Statistics for engineers and scientists

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Ban co the xoa dong chu nav76337_fm.qxd 12/10/09 1:35 PM Page i Statistics for Engineers and Scientists Third Edition This page intentionally left blank nav76337_fm.qxd 12/10/09 1:35 PM Page iii Statistics for Engineers and Scientists Third Edition William Navidi Colorado School of Mines nav76337_fm.qxd 12/10/09 1:35 PM Page iv STATISTICS FOR ENGINEERS AND SCIENTISTS, THIRD EDITION Published by McGraw-Hill, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of the Americas, New York, NY 10020 Copyright © 2011 by The McGraw-Hill Companies, Inc All rights reserved Previous editions © 2008 and 2006 No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written consent of The McGraw-Hill Companies, Inc., including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning Some ancillaries, including electronic and print components, may not be available to customers outside the United States This book is printed on acid-free paper DOC/DOC ISBN 978-0-07-337633-2 MHID 0-07-337633-7 Global Publisher: Raghothaman Srinivasan Sponsoring Editor: Debra B Hash Director of Development: Kristine Tibbetts Developmental Editor: Lora Neyens Senior Marketing Manager: Curt Reynolds Project Manager: Melissa M Leick Production Supervisor: Susan K Culbertson Design Coordinator: Brenda A Rolwes Cover Designer: Studio Montage, St Louis, Missouri (USE) Cover Image: Figure 4.20 from interior Compositor: MPS Limited Typeface: 10.5/12 Times Printer: R.R Donnelley Library of Congress Cataloging-in-Publication Data Navidi, William Cyrus Statistics for engineers and scientists / William Navidi – 3rd ed p cm Includes bibliographical references and index ISBN-13: 978-0-07-337633-2 (alk paper) ISBN-10: 0-07-337633-7 (alk paper) Mathematical statistics—Simulation methods Bootstrap (Statistics) Linear models (Statistics) I Title QA276.4.N38 2010 519.5—dc22 2009038985 www.mhhe.com nav76337_fm.qxd 12/10/09 1:35 PM Page v To Catherine, Sarah, and Thomas nav76337_fm.qxd 12/10/09 1:35 PM Page vi ABOUT THE AUTHOR William Navidi is Professor of Mathematical and Computer Sciences at the Colorado School of Mines He received his B.A degree in mathematics from New College, his M.A in mathematics from Michigan State University, and his Ph.D in statistics from the University of California at Berkeley Professor Navidi has authored more than 50 research papers both in statistical theory and in a wide variety of applications including computer networks, epidemiology, molecular biology, chemical engineering, and geophysics vi nav76337_fm.qxd 12/10/09 1:35 PM Page vii BRIEF CONTENTS Preface xiii Acknowledgments of Reviewers and Contributors xvii Key Features xix Supplements for Students and Instructors xx Sampling and Descriptive Statistics Probability 48 Propagation of Error 164 Commonly Used Distributions 200 Confidence Intervals 322 Hypothesis Testing 396 Correlation and Simple Linear Regression 505 Multiple Regression 592 Factorial Experiments 658 10 Statistical Quality Control 761 Appendix A: Tables 800 Appendix B: Partial Derivatives 825 Appendix C: Bibliography 827 Answers to Odd-Numbered Exercises 830 Index 898 vii This page intentionally left blank nav76337_fm.qxd 12/10/09 1:35 PM Page ix CONTENTS Preface xiii Acknowledgments of Reviewers and Contributors xvii Key Features xix Supplements for Students and Instructors xx Chapter Sampling and Descriptive Statistics Introduction 1.1 1.2 Sampling Summary Statistics 13 1.3 Graphical Summaries 25 Chapter Probability 48 Introduction 48 2.1 2.2 Basic Ideas 48 Counting Methods 62 2.3 Conditional Probability and Independence 69 Random Variables 90 2.4 2.5 2.6 Linear Functions of Random Variables 116 Jointly Distributed Random Variables 127 Chapter Propagation of Error 164 Introduction 164 3.1 Measurement Error 164 3.2 Linear Combinations of Measurements 170 3.3 3.4 Uncertainties for Functions of One Measurement 180 Uncertainties for Functions of Several Measurements 186 Chapter Commonly Used Distributions 200 Introduction 200 4.1 The Bernoulli Distribution 200 4.2 The Binomial Distribution 203 4.3 The Poisson Distribution 215 4.4 Some Other Discrete Distributions 230 4.5 The Normal Distribution 241 4.6 The Lognormal Distribution 256 4.7 The Exponential Distribution 262 4.8 Some Other Continuous Distributions 271 4.9 Some Principles of Point Estimation 280 4.10 Probability Plots 285 4.11 The Central Limit Theorem 290 4.12 Simulation 302 Chapter Confidence Intervals 322 Introduction 322 5.1 Large-Sample Confidence Intervals for a Population Mean 323 5.2 Confidence Intervals for Proportions 338 5.3 Small-Sample Confidence Intervals for a Population Mean 344 ix Navidi-1820036 book November 16, 2009 Chapter 8:49 10 Statistical Quality Control Introduction As the marketplace for industrial goods has become more global, manufacturers have realized that the quality and reliability of their products must be as high as possible for them to be competitive It is now generally recognized that the most cost-effective way to maintain high quality is through constant monitoring of the production process This monitoring is often done by sampling units of production and measuring some quality characteristic Because the units are sampled from some larger population, these methods are inherently statistical in nature One of the early pioneers in the area of statistical quality control was Dr Walter A Shewart of the Bell Telephone Laboratories In 1924, he developed the modern control chart, which remains one of the most widely used tools for quality control to this day After World War II, W Edwards Deming was instrumental in stimulating interest in quality control; first in Japan, and then in the United States and other countries The Japanese scientist Genichi Taguchi played a major role as well, developing methods of experimental design with a view toward improving quality In this chapter, we will focus on the Shewart control charts and on cumulative sum (CUSUM) charts, since these are among the most powerful of the commonly used tools for statistical quality control 10.1 Basic Ideas The basic principle of control charts is that in any process there will always be variation in the output Some of this variation will be due to causes that are inherent in the process and are difficult or impossible to specify These causes are called common causes or chance causes When common causes are the only causes of variation, the process is said to be in a state of statistical control, or, more simply, in control 761 Navidi-1820036 762 book November 16, 2009 CHAPTER 10 8:49 Statistical Quality Control Sometimes special factors are present that produce additional variability Machines that are malfunctioning, operator error, fluctuations in ambient conditions, and variations in the properties of raw materials are among the most common of these factors These are called special causes or assignable causes Special causes generally produce a higher level of variability than common causes; this variability is considered to be unacceptable When a process is operating in the presence of one or more special causes, it is said to be out of statistical control Control charts enable the quality engineer to decide whether a process appears to be in control, or whether one or more special causes are present If the process is found to be out of control, the nature of the special cause must be determined and corrected, so as to return the process to a state of statistical control There are several types of control charts; which ones are used depend on whether the quality characteristic being measured is a continuous variable, a binary variable, or a count variable For example, when monitoring a process that manufactures aluminum beverage cans, the height of each can in a sample might be measured Height is a continuous variable In some situations, it might be sufficient simply to determine whether the height falls within some specification limits In this case the quality measurement takes on one of only two values: conforming (within the limits) or nonconforming (not within the limits) This measurement is a binary variable, since it has two possible values Finally, we might be interested in counting the number of flaws on the surface of the can This is a count variable Control charts used for continuous variables are called variables control charts Examples include the X chart, the R chart, and the S chart Control charts used for binary or count variables are called attribute control charts The p chart is most commonly used for binary variables, while the c chart is commonly used for count variables Collecting Data -Rational Subgroups Data to be used in the construction of a control chart are collected in a number of samples, taken over a period of time These samples are called rational subgroups There are many different strategies for choosing rational subgroups The basic principle to be followed is that all the variability within the units in a rational subgroup should be due to common causes, and none should be due to special causes In general, a good way to choose rational subgroups is to decide which special causes are most important to detect, and then choose the rational subgroups to provide the best chance to detect them The two most commonly used methods are ■ ■ Sample at regular time intervals, with all the items in each sample manufactured near the time the sampling is done Sample at regular time intervals, with the items in each sample drawn from all the units produced since the last sample was taken For variables data, the number of units in each sample is typically small, often between three and eight The number of samples should be at least 20 In general, many small samples taken frequently are better than a few large samples taken less frequently For binary and for count data, samples must in general be larger Navidi-1820036 book November 16, 2009 8:49 10.1 Basic Ideas 763 Control versus Capability It is important to understand the difference between process control and process capability A process is in control if there are no special causes operating The distinguishing feature of a process that is in control is that the values of the quality characteristic vary without any trend or pattern, since the common causes not change over time However, it is quite possible for a process to be in control, and yet to be producing output that does not meet a given specification For example, assume that a process produces steel rods whose lengths vary randomly between 19.9 and 20.1 cm, with no apparent pattern to the fluctuation This process is in a state of control However, if the design specification calls for a length between 21 and 21.2 cm, very little of the output would meet the specification The ability of a process to produce output that meets a given specification is called the capability of the process We will discuss the measurement of process capability in Section 10.5 Process Control Must Be Done Continually There are three basic phases to the use of control charts First, data are collected Second, these data are plotted to determine whether the process is in control Third, once the process is brought into control, its capability may be assessed Of course, a process that is in control and capable at a given time may go out of control at a later time, as special causes re-occur For this reason processes must be continually monitored Similarities Between Control Charts and Hypothesis Tests Control charts function in many ways like hypothesis tests The null hypothesis is that the process is in control The control chart presents data that provide evidence about the truth of this hypothesis If the evidence against the null hypothesis is sufficiently strong, the process is declared out of control Understanding how to use control charts involves knowing what data to collect and knowing how to organize those data to measure the strength of the evidence against the hypothesis that the process is in control Exercises for Section 10.1 Indicate whether each of the following quality characteristics is a continuous, binary, or count variable a The number of flaws in a plate glass window b The length of time taken to perform a final inspection of a finished product c Whether the breaking strength of a bolt meets a specification d The diameter of a rivet head True or false: a Control charts are used to determine whether special causes are operating b If no special causes are operating, then most of the output produced will meet specifications c Variability due to common causes does not increase or decrease much over short periods of time d Variability within the items sampled in a rational subgroup is due to special causes e If a process is in a state of statistical control, there will be almost no variation in the output Fill in the blank The choices are: is in control; has high capability Navidi-1820036 764 book November 16, 2009 CHAPTER 10 8:49 Statistical Quality Control a If the variability in a process is approximately constant over time, the process b If most units produced conform to specifications, the process Fill in the blank: Once a process has been brought into a state of statistical control, i It must still be monitored continually ii Monitoring can be stopped for a while, since it is unlikely that the process will go out of control again right away iii The process need not be monitored again, unless it is redesigned True or false: a When a process is in a state of statistical control, then most of the output will meet specifications b When a process is out of control, an unacceptably large proportion of the output will not meet specifications c When a process is in a state of statistical control, all the variation in the process is due to causes that are inherent in the process itself d When a process is out of control, some of the variation in the process is due to causes that are outside of the process Fill in the blank: When sampling units for rational subgroups, i it is more important to choose large samples than to sample frequently, since large samples provide more precise information about the process ii it is more important to sample frequently than to choose large samples, so that special causes can be detected more quickly 10.2 Control Charts for Variables When a quality measurement is made on a continuous scale, the data are called variables data For these data an R chart or S chart is first used to control the variability in the process, and then an X -chart is used to control the process mean The methods described in this section assume that the measurements follow an approximately normal distribution We illustrate with an example The quality engineer in charge of a salt packaging process is concerned about the moisture content in packages of salt To determine whether the process is in statistical control, it is first necessary to define the rational subgroups, and then to collect some data Assume that for the salt packaging process, the primary concern is that variation in the ambient humidity in the plant may be causing variation in the mean moisture content in the packages over time Recall that rational subgroups should be chosen so that the variation within each sample is due only to common causes, not to special causes Therefore a good choice for the rational subgroups in this case is to draw samples of several packages each at regular time intervals The packages in each sample will be produced as close to each other in time as possible In this way, the ambient humidity will be nearly the same for each package in the sample, so the within-group variation will not be affected by this special cause Assume that five packages of salt are sampled every 15 minutes for eight hours, and that the moisture content in each package is measured as a percentage of total weight The data are presented in Table 10.1 Since moisture is measured on a continuous scale, these are variables data Each row of Table 10.1 presents the five moisture measurements in a given sample, along with their sample mean X , their sample standard deviation s, and their sample range R (the difference between the largest and smallest value) The last row of the table contains Navidi-1820036 book November 16, 2009 8:49 10.2 Control Charts for Variables 765 TABLE 10.1 Moisture content for salt packages, as a percentage of total weight Sample 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 Sample Values 2.53 2.69 2.67 2.10 2.64 2.64 2.58 2.31 3.03 2.86 2.71 2.95 3.14 2.85 2.82 3.17 2.81 2.99 3.11 2.83 2.76 2.54 2.27 2.40 2.41 2.40 2.56 2.21 2.56 2.42 2.62 2.21 2.66 2.38 2.23 2.26 2.42 1.63 2.69 2.39 2.68 3.22 2.80 3.54 2.84 3.29 3.71 3.07 3.21 2.65 2.74 2.74 2.85 2.63 2.54 2.62 2.72 2.33 2.47 2.61 2.26 2.37 2.11 2.15 1.88 2.34 2.10 2.51 2.56 2.95 3.01 2.60 2.27 2.72 3.09 2.59 3.77 3.25 3.36 3.14 2.95 2.79 2.59 3.03 2.59 2.32 2.82 2.84 2.29 2.40 2.11 2.59 1.95 2.13 2.47 2.18 2.21 2.47 2.43 2.58 2.51 2.12 3.01 2.40 2.54 3.09 2.60 3.31 2.80 3.35 2.95 3.63 3.04 2.80 3.01 2.68 2.23 2.48 2.11 2.50 2.35 2.02 2.43 2.24 2.26 2.09 2.27 2.59 Mean (X) 2.26 2.61 2.54 2.28 2.36 2.67 2.23 2.46 2.63 2.48 3.39 2.87 3.22 3.59 3.37 3.70 2.85 2.95 3.03 2.49 2.87 2.93 2.69 2.51 2.63 2.43 2.85 2.34 2.40 2.41 2.49 2.61 2.308 2.498 2.394 2.346 2.498 2.402 2.704 2.432 2.630 2.874 2.918 3.052 3.154 3.266 3.242 3.342 2.972 2.836 2.896 2.754 2.660 2.580 2.486 2.574 2.480 2.316 2.484 2.398 2.286 2.284 2.392 2.348 X = 2.6502 Range (R) 0.780 0.350 0.570 0.480 0.280 1.320 0.780 0.290 0.760 0.740 0.790 0.950 0.970 0.740 0.890 0.630 0.400 0.340 0.520 0.540 0.640 0.610 0.710 0.440 0.430 0.410 0.740 0.400 0.610 0.330 0.510 0.460 R = 0.6066 SD (s) 0.303 0.149 0.230 0.196 0.111 0.525 0.327 0.108 0.274 0.294 0.320 0.375 0.390 0.267 0.358 0.298 0.160 0.137 0.221 0.198 0.265 0.226 0.293 0.168 0.186 0.169 0.266 0.191 0.225 0.161 0.201 0.231 s = 0.2445 the mean of the sample means (X ), the mean of the sample ranges (R), and the mean of the sample standard deviations (s) We assume that each of the 32 samples in Table 10.1 is a sample from a normal population with mean μ and standard deviation σ The quantity μ is called the process mean, and σ is called the process standard deviation The idea behind control charts is that each value of X approximates the process mean during the time its sample was taken, while the values of R and s can be used to approximate the process standard deviation If the process is in control, then the process mean and standard deviation are the same for each sample If the process is out of control, the process mean μ or the process standard deviation σ , or both, will differ from sample to sample Therefore the values of X , R, and s will vary less when the process is in control than when the process is out of control If the process is in control, the values of X , R, and s will almost 766 book November 16, 2009 CHAPTER 10 8:49 Statistical Quality Control always be contained within computable limits, called control limits If the process is out of control, the values of X , R, and s will be more likely to exceed these limits A control chart plots the values of X , R, or s, along with the control limits, so that it can be easily seen whether the variation is large enough to conclude that the process is out of control Now let’s see how to determine whether the salt packaging process is in a state of statistical control with respect to moisture content Since the variation within each sample is supposed to be due only to common causes, this variation should not be too different from one sample to another Therefore the first thing to is to check to make sure that the amount of variation within each sample, measured either by the sample range or the sample standard deviation, does not vary too much from sample to sample For this purpose the R chart can be used to assess variation in the sample range, or the S chart can be used to assess variation in the sample standard deviation We will discuss the R chart first, since it is the more traditional We will discuss the S chart at the end of this section Figure 10.1 presents the R chart for the moisture data The horizontal axis represents the samples, numbered from to 32 The sample ranges are plotted on the vertical axis Most important are the three horizontal lines The line in the center is plotted at the value R and is called the center line The upper and lower lines indicate the 3σ upper and lower control limits (UCL and LCL, respectively) The control limits are drawn so that when the process is in control, almost all the points will lie within the limits A point plotting outside the control limits is evidence that the process is out of control R chart UCL = 1.282 Sample range Navidi-1820036 R = 0.607 0.5 LCL = 0 10 15 20 Sample number 25 30 FIGURE 10.1 R chart for the moisture data To understand where the control limits are plotted, assume that the 32 sample ranges come from a population with mean μ R and standard deviation σ R The values of μ R and σ R will not be known exactly, but it is known that for most populations, it is unusual to observe a value that differs from the mean by more than three standard deviations For this reason, it is conventional to plot the control limits at points that approximate the values μ R ± 3σ R It can be shown by advanced methods that the quantities μ R ± 3σ R can be estimated with multiples of R; these multiples are denoted D3 and D4 The quantity μ R − 3σ R is estimated with D3 R, and the quantity μ R + 3σ R is estimated with D4 R The quantities D3 and D4 are constants whose values depend on the sample size n A brief table of values of D3 and D4 follows A more extensive tabulation is provided in Navidi-1820036 book November 16, 2009 8:49 10.2 Control Charts for Variables 767 Table A.10 (in Appendix A) Note that for sample sizes of or less, the value of D3 is For these small sample sizes, the quantity μ R − 3σ R is negative In these cases the lower control limit is set to 0, because it is impossible for the range to be negative Example 10.1 n D3 D4 3.267 2.575 2.282 2.114 2.004 0.076 1.924 0.136 1.864 Compute the 3σ R chart upper and lower control limits for the moisture data in Table 10.1 Solution The value of R is 0.6066 (Table 10.1) The sample size is n = From the table, D3 = and D4 = 2.114 Therefore the upper control limit is (2.114)(0.6066) = 1.282, and the lower control limit is (0)(0.6066) = Summary In an R chart, the center line and the 3σ upper and lower control limits are given by 3σ upper limit = D4 R Center line = R 3σ lower limit = D3 R The values D3 and D4 depend on the sample size Values are tabulated in Table A.10 Once the control limits have been calculated and the points plotted, the R chart can be used to assess whether the process is in control with respect to variation Figure 10.1 shows that the range for sample number is above the upper control limit, providing evidence that a special cause was operating and that the process variation is not in control The appropriate action is to determine the nature of the special cause, and then delete the out-of-control sample and recompute the control limits Assume it is discovered that a technician neglected to close a vent, causing greater than usual variation in moisture content during the time period when the sample was chosen Retraining the technician will correct that special cause We delete sample from the data and recompute the R chart The results are shown in Figure 10.2 The process variation is now in control Now that the process variation has been brought into control, we can assess whether the process mean is in control by plotting the X chart The X chart is presented in Figure 10.3 The sample means are plotted on the vertical axis Note that sample has not been used in this chart since it had to be deleted in order to bring the process variation November 16, 2009 CHAPTER 10 8:49 Statistical Quality Control R chart UCL = 1.234 Sample range 768 book R = 0.584 0.5 LCL = 0 10 15 20 Sample number 25 30 FIGURE 10.2 R chart for the moisture data, after deleting the out-of-control sample X chart 3.5 Sample mean Navidi-1820036 UCL = 2.995 X = 2.658 2.5 LCL = 2.321 10 15 20 Sample number 25 30 FIGURE 10.3 X chart for the moisture data Sample has been deleted to bring the process variation under control However, the X chart shows that the process mean is out of control under control Like all control charts, the X chart has a center line and upper and lower control limits To compute the center line and the control limits, we can assume that the process standard deviation is the same for all samples, since the R chart has been used to bring the process variation into control If the process mean μ is in control as well, then it too is the same for all samples In that case the 32 sample means are drawn √ from a normal population with mean μ X = μ and standard deviation σ X = σ/ n, where n is the sample size, equal to in this case Ideally, we would like to plot the center line at μ and the 3σ control limits at μ ± 3σ X However, the values of μ and σ X are usually unknown and have to be estimated from the data We estimate μ with X , the average of the sample means The center line is therefore plotted at X The quantity σ X can be estimated by using either the average range R or by using the sample standard deviations We will use R here and discuss the methods based on the standard deviation at the end of the section, Navidi-1820036 book November 16, 2009 8:49 10.2 Control Charts for Variables 769 in conjunction with the discussion of S charts It can be shown by advanced methods that the quantity 3σ X can be estimated with A2 R, where A2 is a constant whose value depends on the sample size A short table of values of A2 follows A more extensive tabulation is provided in Table A.10 n A2 1.880 1.023 0.729 0.577 0.483 0.419 0.373 Summary In an X chart, when R is used to estimate σ X , the center line and the 3σ upper and lower control limits are given by 3σ upper limit = X + A2 R Center line = X 3σ lower limit = X − A2 R The value A2 depends on the sample size Values are tabulated in Table A.10 Example 10.2 Compute the 3σ X chart upper and lower control limits for the moisture data in Table 10.1 Solution With sample deleted, the value of X is 2.658, and the value of R is 0.5836 The sample size is n = From the table, A2 = 0.577 Therefore the upper control limit is 2.658 + (0.577)(0.5836) = 2.995, and the lower control limit is 2.658 − (0.577)(0.5836) = 2.321 The X chart clearly shows that the process mean is not in control, as there are several points plotting outside the control limits The production manager installs a hygrometer to monitor the ambient humidity and determines that the fluctuations in moisture content are caused by fluctuations in ambient humidity A dehumidifier is installed to stabilize the ambient humidity After this special cause is remedied, more data are collected, and a new R chart and X chart are constructed Figure 10.4 presents the results The process is now in a state of statistical control Of course, the process must be continually monitored, since new special causes are bound to crop up from time to time and will need to be detected and corrected Note that while control charts can detect the presence of a special cause, they cannot determine its nature, nor how to correct it It is necessary for the process engineer to have a good understanding of the process, so that special causes detected by control charts can be diagnosed and corrected November 16, 2009 CHAPTER 10 8:49 Statistical Quality Control R chart UCL = 1.223 Sample range 770 book R = 0.578 0.5 LCL = 0 10 15 20 Sample number 25 30 X chart 2.8 UCL = 2.609 2.6 Sample mean Navidi-1820036 2.4 X = 2.275 2.2 LCL = 1.942 1.8 10 15 20 Sample number 25 30 FIGURE 10.4 R chart and X chart after special cause is remedied The process is now in a state of statistical control Summary The steps in using the R chart and X chart are Choose rational subgroups Compute the R chart Determine the special causes for any out-of-control points Recompute the R chart, omitting samples that resulted in out-of-control points Once the R chart indicates a state of control, compute the X chart, omitting samples that resulted in out-of-control points on the R chart If the X chart indicates that the process is not in control, identify and correct any special causes Continue to monitor X and R Navidi-1820036 book November 16, 2009 8:49 10.2 Control Charts for Variables 771 Control Chart Performance There is a close connection between control charts and hypothesis tests The null hypothesis is that the process is in a state of control A point plotting outside the 3σ control limits presents evidence against the null hypothesis As with any hypothesis test, it is possible to make an error For example, a point will occasionally plot outside the 3σ limits even when the process is in control This is called a false alarm It can also happen that a process that is not in control may not exhibit any points outside the control limits, especially if it is not observed for a long enough time This is called a failure to detect It is desirable for these errors to occur as infrequently as possible We describe the frequency with which these errors occur with a quantity called the average run length (ARL) The ARL is the number of samples that must be observed, on average, before a point plots outside the control limits We would like the ARL to be large when the process is in control, and small when the process is out of control We can compute the ARL for an X chart if we assume that process mean μ and the process standard deviation σ are known Then the center line is located at the process mean μ and the control limits are at μ ± 3σ X We must also assume, as is always the case with the X chart, that the quantity being measured is approximately normally distributed Examples 10.3 through 10.6 show how to compute the ARL Example 10.3 For an X chart with control limits at μ ± 3σ X , compute the ARL for a process that is in control Solution Let X be the mean of a sample Then X ∼ N (μ, σ X2 ) The probability that a point plots outside the control limits is equal to P(X < μ − 3σ X ) + P(X > μ + 3σ X ) This probability is equal to 0.00135 + 0.00135 = 0.0027 (see Figure 10.5) Therefore, on the average, 27 out of every 10,000 points will plot outside the control limits This is equivalent to every 10,000/27 = 370.4 points The average run length is therefore equal to 370.4 0.00135 m ⫺ 3s⫺ X z=⫺3 0.00135 m m ⫹ 3s⫺ X z=3 FIGURE 10.5 The probability that a point plots outside the 3σ control limits, when the process is in control, is 0.0027 (0.00135 + 0.00135) The result of Example 10.3 can be interpreted as follows: If a process is in control, we expect to observe about 370 samples, on the average, before finding one that plots Navidi-1820036 772 book November 16, 2009 CHAPTER 10 8:49 Statistical Quality Control outside the control limits, causing a false alarm Note also that the ARL in Example 10.3 was 10,000/27, which is equal to 1/0.0027, where 0.0027 is the probability that any given sample plots outside the control limits This is true in general Summary The average run length (ARL) is the number of samples that will be observed, on the average, before a point plots outside the control limits If p is the probability that any given point plots outside the control limits, then ARL = (10.1) p If a process is out of control, the ARL will be less than 370.4 Example 10.4 shows how to compute the ARL for a situation where the process shifts to an out-of-control condition Example 10.4 A process has mean μ = and standard deviation σ = Samples of size n = are taken If a special cause shifts the process mean to a value of 3.5, find the ARL Solution We first compute the probability p that a given point plots outside the control limits Then ARL = 1/ p The control limits are plotted on the basis of√ a process√that is in control Therefore they are at μ±3σ X , where μ = and σ X = σ/ n = 1/ = 0.5 The lower control limit is thus at 1.5, and the upper control limit is at 4.5 If X is the mean of a sample taken after the process mean has shifted, then X ∼ N (3.5, 0.52 ) The probability that X plots outside the control limits is equal to P(X < 1.5) + P(X > 4.5) This probability is 0.0228 (see Figure 10.6) The ARL is therefore equal to 1/0.0228 = 43.9 We will have to observe about 44 samples, on the average, before detecting this shift ≈0 1.5 z=⫺4 0.0228 3.5 4.5 z=2 FIGURE 10.6 The process mean has shifted from μ = to μ = 3.5 The upper control limit of 4.5 is now only 2σ X above the mean, indicated by the fact that z = The lower limit is now 4σ X below the mean The probability that a point plots outside the control limits is 0.0228 (0 + 0.0228) Navidi-1820036 book November 16, 2009 8:49 10.2 Control Charts for Variables Example 10.5 773 Refer to Example 10.4 An upward shift to what value can be detected with an ARL of 20? Solution Let m be the new mean to which the process has shifted Since we have specified an upward shift, m > In Example 10.4 we computed the control limits to be 1.5 and 4.5 If X is the mean of a sample taken after the process mean has shifted, then X ∼ N (m, 0.52 ) The probability that X plots outside the control limits is equal to P(X < 1.5) + P(X > 4.5) (see Figure 10.7) This probability is equal to 1/ARL = 1/20 = 0.05 Since m > 3, m is closer to 4.5 than to 1.5 We will begin by assuming that the area to the left of 1.5 is negligible and that the area to the right of 4.5 is equal to 0.05 The z-score of 4.5 is then 1.645, so (4.5 − m)/0.5 = 1.645 Solving for m, we have m = 3.68 We finish by checking our assumption that the area to the left of 1.5 is negligible With m = 3.68, the z-score for 1.5 is (1.5 − 3.68)/0.5 = − 4.36 The area to the left of 1.5 is indeed negligible ≈0 0.05 1.5 z = ⫺ 4.36 m 4.5 z = 1.645 FIGURE 10.7 Solution to Example 10.5 Example 10.6 Refer to Example 10.4 If the sample size remains at n = 4, what must the value of the process standard deviation σ be to produce an ARL of 10 when the process mean shifts to 3.5? Solution Let √ σ denote the new process standard deviation The new control limits are ± 3σ/ n, or ± 3σ/2 If the process mean shifts to 3.5, then X ∼ N (3.5, σ /4) The probability that X plots outside the control limits is equal to P(X < − 3σ/2) + P(X > 3+3σ/2) This probability is equal to 1/ARL = 1/10 = 0.10 (see Figure 10.8, page 774) The process mean, 3.5, is closer to + 3σ/2 than to − 3σ/2 We will assume that the area to the left of − 3σ/2 is negligible and that the area to the right of + 3σ/2 is equal to 0.10 The z-score for + 3σ/2 is then 1.28, so (3 + 3σ/2) − 3.5 = 1.28 σ/2 Navidi-1820036 774 book November 16, 2009 CHAPTER 10 8:49 Statistical Quality Control Solving for σ , we obtain σ = 0.58 We finish by checking that the area to the left of − 3σ/2 is negligible Substituting σ = 0.58, we obtain − 3σ/2 = 2.13 The z-score is (2.13 − 3.5)/(0.58/2) = − 4.72 The area to the left of − 3σ/2 is indeed negligible ≈0 0.10 ⫺ 3s/2 z = ⫺ 4.72 3.5 ⫹ 3s/2 z = 1.28 FIGURE 10.8 Solution to Example 10.6 Examples 10.4 through 10.6 show that X charts not usually detect small shifts quickly In other words, the ARL is high when shifts in the process mean are small In principle, one could reduce the ARL by moving the control limits closer to the centerline This would reduce the size of the shift needed to detect an out-of-control condition, so that changes in the process mean would be detected more quickly However, there is a trade-off The false alarm rate would increase as well, because shifts outside the control limits would be more likely to occur by chance The situation is much like that in fixedlevel hypothesis testing The null hypothesis is that the process is in control The control chart performs a hypothesis test on each sample When a point plots outside the control limits, the null hypothesis is rejected With the control limits at ± 3σ X , a type I error (rejection of a true null hypothesis) will occur about once in every 370 samples The price to pay for this low false alarm rate is lack of power to reject the null hypothesis when it is false Moving the control limits closer together is not the answer Although it will increase the power, it will also increase the false alarm rate Two of the ways in which practitioners have attempted to improve their ability to detect small shifts quickly are by using the Western Electric rules to interpret the control chart and by using CUSUM charts The Western Electric rules are described next CUSUM charts are described in Section 10.4 The Western Electric Rules Figure 10.9 presents an X chart While none of the points fall outside the 3σ control limits, the process is clearly not in a state of control, since there is a nonrandom pattern to the sample means In recognition of the fact that a process can fail to be in control even when no points plot outside the control limits, engineers at the Western Electric company in 1956 suggested a list of conditions, any one of which could be used as evidence that a process is out of control The idea behind these conditions is that if a trend or pattern in the control chart persists for long enough, it can indicate the absence of control, even if no point plots outside the 3σ control limits To apply the Western Electric rules, it is necessary to compute the 1σ and 2σ control limits The 1σ control limits are given by X ± A2 R/3, and the 2σ control limits are given by X ± 2A2 R/3 book November 16, 2009 8:49 10.2 Control Charts for Variables 775 X chart 3s = 4.725 Sample mean Navidi-1820036 2s = 3.987 1s = 3.249 X = 2.511 ⫺1s = 1.773 ⫺2s = 1.035 ⫺3s = 0.297 10 15 20 Sample number 25 30 FIGURE 10.9 This X chart exhibits nonrandom patterns, indicating a lack of statistical control, even though no points plot outside the 3σ control limits The 1σ and 2σ control limits are shown on this plot, so that the Western Electric rules can be applied The Western Electric Rules Any one of the following conditions is evidence that a process is out of control: Any point plotting outside the 3σ control limits Two out of three consecutive points plotting above the upper 2σ limit, or two out of three consecutive points plotting below the lower 2σ limit Four out of five consecutive points plotting above the upper 1σ limit, or four out of five consecutive points plotting below the lower 1σ limit Eight consecutive points plotting on the same side of the center line In Figure 10.9, the Western Electric rules indicate that the process is out of control at sample number 8, at which time four out of five consecutive points have plotted above the upper 1σ control limit For more information on using the Western Electric rules to interpret control charts, see Montgomery (2005b) The S chart The S chart is an alternative to the R chart Both the S chart and the R chart are used to control the variability in a process While the R chart assesses variability with the sample range, the S chart uses the sample standard deviation Figure 10.10 presents the S chart for the moisture data in Table 10.1