Statistics and Probability for Engineering Applications

408 Pages·2003·2.56 MB·English

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
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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

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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

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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

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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.
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