《概率论与数理统计》课程教学资源（阅读材料）The Role of Statistics in Engineering.pdf_大学文库

2 CHAPTER I THE ROLE OF STATISTICS IN ENGINEERING available for some of the text sections that appear on CD only.These exercises may be found within the e-Text immediately following the section they accompany. 1-1 THE ENGINEERING METHOD AND STATISTICAL THINKING An engineer is someone who solves problems of interest to society by the efficient application of scientific principles.Engineers accomplish this by either refining an existing product or process or by designing a new product or process that meets customers'needs.The engineering, or scientific,method is the approach to formulating and solving these problems.The steps in the engineering method are as follows: 1.Develop a clear and concise description of the problem. 2.Identify,at least tentatively,the important factors that affect this problem or that may play a role in its solution. 3.Propose a model for the problem,using scientific or engineering knowledge of the phenomenon being studied.State any limitations or assumptions of the model. 4. Conduct appropriate experiments and collect data to test or validate the tentative model or conclusions made in steps 2 and 3. 5.Refine the model on the basis of the observed data 6.Manipulate the model to assist in developing a solution to the problem. 7.Conduct an appropriate experiment to confirm that the proposed solution to the prob- lem is both effective and efficient. 8.Draw conclusions or make recommendations based on the problem solution. The steps in the engineering method are shown in Fig.1-1.Notice that the engineering method features a strong interplay between the problem,the factors that may influence its solution,a model of the phenomenon,and experimentation to verify the adequacy of the model and the proposed solution to the problem.Steps 2-4 in Fig.1-1 are enclosed in a box,indicating that several cycles or iterations of these steps may be required to obtain the final solution. Consequently,engineers must know how to efficiently plan experiments,collect data,analyze and interpret the data,and understand how the observed data are related to the model they have proposed for the problem under study. The field of statistics deals with the collection,presentation,analysis,and use of data to make decisions,solve problems,and design products and processes.Because many aspects of engineering practice involve working with data,obviously some knowledge of statistics is important to any engineer.Specifically,statistical techniques can be a powerful aid in design- ing new products and systems,improving existing designs,and designing,developing,and improving production processes. Develop a Identify the Propose or Manipulate Confirm Conclusions clear important refine a the the and description factors model model solution recommendations Conduct experiments Figure 1-1 The engineering method

2 CHAPTER 1 THE ROLE OF STATISTICS IN ENGINEERING available for some of the text sections that appear on CD only. These exercises may be found within the e-Text immediately following the section they accompany. 1-1 THE ENGINEERING METHOD AND STATISTICAL THINKING An engineer is someone who solves problems of interest to society by the efficient application of scientific principles. Engineers accomplish this by either refining an existing product or process or by designing a new product or process that meets customers’needs. The engineering, or scientific, method is the approach to formulating and solving these problems. The steps in the engineering method are as follows: 1. Develop a clear and concise description of the problem. 2. Identify, at least tentatively, the important factors that affect this problem or that may play a role in its solution. 3. Propose a model for the problem, using scientific or engineering knowledge of the phenomenon being studied. State any limitations or assumptions of the model. 4. Conduct appropriate experiments and collect data to test or validate the tentative model or conclusions made in steps 2 and 3. 5. Refine the model on the basis of the observed data. 6. Manipulate the model to assist in developing a solution to the problem. 7. Conduct an appropriate experiment to confirm that the proposed solution to the problem is both effective and efficient. 8. Draw conclusions or make recommendations based on the problem solution. The steps in the engineering method are shown in Fig. 1-1. Notice that the engineering method features a strong interplay between the problem, the factors that may influence its solution, a model of the phenomenon, and experimentation to verify the adequacy of the model and the proposed solution to the problem. Steps 2–4 in Fig. 1-1 are enclosed in a box, indicating that several cycles or iterations of these steps may be required to obtain the final solution. Consequently, engineers must know how to efficiently plan experiments, collect data, analyze and interpret the data, and understand how the observed data are related to the model they have proposed for the problem under study. The field of statistics deals with the collection, presentation, analysis, and use of data to make decisions, solve problems, and design products and processes. Because many aspects of engineering practice involve working with data, obviously some knowledge of statistics is important to any engineer. Specifically, statistical techniques can be a powerful aid in designing new products and systems, improving existing designs, and designing, developing, and improving production processes. Figure 1-1 The engineering method. Develop a clear description Identify the important factors Propose or refine a model Conduct experiments Manipulate the model Confirm the solution Conclusions and recommendations c01.qxd 5/9/02 1:27 PM Page 2 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:MONTGOMERY:REVISES UPLO D CH 1 12 FIN L:

1-1 THE ENGINEERING METHOD AND STATISTICAL THINKING 3 Statistical methods are used to help us describe and understand variability.By variability, we mean that successive observations of a system or phenomenon do not produce exactly the same result.We all encounter variability in our everyday lives,and statistical thinking can give us a useful way to incorporate this variability into our decision-making processes.For example,consider the gasoline mileage performance of your car.Do you always get exactly the same mileage performance on every tank of fuel?Of course not-in fact,sometimes the mileage performance varies considerably.This observed variability in gasoline mileage depends on many factors,such as the type of driving that has occurred most recently(city versus highway), the changes in condition of the vehicle over time (which could include factors such as tire inflation,engine compression,or valve wear),the brand and/or octane number of the gasoline used,or possibly even the weather conditions that have been recently experienced.These factors represent potential sources of variability in the system.Statistics gives us a framework for describing this variability and for learning about which potential sources of variability are the most important or which have the greatest impact on the gasoline mileage performance. We also encounter variability in dealing with engineering problems.For example,sup- pose that an engineer is designing a nylon connector to be used in an automotive engine application.The engineer is considering establishing the design specification on wall thick- ness at 3/32 inch but is somewhat uncertain about the effect of this decision on the connector pull-off force.If the pull-off force is too low,the connector may fail when it is installed in an engine.Eight prototype units are produced and their pull-off forces measured,resulting in the following data (in pounds):12.6,12.9,13.4,12.3,13.6,13.5,12.6,13.1.As we anticipated, not all of the prototypes have the same pull-off force.We say that there is variability in the pull-off force measurements.Because the pull-off force measurements exhibit variability,we consider the pull-off force to be a random variable.A convenient way to think of a random variable,say X,that represents a measurement,is by using the model X=μ十e (1-1) where u is a constant and e is a random disturbance.The constant remains the same with every measurement,but small changes in the environment,test equipment,differences in the indi- vidual parts themselves,and so forth change the value of e.If there were no disturbances,e would always equal zero and X would always be equal to the constant u.However,this never happens in the real world,so the actual measurements X exhibit variability.We often need to describe,quantify and ultimately reduce variability. Figure 1-2 presents a dot diagram of these data.The dot diagram is a very useful plot for displaying a small body of data-say,up to about 20 observations.This plot allows us to see eas- ily two features of the data:the location,or the middle,and the scatter or variability.When the number of observations is small,it is usually difficult to identify any specific patterns in the vari- ability,although the dot diagram is a convenient way to see any unusual data features. The need for statistical thinking arises often in the solution of engineering problems. Consider the engineer designing the connector.From testing the prototypes,he knows that the average pull-off force is 13.0 pounds.However,he thinks that this may be too low for the 。80880●8，°00 =最inch 12 13 14 1512 13 14 15o=inch Pull-off force Pull-off force Figure 1-2 Dot diagram of the pull-off force Figure 1-3 Dot diagram of pull-off force for two wall data when wall thickness is 3/32 inch. thicknesses

1-1 THE ENGINEERING METHOD AND STATISTICAL THINKING 3 Statistical methods are used to help us describe and understand variability. By variability, we mean that successive observations of a system or phenomenon do not produce exactly the same result. We all encounter variability in our everyday lives, and statistical thinking can give us a useful way to incorporate this variability into our decision-making processes. For example, consider the gasoline mileage performance of your car. Do you always get exactly the same mileage performance on every tank of fuel? Of course not—in fact, sometimes the mileage performance varies considerably. This observed variability in gasoline mileage depends on many factors, such as the type of driving that has occurred most recently (city versus highway), the changes in condition of the vehicle over time (which could include factors such as tire inflation, engine compression, or valve wear), the brand and/or octane number of the gasoline used, or possibly even the weather conditions that have been recently experienced. These factors represent potential sources of variability in the system. Statistics gives us a framework for describing this variability and for learning about which potential sources of variability are the most important or which have the greatest impact on the gasoline mileage performance. We also encounter variability in dealing with engineering problems. For example, suppose that an engineer is designing a nylon connector to be used in an automotive engine application. The engineer is considering establishing the design specification on wall thickness at 332 inch but is somewhat uncertain about the effect of this decision on the connector pull-off force. If the pull-off force is too low, the connector may fail when it is installed in an engine. Eight prototype units are produced and their pull-off forces measured, resulting in the following data (in pounds): 12.6, 12.9, 13.4, 12.3, 13.6, 13.5, 12.6, 13.1. As we anticipated, not all of the prototypes have the same pull-off force. We say that there is variability in the pull-off force measurements. Because the pull-off force measurements exhibit variability, we consider the pull-off force to be a random variable. A convenient way to think of a random variable, say X, that represents a measurement, is by using the model (1-1) where is a constant and is a random disturbance. The constant remains the same with every measurement, but small changes in the environment, test equipment, differences in the individual parts themselves, and so forth change the value of . If there were no disturbances, would always equal zero and X would always be equal to the constant . However, this never happens in the real world, so the actual measurements X exhibit variability. We often need to describe, quantify and ultimately reduce variability. Figure 1-2 presents a dot diagram of these data. The dot diagram is a very useful plot for displaying a small body of data—say, up to about 20 observations. This plot allows us to see easily two features of the data; the location, or the middle, and the scatter or variability. When the number of observations is small, it is usually difficult to identify any specific patterns in the variability, although the dot diagram is a convenient way to see any unusual data features. The need for statistical thinking arises often in the solution of engineering problems. Consider the engineer designing the connector. From testing the prototypes, he knows that the average pull-off force is 13.0 pounds. However, he thinks that this may be too low for the X 12 13 14 15 Pull-off force Figure 1-2 Dot diagram of the pull-off force data when wall thickness is 3/32 inch. 12 13 14 15 Pull-off force 3 32 inch inch = 1 8 = Figure 1-3 Dot diagram of pull-off force for two wall thicknesses. c01.qxd 5/9/02 1:28 PM Page 3 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:MONTGOMERY:REVISES UPLO D CH 1 12 FIN L:

4 CHAPTER I THE ROLE OF STATISTICS IN ENGINEERING intended application,so he decides to consider an alternative design with a greater wall thickness,1/8 inch.Eight prototypes of this design are built,and the observed pull-off force measurements are12.9,13.7,12.8,13.9,14.2,13.2,13.5,and13.1.The average is13.4. Results for both samples are plotted as dot diagrams in Fig.1-3,page 3.This display gives the impression that increasing the wall thickness has led to an increase in pull-off force. However,there are some obvious questions to ask.For instance,how do we know that an- other sample of prototypes will not give different results?Is a sample of eight prototypes adequate to give reliable results?If we use the test results obtained so far to conclude that increasing the wall thickness increases the strength,what risks are associated with this de- cision?For example,is it possible that the apparent increase in pull-off force observed in the thicker prototypes is only due to the inherent variability in the system and that increas- ing the thickness of the part(and its cost)really has no effect on the pull-off force? Often,physical laws(such as Ohm's law and the ideal gas law)are applied to help design products and processes.We are familiar with this reasoning from general laws to specific cases.But it is also important to reason from a specific set of measurements to more general cases to answer the previous questions.This reasoning is from a sample(such as the eight con- nectors)to a population(such as the connectors that will be sold to customers).The reasoning is referred to as statistical inference.See Fig.1-4.Historically,measurements were obtained from a sample of people and generalized to a population,and the terminology has remained. Clearly,reasoning based on measurements from some objects to measurements on all objects can result in errors(called sampling errors).However,if the sample is selected properly,these risks can be quantified and an appropriate sample size can be determined. In some cases,the sample is actually selected from a well-defined population.The sam- ple is a subset of the population.For example,in a study of resistivity a sample of three wafers might be selected from a production lot of wafers in semiconductor manufacturing.Based on the resistivity data collected on the three wafers in the sample,we want to draw a conclusion about the resistivity of all of the wafers in the lot. In other cases,the population is conceptual(such as with the connectors),but it might be thought of as future replicates of the objects in the sample.In this situation,the eight proto- type connectors must be representative,in some sense,of the ones that will be manufactured in the future.Clearly,this analysis requires some notion of stability as an additional assump- tion.For example,it might be assumed that the sources of variability in the manufacture of the prototypes(such as temperature,pressure,and curing time)are the same as those for the con- nectors that will be manufactured in the future and ultimately sold to customers. Time Physical Population Population laws 2 Future population Types of Statistical inference Sample Sample reasoning Product Sample x1 X2.....xn x1x21..xm designs Enumerative Analytic study study Figure 1-4 Statistical inference is one type of Figure 1-5 Enumerative versus analytic study. reasoning

4 CHAPTER 1 THE ROLE OF STATISTICS IN ENGINEERING Figure 1-5 Enumerative versus analytic study. Time Future population ? Population ? Enumerative study Analytic study Sample Sample x1, x2,…, xn x1, x2,…, xn intended application, so he decides to consider an alternative design with a greater wall thickness, 18 inch. Eight prototypes of this design are built, and the observed pull-off force measurements are 12.9, 13.7, 12.8, 13.9, 14.2, 13.2, 13.5, and 13.1. The average is 13.4. Results for both samples are plotted as dot diagrams in Fig. 1-3, page 3. This display gives the impression that increasing the wall thickness has led to an increase in pull-off force. However, there are some obvious questions to ask. For instance, how do we know that another sample of prototypes will not give different results? Is a sample of eight prototypes adequate to give reliable results? If we use the test results obtained so far to conclude that increasing the wall thickness increases the strength, what risks are associated with this decision? For example, is it possible that the apparent increase in pull-off force observed in the thicker prototypes is only due to the inherent variability in the system and that increasing the thickness of the part (and its cost) really has no effect on the pull-off force? Often, physical laws (such as Ohm’s law and the ideal gas law) are applied to help design products and processes. We are familiar with this reasoning from general laws to specific cases. But it is also important to reason from a specific set of measurements to more general cases to answer the previous questions. This reasoning is from a sample (such as the eight connectors) to a population (such as the connectors that will be sold to customers). The reasoning is referred to as statistical inference. See Fig. 1-4. Historically, measurements were obtained from a sample of people and generalized to a population, and the terminology has remained. Clearly, reasoning based on measurements from some objects to measurements on all objects can result in errors (called sampling errors). However, if the sample is selected properly, these risks can be quantified and an appropriate sample size can be determined. In some cases, the sample is actually selected from a well-defined population. The sample is a subset of the population. For example, in a study of resistivity a sample of three wafers might be selected from a production lot of wafers in semiconductor manufacturing. Based on the resistivity data collected on the three wafers in the sample, we want to draw a conclusion about the resistivity of all of the wafers in the lot. In other cases, the population is conceptual (such as with the connectors), but it might be thought of as future replicates of the objects in the sample. In this situation, the eight prototype connectors must be representative, in some sense, of the ones that will be manufactured in the future. Clearly, this analysis requires some notion of stability as an additional assumption. For example, it might be assumed that the sources of variability in the manufacture of the prototypes (such as temperature, pressure, and curing time) are the same as those for the connectors that will be manufactured in the future and ultimately sold to customers. Physical laws Types of reasoning Product designs Population Statistical inference Sample Figure 1-4 Statistical inference is one type of reasoning. c01.qxd 5/9/02 1:28 PM Page 4 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:MONTGOMERY:REVISES UPLO D CH 1 12 FIN L:

1-2 COLLECTING ENGINEERING DATA 5 The wafers-from-lots example is called an enumerative study.A sample is used to make an inference to the population from which the sample is selected.The connector example is called an analytic study.A sample is used to make an inference to a conceptual (future) population.The statistical analyses are usually the same in both cases,but an analytic study clearly requires an assumption of stability.See Fig.1-5,on page 4. 1-2 COLLECTING ENGINEERING DATA 1-2.1 Basic Principles In the previous section,we illustrated some simple methods for summarizing data.In the en- gineering environment,the data is almost always a sample that has been selected from some population.Three basic methods of collecting data are A retrospective study using historical data An observational study A designed experiment An effective data collection procedure can greatly simplify the analysis and lead to improved understanding of the population or process that is being studied.We now consider some ex- amples of these data collection methods. 1-2.2 Retrospective Study Montgomery,Peck,and Vining(2001)describe an acetone-butyl alcohol distillation column for which concentration of acetone in the distillate or output product stream is an important variable.Factors that may affect the distillate are the reboil temperature,the con- densate temperature,and the reflux rate.Production personnel obtain and archive the following records: The concentration of acetone in an hourly test sample of output product The reboil temperature log,which is a plot of the reboil temperature over time The condenser temperature controller log The nominal reflux rate each hour The reflux rate should be held constant for this process.Consequently,production personnel change this very infrequently. A retrospective study would use either all or a sample of the historical process data archived over some period of time.The study objective might be to discover the relationships among the two temperatures and the reflux rate on the acetone concentration in the output product stream.However,this type of study presents some problems: 1.We may not be able to see the relationship between the reflux rate and acetone con- centration,because the reflux rate didn't change much over the historical period. 2.The archived data on the two temperatures (which are recorded almost continu- ously)do not correspond perfectly to the acetone concentration measurements (which are made hourly).It may not be obvious how to construct an approximate correspondence

1-2 COLLECTING ENGINEERING DATA 5 The wafers-from-lots example is called an enumerative study. A sample is used to make an inference to the population from which the sample is selected. The connector example is called an analytic study. A sample is used to make an inference to a conceptual (future) population. The statistical analyses are usually the same in both cases, but an analytic study clearly requires an assumption of stability. See Fig. 1-5, on page 4. 1-2 COLLECTING ENGINEERING DATA 1-2.1 Basic Principles In the previous section, we illustrated some simple methods for summarizing data. In the engineering environment, the data is almost always a sample that has been selected from some population. Three basic methods of collecting data are A retrospective study using historical data An observational study A designed experiment An effective data collection procedure can greatly simplify the analysis and lead to improved understanding of the population or process that is being studied. We now consider some examples of these data collection methods. 1-2.2 Retrospective Study Montgomery, Peck, and Vining (2001) describe an acetone-butyl alcohol distillation column for which concentration of acetone in the distillate or output product stream is an important variable. Factors that may affect the distillate are the reboil temperature, the condensate temperature, and the reflux rate. Production personnel obtain and archive the following records: The concentration of acetone in an hourly test sample of output product The reboil temperature log, which is a plot of the reboil temperature over time The condenser temperature controller log The nominal reflux rate each hour The reflux rate should be held constant for this process. Consequently, production personnel change this very infrequently. A retrospective study would use either all or a sample of the historical process data archived over some period of time. The study objective might be to discover the relationships among the two temperatures and the reflux rate on the acetone concentration in the output product stream. However, this type of study presents some problems: 1. We may not be able to see the relationship between the reflux rate and acetone concentration, because the reflux rate didn’t change much over the historical period. 2. The archived data on the two temperatures (which are recorded almost continuously) do not correspond perfectly to the acetone concentration measurements (which are made hourly). It may not be obvious how to construct an approximate correspondence. c01.qxd 5/9/02 1:28 PM Page 5 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:MONTGOMERY:REVISES UPLO D CH 1 12 FIN L:

6 CHAPTER I THE ROLE OF STATISTICS IN ENGINEERING 3.Production maintains the two temperatures as closely as possible to desired targets or set points.Because the temperatures change so little,it may be difficult to assess their real impact on acetone concentration. 4.Within the narrow ranges that they do vary,the condensate temperature tends to in- crease with the reboil temperature.Consequently,the effects of these two process variables on acetone concentration may be difficult to separate. As you can see,a retrospective study may involve a lot of data,but that data may contain relatively little useful information about the problem.Furthermore,some of the relevant data may be missing,there may be transcription or recording errors resulting in outliers (or unusual values),or data on other important factors may not have been collected and archived.In the distillation column,for example,the specific concentrations of butyl alco- hol and acetone in the input feed stream are a very important factor,but they are not archived because the concentrations are too hard to obtain on a routine basis.As a result of these types of issues,statistical analysis of historical data sometimes identify interesting phenomena,but solid and reliable explanations of these phenomena are often difficult to obtain. 1-2.3 Observational Study In an observational study,the engineer observes the process or population,disturbing it as lit- tle as possible,and records the quantities of interest.Because these studies are usually con- ducted for a relatively short time period,sometimes variables that are not routinely measured can be included.In the distillation column,the engineer would design a form to record the two temperatures and the reflux rate when acetone concentration measurements are made.It may even be possible to measure the input feed stream concentrations so that the impact of this fac- tor could be studied.Generally,an observational study tends to solve problems 1 and 2 above and goes a long way toward obtaining accurate and reliable data.However,observational studies may not help resolve problems 3 and 4. 1.2.4 Designed Experiments In a designed experiment the engineer makes deliberate or purposeful changes in the control- lable variables of the system or process,observes the resulting system output data,and then makes an inference or decision about which variables are responsible for the observed changes in output performance.The nylon connector example in Section 1-1 illustrates a designed ex- periment;that is,a deliberate change was made in the wall thickness of the connector with the objective of discovering whether or not a greater pull-off force could be obtained.Designed experiments play a very important role in engineering design and development and in the improvement of manufacturing processes.Generally,when products and processes are designed and developed with designed experiments,they enjoy better performance,higher reliability,and lower overall costs.Designed experiments also play a crucial role in reducing the lead time for engineering design and development activities. For example,consider the problem involving the choice of wall thickness for the nylon connector.This is a simple illustration of a designed experiment.The engineer chose two wall thicknesses for the connector and performed a series of tests to obtain pull-off force measurements at each wall thickness.In this simple comparative experiment,the

6 CHAPTER 1 THE ROLE OF STATISTICS IN ENGINEERING 3. Production maintains the two temperatures as closely as possible to desired targets or set points. Because the temperatures change so little, it may be difficult to assess their real impact on acetone concentration. 4. Within the narrow ranges that they do vary, the condensate temperature tends to increase with the reboil temperature. Consequently, the effects of these two process variables on acetone concentration may be difficult to separate. As you can see, a retrospective study may involve a lot of data, but that data may contain relatively little useful information about the problem. Furthermore, some of the relevant data may be missing, there may be transcription or recording errors resulting in outliers (or unusual values), or data on other important factors may not have been collected and archived. In the distillation column, for example, the specific concentrations of butyl alcohol and acetone in the input feed stream are a very important factor, but they are not archived because the concentrations are too hard to obtain on a routine basis. As a result of these types of issues, statistical analysis of historical data sometimes identify interesting phenomena, but solid and reliable explanations of these phenomena are often difficult to obtain. 1-2.3 Observational Study In an observational study, the engineer observes the process or population, disturbing it as little as possible, and records the quantities of interest. Because these studies are usually conducted for a relatively short time period, sometimes variables that are not routinely measured can be included. In the distillation column, the engineer would design a form to record the two temperatures and the reflux rate when acetone concentration measurements are made. It may even be possible to measure the input feed stream concentrations so that the impact of this factor could be studied. Generally, an observational study tends to solve problems 1 and 2 above and goes a long way toward obtaining accurate and reliable data. However, observational studies may not help resolve problems 3 and 4. 1-2.4 Designed Experiments In a designed experiment the engineer makes deliberate or purposeful changes in the controllable variables of the system or process, observes the resulting system output data, and then makes an inference or decision about which variables are responsible for the observed changes in output performance. The nylon connector example in Section 1-1 illustrates a designed experiment; that is, a deliberate change was made in the wall thickness of the connector with the objective of discovering whether or not a greater pull-off force could be obtained. Designed experiments play a very important role in engineering design and development and in the improvement of manufacturing processes. Generally, when products and processes are designed and developed with designed experiments, they enjoy better performance, higher reliability, and lower overall costs. Designed experiments also play a crucial role in reducing the lead time for engineering design and development activities. For example, consider the problem involving the choice of wall thickness for the nylon connector. This is a simple illustration of a designed experiment. The engineer chose two wall thicknesses for the connector and performed a series of tests to obtain pull-off force measurements at each wall thickness. In this simple comparative experiment, the c01.qxd 5/9/02 1:28 PM Page 6 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:MONTGOMERY:REVISES UPLO D CH 1 12 FIN L:

1-2 COLLECTING ENGINEERING DATA 7 engineer is interested in determining if there is any difference between the 3/32-and 1/8-inch designs.An approach that could be used in analyzing the data from this experi- ment is to compare the mean pull-off force for the 3/32-inch design to the mean pull-off force for the 1/8-inch design using statistical hypothesis testing,which is discussed in detail in Chapters 9 and 10.Generally,a hypothesis is a statement about some aspect of the system in which we are interested.For example,the engineer might want to know if the mean pull-off force of a 3/32-inch design exceeds the typical maximum load expected to be encountered in this application,say 12.75 pounds.Thus,we would be interested in test- ing the hypothesis that the mean strength exceeds 12.75 pounds.This is called a single- sample hypothesis testing problem.It is also an example of an analytic study.Chapter 9 presents techniques for this type of problem.Alternatively,the engineer might be inter- ested in testing the hypothesis that increasing the wall thickness from 3/32-to 1/8-inch results in an increase in mean pull-off force.Clearly,this is an analytic study;it is also an example of a two-sample hypothesis testing problem.Two-sample hypothesis testing problems are discussed in Chapter 10. Designed experiments are a very powerful approach to studying complex systems,such as the distillation column.This process has three factors,the two temperatures and the reflux rate,and we want to investigate the effect of these three factors on output acetone concentra- tion.A good experimental design for this problem must ensure that we can separate the effects of all three factors on the acetone concentration.The specified values of the three factors used in the experiment are called factor levels.Typically,we use a small number of levels for each factor,such as two or three.For the distillation column problem,suppose we use a"high,"and "low,"level(denoted +1 and-1,respectively)for each of the factors.We thus would use two levels for each of the three factors.A very reasonable experiment design strategy uses every possible combination of the factor levels to form a basic experiment with eight different set- tings for the process.This type of experiment is called a factorial experiment.Table 1-1 pres- ents this experimental design. Figure 1-6,on page 8,illustrates that this design forms a cube in terms of these high and low levels.With each setting of the process conditions,we allow the column to reach equilib- rium,take a sample of the product stream,and determine the acetone concentration.We then can draw specific inferences about the effect of these factors.Such an approach allows us to proactively study a population or process.Designed experiments play a very important role in engineering and science.Chapters 13 and 14 discuss many of the important principles and techniques of experimental design. Table 1-1 The Designed Experiment(Factorial Design)for the Distillation Column Reboil Temp. Condensate Temp. Reflux Rate -1 -1 -1 +1 -1 -1 -1 +1 -1 +1 +1 -1 -1 -1 +1 +1 -1 -1 +1 +1 +1 +1 +1

engineer is interested in determining if there is any difference between the 332- and 18-inch designs. An approach that could be used in analyzing the data from this experiment is to compare the mean pull-off force for the 332-inch design to the mean pull-off force for the 18-inch design using statistical hypothesis testing, which is discussed in detail in Chapters 9 and 10. Generally, a hypothesis is a statement about some aspect of the system in which we are interested. For example, the engineer might want to know if the mean pull-off force of a 332-inch design exceeds the typical maximum load expected to be encountered in this application, say 12.75 pounds. Thus, we would be interested in testing the hypothesis that the mean strength exceeds 12.75 pounds. This is called a singlesample hypothesis testing problem. It is also an example of an analytic study. Chapter 9 presents techniques for this type of problem. Alternatively, the engineer might be interested in testing the hypothesis that increasing the wall thickness from 332- to 18-inch results in an increase in mean pull-off force. Clearly, this is an analytic study; it is also an example of a two-sample hypothesis testing problem. Two-sample hypothesis testing problems are discussed in Chapter 10. Designed experiments are a very powerful approach to studying complex systems, such as the distillation column. This process has three factors, the two temperatures and the reflux rate, and we want to investigate the effect of these three factors on output acetone concentration. A good experimental design for this problem must ensure that we can separate the effects of all three factors on the acetone concentration. The specified values of the three factors used in the experiment are called factor levels. Typically, we use a small number of levels for each factor, such as two or three. For the distillation column problem, suppose we use a “high,’’ and “low,’’ level (denoted +1 and 1, respectively) for each of the factors. We thus would use two levels for each of the three factors. A very reasonable experiment design strategy uses every possible combination of the factor levels to form a basic experiment with eight different settings for the process. This type of experiment is called a factorial experiment. Table 1-1 presents this experimental design. Figure 1-6, on page 8, illustrates that this design forms a cube in terms of these high and low levels. With each setting of the process conditions, we allow the column to reach equilibrium, take a sample of the product stream, and determine the acetone concentration. We then can draw specific inferences about the effect of these factors. Such an approach allows us to proactively study a population or process. Designed experiments play a very important role in engineering and science. Chapters 13 and 14 discuss many of the important principles and techniques of experimental design. 1-2 COLLECTING ENGINEERING DATA 7 Table 1-1 The Designed Experiment (Factorial Design) for the Distillation Column Reboil Temp. Condensate Temp. Reflux Rate 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 c01.qxd 5/9/02 1:28 PM Page 7 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:MONTGOMERY:REVISES UPLO D CH 1 12 FIN L:

8 CHAPTER I THE ROLE OF STATISTICS IN ENGINEERING *1 Condensate 1 Figure 1-6 The fac- temperature torial design for the -1 +1 distillation column Reboil temperature 1-2.5 A Factorial Experiment for the Connector Pull-off Force Problem(CD Only) 1-2.6 Observing Processes Over Time Often data are collected over time.In this case,it is usually very helpful to plot the data ver- sus time in a time series plot.Phenomena that might affect the system or process often be- come more visible in a time-oriented plot and the concept of stability can be better judged. Figure 1-7 is a dot diagram of acetone concentration readings taken hourly from the distillation column described in Section 1-2.2.The large variation displayed on the dot diagram indicates a lot of variability in the concentration,but the chart does not help explain the reason for the variation.The time series plot is shown in Figure 1-8,on page 9.A shift in the process mean level is visible in the plot and an estimate of the time of the shift can be obtained. W.Edwards Deming,a very influential industrial statistician,stressed that it is important to understand the nature of variability in processes and systems over time.He conducted an experiment in which he attempted to drop marbles as close as possible to a target on a table. He used a funnel mounted on a ring stand and the marbles were dropped into the funnel.See Fig.1-9.The funnel was aligned as closely as possible with the center of the target.He then used two different strategies to operate the process.(1)He never moved the funnel.He just dropped one marble after another and recorded the distance from the target.(2)He dropped the first marble and recorded its location relative to the target.He then moved the funnel an equal and opposite distance in an attempt to compensate for the error.He continued to make this type of adjustment after each marble was dropped. After both strategies were completed,he noticed that the variability of the distance from the target for strategy 2 was approximately 2 times larger than for strategy 1.The ad- justments to the funnel increased the deviations from the target.The explanation is that the error(the deviation of the marble's position from the target)for one marble provides no information about the error that will occur for the next marble.Consequently,adjustments to the funnel do not decrease future errors.Instead,they tend to move the funnel farther from the target. This interesting experiment points out that adjustments to a process based on random dis- turbances can actually increase the variation of the process.This is referred to as overcontrol 8 Figure 1-7 The dot diagram illustrates 。。。。。8。88.。。。 variation but does not 80.584.0 87.591.094.598.0 identify the problem. Acetone concentration

8 CHAPTER 1 THE ROLE OF STATISTICS IN ENGINEERING 1-2.5 A Factorial Experiment for the Connector Pull-off Force Problem (CD Only) 1-2.6 Observing Processes Over Time Often data are collected over time. In this case, it is usually very helpful to plot the data versus time in a time series plot. Phenomena that might affect the system or process often become more visible in a time-oriented plot and the concept of stability can be better judged. Figure 1-7 is a dot diagram of acetone concentration readings taken hourly from the distillation column described in Section 1-2.2. The large variation displayed on the dot diagram indicates a lot of variability in the concentration, but the chart does not help explain the reason for the variation. The time series plot is shown in Figure 1-8, on page 9. A shift in the process mean level is visible in the plot and an estimate of the time of the shift can be obtained. W. Edwards Deming, a very influential industrial statistician, stressed that it is important to understand the nature of variability in processes and systems over time. He conducted an experiment in which he attempted to drop marbles as close as possible to a target on a table. He used a funnel mounted on a ring stand and the marbles were dropped into the funnel. See Fig. 1-9. The funnel was aligned as closely as possible with the center of the target. He then used two different strategies to operate the process. (1) He never moved the funnel. He just dropped one marble after another and recorded the distance from the target. (2) He dropped the first marble and recorded its location relative to the target. He then moved the funnel an equal and opposite distance in an attempt to compensate for the error. He continued to make this type of adjustment after each marble was dropped. After both strategies were completed, he noticed that the variability of the distance from the target for strategy 2 was approximately 2 times larger than for strategy 1. The adjustments to the funnel increased the deviations from the target. The explanation is that the error (the deviation of the marble’s position from the target) for one marble provides no information about the error that will occur for the next marble. Consequently, adjustments to the funnel do not decrease future errors. Instead, they tend to move the funnel farther from the target. This interesting experiment points out that adjustments to a process based on random disturbances can actually increase the variation of the process. This is referred to as overcontrol Reflux rate Reboil temperature temperature Condensate –1 +1 –1 –1 +1 +1 Figure 1-6 The factorial design for the distillation column. 80.5 84.0 87.5 91.0 94.5 98.0x Acetone concentration Figure 1-7 The dot diagram illustrates variation but does not identify the problem. c01.qxd 5/9/02 1:28 PM Page 8 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:MONTGOMERY:REVISES UPLO D CH 1 12 FIN L:

10 CHAPTER 1 THE ROLE OF STATISTICS IN ENGINEERING Without adjustment With adjustment 0 2 4 6 8 y 10 12 14 16 1 11 21 31 41 51 61 71 81 91 Observation number Process mean shift is detected. Figure 1-11 Process mean shift is detected at observation number 57, and one adjustment (a decrease of two units) reduces the deviations from target. 80 5 15 25 10 90 Acetone concentration 100 0 20 1 1 Observation number (hour) 30 x = 91.50 Lower control limit = 82.54 Upper control limit = 100.5 Figure 1-12 A control chart for the chemical process concentration data. two units) is applied to the mean after the shift is detected (at observation number 57). Note that this adjustment decreases the deviations from target. The question of when to apply adjustments (and by what amounts) begins with an understanding of the types of variation that affect a process. A control chart is an invaluable way to examine the variability in time-oriented data. Figure 1-12 presents a control chart for the concentration data from Fig. 1-8. The center line on the control chart is just the average of the concentration measurements for the first 20 samples ( ) when the process is stable. The upper control limit and the lower control limit are a pair of statistically derived limits that reflect the inherent or natural variability in the process. These limits are located three standard deviations of the concentration values above and below the center line. If the process is operating as it should, without any external sources of variability present in the system, the concentration measurements should fluctuate randomly around the center line, and almost all of them should fall between the control limits. In the control chart of Fig. 1-12, the visual frame of reference provided by the center line and the control limits indicates that some upset or disturbance has affected the process around sample 20 because all of the following observations are below the center line and two of them x 91.5 gl c01.qxd 5/10/02 10:15 M Page 10 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:MONTGOMERY:REVISES UPLO D CH 1 14 FIN L: