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

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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:

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