Table Of Content

Practical Engineering, Process, and Reliability Statistics Second Edition Mark Allen Durivage Milwaukee, Wisconsin Table of Contents List of Figures and Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv Chapter 1 Point Estimates and Measures of Dispersion . . . . . . . . . . . . . . . . 1 Estimates of Central Tendency for Variables Data ...................... 1 Range for Variables Data.......................................... 4 Variance and Standard Deviation for Variables Data .................... 4 Skewness and Kurtosis for Variables Data ............................ 7 Estimates of Central Tendency for Attributes Data...................... 9 Estimates of Dispersion for Attributes Data ........................... 10 Standard Error.................................................. 11 Chapter 2 Confidence Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Confidence Interval for the Mean of Continuous Data ................... 14 Confidence Interval for the Variance and Standard Deviation ............. 15 Confidence Interval for the Fraction Nonconforming—Normal Distribution................................................. 17 Confidence Interval for Proportion (Normal Approximation of the Binomial Confidence Interval) .................................. 18 Small Sample Size Confidence Intervals ............................. 19 Confidence Interval for the Poisson Distributed Data ................... 20 Chapter 3 Tolerance and Prediction Intervals . . . . . . . . . . . . . . . . . . . . . . . . 21 Tolerance Intervals............................................... 21 Prediction Intervals .............................................. 22 Prediction Intervals When the Variance (σ2) Is Known................... 22 Prediction Intervals When the Variance (σ2) Is Unknown................. 24 Chapter 4 Correlation and Regression Analysis . . . . . . . . . . . . . . . . . . . . . . 27 Correlation Analysis ............................................. 27 Regression Analysis.............................................. 30 Normal Probability Plots.......................................... 31 iii iv Table of Contents Chapter 5 Outliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Outlier Detection Based on the Standard Deviation for a Normal Distribution .......................................... 35 Discordance Outlier Test.......................................... 36 Outlier Detection Based on the Standard Deviation for an Unknown Distribution......................................... 37 Outlier Detection Based on the Interquartile Range ..................... 39 Dixon’s Q Test .................................................. 39 Dean and Dixon Outlier Test....................................... 40 Grubbs’ Outlier Test ............................................. 41 Walsh’s Outlier Test.............................................. 42 Hampel’s Method for Outlier Detection .............................. 44 Chapter 6 Hypothesis Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Type I and Type II Errors ......................................... 47 Alpha (α) and Beta (β) Risks....................................... 47 The Effect Size Index ............................................ 48 Apportionment of Risk in Hypothesis Testing.......................... 49 The Hypothesis Test for a One-Tail (Upper-Tailed) Test . . . . . . . . . . . . . . . . . . 49 The Hypothesis Test for a One-Tail (Lower-Tailed) Test.................. 50 The Hypothesis Test for a Two-Tail Test .............................. 50 The Hypothesis Test Conclusion Statements........................... 51 Chapter 7 Sample Size Determination for Tests of Hypotheses . . . . . . . . . . 53 Sample Size Required to Test an Observed Mean versus a Hypothesized Mean When Standard Deviation (σ) Is Known .......... 53 Sample Size Required to Test an Observed Mean versus a Hypothesized Mean When the Standard Deviation (σ) Is Estimated from Observed Values......................................... 55 Sample Size Required to Test for Differences in Two Observed Means When Standard Deviation (σ) for Each Population Is Known..... 55 Sample Size Required to Test for Differences in Two Observed Means When the Standard Deviation (σ) Is Estimated from the Observed Data............................................... 56 Sample Size Required to Detect a Difference in the Mean with a Given Confidence....................................... 57 Paired Sample t-Test Requirements.................................. 57 Sample Size Required for Chi-Square Test of Observed Variance to a Hypothesized Variance....................................... 59 Sample Size Required for F-Test of Two Observed Variances ............. 61 Chapter 8 Hypothesis Testing for a Difference in Means . . . . . . . . . . . . . . . 63 Testing a Sample Mean versus a Hypothesized Mean When the Standard Deviation (σ) Is Known ................................ 63 Table of Contents v Testing a Sample Mean versus a Hypothesized Mean When the Standard Deviation (σ) Is Estimated from the Sample Data............ 66 Testing for a Difference in Two Population Means—Standard Deviations (σ) Known......................................... 68 Testing a Sample Mean versus a Hypothesized Mean When Standard Deviation (σ) Is Estimated from the Sample Data.................... 71 Testing for a Difference in Two Population Means—Standard Deviations (σ) Not Known and Not Assumed Equal.................. 74 Testing for Differences in Means of Paired Samples..................... 77 Hypothesis Test One Proportion .................................... 80 Testing for Differences in Two Proportions............................ 82 Testing for Differences in Count Data—Equal Sample Sizes .............. 85 Testing for Differences in Count Data—Unequal Sample Sizes ............ 87 Hypothesis Testing for Differences in Means—Confidence Interval Approach (Standard Deviations Known)........................... 89 Hypothesis Testing for Differences in Means—Confidence Interval Approach (Standard Deviations Not Known but Assumed Equal)....... 91 Kruskal-Wallis Means Test ........................................ 93 Chapter 9 Hypothesis Testing for a Difference in Variances . . . . . . . . . . . . 97 Testing a Variance Calculated from a Sample against a Hypothesized Variance ................................................... 97 Testing an Observed Variance against a Hypothesized Variance— Large Samples............................................... 99 Testing for a Difference between Two Observed Variances Using Sample Data ................................................ 102 Testing for a Difference between Two Observed Variances Using Large Samples............................................... 104 Levene’s Test for Equality of Variances............................... 106 Chapter 10 Discrete Probability Distributions . . . . . . . . . . . . . . . . . . . . . . . . 109 Binomial Distribution ............................................ 109 Poisson Distribution.............................................. 111 Hypergeometric Distribution....................................... 113 Geometric Distribution ........................................... 114 Negative Binomial Distribution..................................... 115 Chapter 11 Control Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Control Chart Types and Selection .................................. 117 Control Chart Interpretation ....................................... 118 X-bar and R Control Charts........................................ 121 X-bar and s Control Charts ........................................ 124 c-Charts....................................................... 128 u-Charts....................................................... 130 vi Table of Contents np-Charts...................................................... 133 p-Charts....................................................... 135 g-Charts .................................................... 137 X and mR (Moving Range) Control Charts ........................... 140 Pre-control Charts............................................... 143 Chapter 12 Process Capability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Process Capability for Variables Data................................ 147 Process Capability Confidence Intervals.............................. 150 Process Capability for Attributes Data ............................... 152 Increasing Process Capability ...................................... 152 Fall-Out Rates .................................................. 153 Defects per Million Opportunities (DPMO) ........................... 154 Chapter 13 Acceptance Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 C = 0 Sampling Plan ............................................. 155 Average Outgoing Quality......................................... 155 Upper Risk Level (Confidence Statement)............................. 156 Sample Size Required to Find Percentage Defective with a Given Confidence Level ....................................... 156 AQL Calculations ............................................... 157 LTPD Calculations............................................... 158 Chapter 14 ANOVA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 One-Way ANOVA............................................... 159 Two-Way ANOVA............................................... 162 Chapter 15 The Reliability Bathtub Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 Chapter 16 Availability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 Inherent Availability ............................................. 171 Achieved Availability ............................................ 172 Operational Availability........................................... 173 Chapter 17 Exponential Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Chapter 18 Censoring and MTBF and MCBF Calculations . . . . . . . . . . . . . 177 Type I Censoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 Type II Censoring ............................................... 178 Chapter 19 Confidence Intervals for MTBF/MCBF . . . . . . . . . . . . . . . . . . . 181 Testing to a Predetermined Time/Cycles.............................. 181 Testing to a Predetermined Number of Failures ........................ 182 Failure-Free Testing.............................................. 184 Chapter 20 Nonparametric and Related Test Designs . . . . . . . . . . . . . . . . . 185 Calculating Reliability in Zero-Failure Situations....................... 185 Table of Contents vii Chapter 21 Sample Size, Reliability, and Confidence Determination . . . . . 187 Sample Size Determination Based on Confidence and Reliability with Zero Failures Allowed............................ 187 Reliability Estimate When Sample Size Is Provided..................... 187 Sample Size Calculation with Failures Allowed ........................ 188 Reliability Estimate When Sample Sizes Are Specified.................. 189 Chapter 22 Wear-Out Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 Wear-Out Distribution—Standard Deviation Known .................... 191 Wear-Out Distribution—Standard Deviation Unknown .................. 192 Wear-Out and Chance Failure Combined ............................. 192 Chapter 23 Conditional Probability of Failure . . . . . . . . . . . . . . . . . . . . . . . 195 Chapter 24 System Reliability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 Series Reliability Systems ......................................... 198 Parallel Reliability Systems........................................ 199 Combination Reliability Systems.................................... 200 Standby Parallel Systems.......................................... 201 Equal Failure Rates—Perfect Switching .............................. 202 Unequal Failure Rates—Perfect Switching ............................ 203 Equal Failure Rates—Imperfect Switching............................ 203 Unequal Failure Rates—Imperfect Switching.......................... 204 Shared Load Parallel Systems...................................... 205 k-out-of-n System................................................ 206 Bayes’ Theorem................................................. 207 Chapter 25 Stress-Strength Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Chapter 26 Weibull Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Weibull Distribution Parameters .................................... 213 Weibull Fit Determination......................................... 214 Chapter 27 Log-Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 Chapter 28 Beta Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 Chapter 29 Taguchi Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 Nominal is Best S/N Ratio......................................... 223 Larger is Better S/N Ratio......................................... 223 Smaller is Better S/N Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 Zero is Best (Signed Value) S/N Ratio................................ 224 Fraction Defective S/N Ratio....................................... 225 Ordered Categorical S/N Ratio ..................................... 225 Convert dB to Percentage ......................................... 225 viii Table of Contents Appendix A Normal Distribution Probability Points—Area below Z . . . . . 227 Appendix B Normal Distribution Probability Points—Area above Z . . . . . 229 Appendix C Selected Single-Sided Normal Distribution Probability Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 Appendix D Selected Double-Sided Normal Distribution Probability Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 Appendix E Percentage Points of the Student’s t-Distribution . . . . . . . . . . . 235 Appendix F Distribution of the Chi-Square . . . . . . . . . . . . . . . . . . . . . . . . . . 237 Appendix G Percentages of the F-Distribution . . . . . . . . . . . . . . . . . . . . . . . 239 Appendix H Tolerance Interval Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 Appendix I Critical Values of the Correlation Coeffecient . . . . . . . . . . . . . 257 Appendix J Critical Values of the Dean and Dixon Outlier Test . . . . . . . . . 259 Appendix K Critical Values for the Grubbs’ Outlier Test . . . . . . . . . . . . . . 261 Appendix L Critical Values for the Discordance Outlier Test . . . . . . . . . . . 263 Appendix M The Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 Appendix N The Poisson Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 Appendix O Control Chart Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 Appendix P C = 0 Sampling Plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 Appendix Q Fall-Out Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 Appendix R Beta Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 Appendix S Gamma Function of Γ(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 Appendix T Selected Median Rank Percentages . . . . . . . . . . . . . . . . . . . . . . 307 Appendix U Weibull Graph Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 About the Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 American Society for Quality, Quality Press, Milwaukee, WI 53203 © 2015, 2022 by ASQ All rights reserved. Published 2015, 2022. Printed in the United States of America. 26 25 24 23 22 EA 5 4 3 2 1 Library of Congress Cataloging-in-Publication data Names: Durivage, Mark Allen, author. Title: Practical engineering, process, and reliability statistics, second edition / By Mark Allen Durivage. Description: Includes bibliographical references and index. | Milwaukee, WI: ASQ Quality Press, 2022. Identifiers: LCCN: 2022933438 | ISBN 978-1-63694-015-1 (paperback) | 978-1-63694-016-8 (epub) Subjects: LCSH Engineering—Statistical methods. | Reliability (Engineering)— Statistical methods. | BISAC MATHEMATICS / Probability & Statistics / General | TECHNOLOGY & ENGINEERING / Engineering / General Classification: LCC TA340 .D87 2022 | DDC 620/.0045—dc23 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. ASQ advances individual, organizational, and community excellence worldwide through learning, quality improvement, and knowledge exchange. Bookstores, wholesalers, schools, libraries, and organizations: Quality Press books are available at quantity discounts for bulk purchases for business, trade, or educational uses. For more information, please contact Quality Press at 800-248-1946 or [email protected]. To place orders or browse the selection of all Quality Press titles, visit our website at: http://www.asq.org/quality-press. Acknowledgments would like to acknowledge the previous work of Robert A. Dovich in Quality I Engineering Statistics and Reliability Statistics. This book is an expansion of his efforts in an attempt to continue the method of presenting statistical applications in a simple, easy-to-follow format. Several sections of this book come directly from his previous work. I have made some changes to clarify and augment some of his points and present the topics in a consistent manner. I would like to thank those who have inspired, taught, and trained me throughout my academic and professional career. Lastly, I would like to acknowledge the patience of my wife, Dawn, and my sons, Jack and Sam, which allowed me time to research, write, and update Practical Engineering, Process, and Reliability Statistics, second edition. xv List of Figures and Tables Figure 1.1 Normal distribution ............................................ 3 Figure 1.2 Negatively skewed distribution ................................... 3 Figure 1.3 Positively skewed distribution .................................... 3 Figure 1.4 Skewness .................................................... 8 Figure 1.5 Kurtosis ..................................................... 8 Figure 1.6 Sample size and the standard error relationship ...................... 12 Figure 2.1 Relationship between the confidence interval and sample size (confidence level constant) ...................................... 13 Figure 2.2 Relationship between the confidence interval and confidence level (sample size constant) .......................................... 14 Figure 4.1 Relative degrees of correlation ................................... 28 Table 4.1 Correlation axis table .......................................... 29 Table 4.2 Example calculation summary ................................... 29 Figure 4.2 Nonlinear quadratic and cubic relationships ......................... 31 Figure 4.3 Right-skewed distribution ....................................... 32 Figure 4.4 Left-skewed distribution ........................................ 32 Figure 4.5 Short-tailed distribution ........................................ 32 Figure 4.6 Long-tailed distribution ......................................... 32 Figure 4.7 Normal probability plot for strength ............................... 34 Table 4.3 Calculation summary .......................................... 34 Figure 5.1 Normal distribution ............................................ 36 Table 5.1 Selected k values .............................................. 37 Figure 5.2 Unknown distribution .......................................... 38 Table 5.2 Selected critical Q values ....................................... 40 Table 6.1 Hypothesis truth table .......................................... 47 Figure 6.1 Representation of a one-tail (upper-tailed) test ....................... 50 Figure 6.2 Representation of a one-tail (lower-tailed) test ....................... 50 Figure 6.3 Representation of a two-tail test .................................. 51 Table 7.1 Hypothesis constant table for two-tail and one-tail tests ............... 55 ix

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