applications of the Exponential Distribution 239

8-3 The Gamma Distribution 241 The Gamma Function 241 The Probability Density Function 242 Relation to Poisson Process 242

8-4 Failure-Time Distributions: TheWeibull 243 Exponential Failure-Time Distribution 244 The Failure Rate Function 244 The Weibull Distribution 246 The Gamma as a Failure-Time

Distribution 248 Exponential Series Systems 249 Exponential Parallel Systems: Gamma

Reliability 250

8-5 The Hypergeometric Distribution 251 Finding Hypergeometric Probabilities 253 Expected Value and Variance 255 Binomial Approximation to Hypergeometric

Distribution 255

CHAPTER 9

SAMPLING DISTRIBUTIONS 260

9-1 The Sampling Distribution of the Mean 260 Expected Value and Variance 261 Standard Error of X 262 Theoretical Justification for Results 263

9-2 Sampling Distribution o f X When Population Is Normal 265 The Normal Distribution for X 265 The Role of the Standard Error 267

9-3 Sampling Distribution of X for a General Population 271 Central Limit Theorem 272 Finding Probabilities for X 274 Finding Probabilities with an

Assumed er 275 Sampling from Small Populations 276

9-4 The Student t Distribution 277 The Student t Statistic 278 The Student t and Normal Curves 279

9-5 Sampling Distribution of the Proportion 282 Probabilities for P Using Normal

Approximation 282 Sampling from Small Populations 284

9-6 Sampling Distribution of the Variance:The Chi-Square and F Distributions 285 The Chi-Square Distribution 286 Probabilities for the Sample Variance 287 Assumptions of the Chi-Square

Distribution 289 The F Distribution 289

CHAPTER 10

STATISTICAL ESTIMATION 295

10-1 Estimators and Estimates 295 The Estimation Process 296 Selecting an Estimator 297 Criteria for Statistics Used as

Estimators 297 Commonly Used Estimators 299

10-2 Interval Estimates of the Mean 302 Confidence and Meaning of the Interval

Estimate 303 Confidence Interval for Mean When er Is

Known 304 Confidence Interval Estimate of Mean When

er Is Unknown 305 Confidence Interval When Population Is

Small 309 Determining the Required Sample

Size 309

10-3 Interval Estimates of the Proportion 314 Estimates When Sampling from Large

Populations 314 Estimating the Proportion When Sampling

from Small Populations 315 Required Sample Size 316

10-4 Interval Estimates of the Variance 318 Making Estimates Using the Chi-Square

Distribution 318

10-5 Confidence Intervals for the Difference between Means 319 Independent Samples 320 Matched-Pairs Samples 325 Matched Pairs Compared to Independent

Samples 330

10-6 Bootstrapping Estimation 332 Estimating the Population Mean with

Resampling 333 The Resampling Procedure 333 Resampling with the Computer 334 The Resampling Interval Estimate 335 Bootstrapping versus Traditional

Statistics 336

CHAPTER 11

STATISTICALTESTING 344

11 -1 Basic Concepts of Hypothesis Testing 344 The Structure of a Hypothesis Test 345 Finding the Error Probabilities 347 Determining an Appropriate Decision

Rule 350 The Effect of Sample Size 352 Making the Decision 352 Formulating the Hypotheses 352 Drawing Conclusions with Prob

Values 354

11-2 Procedures for Testing the Mean 356 The Hypothesis-Testing Steps 356 Abbreviated Hypothesis-Testing Steps When

Using Prob Values 360 Upper-Tailed Test Illustrations 360 Lower-Tailed Test Illustrations 366 Two-Sided Test Illustrations 370 Hypothesis Testing and Confidence

Intervals 375

11 -3 Testing the Proportion 379 Testing the Proportion Using the Normal

Approximation 379 A Lower-Tailed Test 381 An Upper-Tailed Test 382 Alternative Procedure for Testing

Proportion 384

Contents ( ix 1

Testing with Binomial Probabilities 385 Abbreviated Tests with the Proportion 387

I 1-4 Hypothesis Tests for ComparingTwo Means 391 Independent Samples 391 Matched Pairs Samples 399 Testing with Bootstrapping Resamples 403

CHAPTER 12

T H E O R Y A N D INFERENCES IN REGRESSION ANALYSIS 412

12-1 Assumptions and Properties of Linear Regression Analysis 412 Assumptions of Linear Regression

Analysis 412 Estimating the True Regression

Equation 414 Rationale for the Method of Least

Squares 414 Appropriateness of Model: Analysis of

Residuais 414 Assumptions of Multiple Regression 417 Pitfalls in Multiple Regression

Analysis 418 Alternative Procedures: Weighted Least

Squares 418

12-2 Assessing the Quality of the Regression 420 The Coefficient of Determination 420 The Coefficient of Determination in

Multiple Regression 423

12-3 Statistical Inferences Using the Regression Line 425 Prediction and Confidence Intervals in

Regression Analysis 425 Confidence Intervals for the Conditional

Mean 425 Prediction Intervals for the Individual Y

GivenX 428 Dangers of Extrapolation 428 Inferences Regarding the Regression

Coefficients 429 Using Bootstrapping to Make

Inferences 430

0 Contents

12-4 Inferences in Multiple Regression Analysis 435 Confidence and Prediction Intervals 435 Inferences Regarding Regression

Coefficients 437

CHAPTER 13

ANALYSIS OFVARIANCE 443

13-1 Framework for a Single-Factor Analysis 443 Testing for Equality of Means 444 The Single-Factor Model 445

13-2 Single-Factor Analysis of Variance 448 Deviations about Sample Means 448 Summarizing Variation in Sample

Data 450 The ANOVA Table 451 A Basis for Comparison: Mean

Squares 452 The F Statistic 453 The Hypothesis-Testing Steps 454 Deciding What to Do: Comparing Treatment

Means 455 More Discriminating Testing

Procedures 455 The Type II Error 456 Violation of the Underlying Model 456

13-3 Comparative Analysis of Treatments 460 Single Inferences for a Mean or a Pairwise

Difference 460 Multiple Comparisons and Collective

Inferences 462

13-4 Designing the Experiment 466

13-5 Two-Factor Analysis ofVariance 468 The Population and Means 469 The Underlying Model 469 The Null Hypotheses 472 Sample Data in Two-Factor

Experiments 473 Analytical Framework 475 The Two-Factor ANOVA Table 475 Deciding What To Do 477

13-6 The Randomized Block Design 484 Analytical Framework and Theoretical

Model 486

CHAPTER 14

EXPERIMENTAL DESIGN 495

14-1 Issues in Experimental Design 495 The Factorial Design 497

14-2 The Two-Level Factorial Design for Experiments 500 Main Effects 501 Interaction Effects 503 Short Cut Calculations 506 The Three-Variable Interaction Effect 507 Evaluating the Results 508 Higher-Dimensional Factorial

Experiments 511

14-3 Other Approaches to Experimental Design 520 Analysis of Variance 520 Optimization of Quality Loss Function:

Taguchi Method 421 Fractional Factorial and Other Experimental

Designs 523

APPENDIX A

TABLES 525

TABLE A Cumulative Values for the Binomial Probability Distribution 526

TABLE B Exponential Functions 531

TABLE C Cumulative Probability Values for the Poisson Distribution 533

TABLE D Cumulative Probability Distribution Function for the Standard Normal Distribution 536

TABLE E Critical Normal Deviate Values 538

TABLE F Random Numbers 539

TABLE G Student t Distribution 541

TABLE H Chi-Square Distribution 542

TABLE I Conversion Table for Correlation Coefficient and Fisher's Z 545

TABLE J CriticalValuesforF Distribution 546

TABLE K Constants for Computing Control Chart Limits 554

Contents 0 APPENDIX ß

BIBLIOGRAPHY 555

ANSWERSTO SELECTED PROBLEMS 559

INDEX 575