A population is not always defined to be a set of existing units. Often we are interested in studying the population of all of the units that will be or could potentially be produced by a process.

Processes produce output over time. For example, this year’s Lincoln Town Car manufacturing process produces Lincoln Town Cars over time. Early in the model year, Ford Motor Company might wish to study the population of the city driving mileages of all Lincoln Town Cars that will be produced during the model year. Or, even more hypothetically, Ford might wish to study the population of the city driving mileages of all Lincoln Town Cars that could potentially be produced by this model year’s manufacturing process. The first population is called a finite populationA population that contains a finite number of units. because only a finite number of cars will be produced during the year. Any population of existing units is also finite. The second population is called an infinite populationA population that is defined so that there is no limit to the number of Units that could potentially belong to the population. because the manufacturing process that produces this year’s model could in theory always be used to build “one more car.” That is, theoretically there is no limit to the number of cars that could be produced by this year’s process. There are a multitude of other examples of finite or infinite hypothetical populations. For instance, we might study the population of all waiting times that will or could potentially be experienced by patients of a hospital emergency room. Or we might study the population of all the amounts of grape jelly that will be or could potentially be dispensed into 16-ounce jars by an automated filling machine. To study a population of potential process observations, we sample the process—usually at equally spaced time points—over time. This is illustrated in the following case.

According to the website of the Association of Trial Lawyers of America,7 Stella Liebeck of Albuquerque, New Mexico, was severely burned by McDonald’s coffee in February 1992. Liebeck, who received third-degree burns over 6 percent of her body, was awarded $160,000 in compensatory damages and $480,000 in punitive damages. A postverdict investigation revealed that the coffee temperature at the local Albuquerque McDonald’s had dropped from about 185°F before the trial to about 158° after the trial. This case concerns coffee temperatures at a fast-food restaurant. Because of the possibility of future litigation and to possibly improve the coffee’s taste, the restaurant wishes to study and monitor the temperature of the coffee it serves. To do this, the restaurant personnel measure the temperature of the coffee being dispensed (in degrees Fahrenheit) at half-hour intervals from 10 A.M. to 9:30 P.M. on a given day. Table 1.7 gives the 24 temperature measurements obtained in the time order that they were observed. Here, time equals 1 at 10 A.M. and 24 at 9:30 P.M.

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TABLE 1.7  24 Coffee Temperatures Observed in Time Order (°F)   <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=jpg::::/sites/dl/free/0072977477/339821/cdicon_small.jpg','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif"> (11.0K)</a>  Coffee

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Examining Table 1.7, we see that the coffee temperatures range from 152° to 170°. Based on this, is it reasonable to conclude that the temperature of most of the coffee that will or could potentially be served by the restaurant will be between 152° and 170°? The answer is yes if the restaurant’s coffee-making process operates consistently over time. That is, this process must be in a state of statistical control.A state in which a process does not exhibit any unusual variations. Often this means that the process displays a uniform amount of variation around a constant, or horizontal, level.

To assess whether a process is in statistical control, we sample the process often enough to detect unusual variations or instabilities. The fast-food restaurant has sampled the coffee-making process every half hour. In other situations, we sample processes with other frequencies—for example, every minute, every hour, or every day. In Chapter 14, where we discuss a systematic method for studying processes called statistical process control (SPC)A method for analyzing process data in which we monitor and study the process variation. The goal is to stabilize (and reduce) the amount of process variation., we consider how to determine the sampling frequency for a process. Using the observed process measurements, we can then construct a runs plotA graph of individual process measurements versus time. (sometimes called a time series plot).

Figure 1.3 shows the MINITAB and Excel outputs of a runs plot of the temperature data. (Some people call such a plot a line chart when the plot points are connected by line segments as in the Excel output.) Here we plot each coffee temperature on the vertical scale versus its corresponding time index on the horizontal scale. For instance, the first temperature (163°) is plotted versus time equals 1, the second temperature (169°) is plotted versus time equals 2, and so forth. The runs plot suggests that the temperatures exhibit a relatively constant amount of variation around a relatively constant level. That is, the center of the temperatures can be pretty much represented by a horizontal line (constant level)—see the line drawn through the plotted points—and the spread of the points around the line is staying about the same (constant variation). Note that the plot points tend to form a horizontal band. Therefore, the temperatures are in statistical control.

In general, assume that we have sampled a process at different (usually equally spaced) time points and made a runs plot of the resulting sample measurements. If the plot indicates that the process is in statistical control, and if it is reasonable to believe that the process will remain in control, then it is probably reasonable to regard the sample measurements as an approximately random sample from the population of all possible process measurements. Furthermore, since the process is remaining in statistical control, the process performance is predictable. This allows us to make statistical inferences about the population of all possible process measurements that will or potentially could result from using the process. For example, assuming that the coffee-making process will remain in statistical control, it is reasonable to conclude that the temperature of most of the coffee that will be or could potentially be served will be between 152° and 170°.

To emphasize the importance of statistical control, suppose that another fast-food restaurant observes the 24 coffee temperatures that are plotted versus time in Figure 1.4. These temperatures also range between 152° and 170°. However, we cannot infer from this that the temperature of most of the coffee that will be or could potentially be served by this other restaurant will be between 152° and 170°. This is because the downward trend in the runs plot of Figure 1.4 indicates that the coffee-making process is out of control and will soon produce temperatures below 152°. Another example of an out-of-control process is illustrated in Figure 1.5. Here, the coffee temperatures seem to fluctuate around a constant level but with increasing variation (notice that the plotted temperatures fan out as time advances). In general, the specific pattern of out-of-control behavior can suggest the reason for this behavior. For example, the downward trend in the runs plot of Figure 1.4 might suggest that the restaurant’s coffeemaker has a defective heating element.

FIGURE 1.4  A Runs Plot of Coffee Temperatures: The Process Level Is Decreasing

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FIGURE 1.5  A Runs Plot of Coffee Temperatures: The Process Variation Is Increasing

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Visually inspecting a runs plot to check for statistical control can be tricky. One reason is that the scale of measurements on the vertical axis can influence whether the data appear to form a horizontal band. We will study better methods for detecting out-of-control behavior in Chapter 14. For now, we will simply emphasize that a process must be in statistical control in order to make valid statistical inferences about the population of all possible process observations. Also, note that being in statistical control does not necessarily imply that a process is capable of producing output that meets our requirements. For example, suppose that marketing research suggests that the fast-food restaurant’s customers feel that coffee tastes best if its temperature is between 153° and 167°. Since Table 1.7 indicates that the temperature of some of the coffee it serves is not in this range (note that two of the temperatures are 152°, one is 169°, and another is 170°), the restaurant might take action to reduce the variation of the coffee temperatures.

The marketing research, and coffee temperature cases are both examples of using the statistical processA method for analyzing process data in which we monitor and study the process variation. The goal is to stabilize (and reduce) the amount of process variation. to make a statistical inference. In the next case, we formally describe and illustrate this process.

In 2005 the U.S. Department of Energy (DOE) and the Environmental Protection Agency (EPA) emphasized the importance of auto fuel economy. The Fuel Economy Guide, available at the DOE website, discusses the effects of gasoline consumption on U.S. energy security and the economy as follows.⁸

Buying a more fuel efficient vehicle can help strengthen our national energy security by reducing our dependence on foreign oil. Half of the oil used to produce the gasoline you put in your tank is imported. The United States uses about 20 million barrels of oil per day, two thirds of which is used for transportation. Petroleum imports cost us about $2 billion a week—that’s money that could be used to fuel our own economy.

The Guide also discusses the effects of gasoline consumption on global warming:⁹

Burning fossil fuels such as gasoline or diesel adds greenhouse gases, including carbon dioxide, to the earth’s atmosphere. Greenhouse gases trap heat and thus warm the earth because they prevent a significant proportion of infrared radiation from escaping into space. Vehicles with lower fuel economy burn more fuel, creating more carbon dioxide. Every gallon of gasoline your vehicle burns puts 20 pounds of carbon dioxide into the atmosphere. You can reduce your contribution to global warming by choosing a vehicle with higher fuel economy. By choosing a vehicle that achieves 25 miles per gallon rather than 20 miles per gallon, you can prevent the release of about 15 tons of greenhouse gas pollution over the lifetime of your vehicle.

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In this case study we consider a tax credit offered by the federal government to automakers for improving the fuel economy of midsize cars. According to The Fuel Economy Guide—2005 Model Year, virtually every midsize car equipped with an automatic transmission has an EPA combined city and highway mileage estimate of 26 miles per gallon (mpg) or less. Furthermore, the EPA has concluded that a 5 mpg increase in fuel economy is significant and feasible.¹⁰ Therefore, the government has decided to offer the tax credit to any automaker selling a midsize model with an automatic transmission that achieves an EPA combined city and highway mileage estimate of at least 31 mpg. To find the combined city and highway mileage estimate for a particular car model, the EPA tests a sample of cars. The steps used to obtain this estimate are those used in the statistical process for making a statistical inference:

1. Describe the practical problem of interest and the associated population or process to be studied. Consider an automaker that has recently introduced a midsize model with an automatic transmission and wishes to demonstrate that this new model qualifies for the tax credit. The automaker will study the population of all cars of this type that will be or could potentially be produced.
2. Describe the variable of interest and how it will be measured. The variable of interest is the EPA combined city and highway mileage of a car. This mileage is obtained by testing the car on a device similar to a giant treadmill. The device is used to simulate a 7.5-mile city driving trip and a 10-mile highway driving trip, and the resulting mileages are used to calculate the EPA combined mileage for the car.¹¹
3. Describe the sampling procedure. The automaker selects a sample of 49 of the new midsize cars by randomly selecting one car from those produced during each of 49 consecutive production shifts. Here the sample size (49) is determined by statistical considerations to be discussed in Chapter 7. Each sampled car is subjected to the EPA test. The resulting sample of 49 combined city and highway mileages is given in Table 1.8 (in time order).




TABLE 1.8  A Sample of 49 Mileages (Time Order Is Given by Reading Down the Columns from Left to Right)   <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=jpg::::/sites/dl/free/0072977477/339821/cdicon_small.jpg','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif"> (11.0K)</a>  GasMiles

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4. Describe the statistical inference of interest. The sample of 49 mileages will be used to estimate the “typical” EPA combined mileage for the population of all possible new midsize cars. The estimate obtained is the EPA combined city and highway mileage estimate for the new midsize model.
5. Describe how the statistical inference will be made and evaluate the reliability of the inference. Figure 1.6 gives the MegaStat output of a runs plot of the 49 mileages. The runs plot indicates that the mileages are in statistical control. If it is reasonable to believe that car mileages for this model will remain in control, we can make statistical inferences. For instance, because the mileages in Table 1.8 range from 29.8 to 33.3 mpg, we might infer that most of the new midsize cars will get combined city and highway mileages between 29.8 and 33.3 mpg. To estimate the “typical” EPA combined mileage for the population of all possible cars, we might visually draw a horizontal line through the “middle” of the plot points in Figure 1.6. When we do this, the horizontal line intersects the vertical axis at about 31.5 mpg. Therefore, we might conclude that the EPA combined city and highway mileage estimate for the new midsize model should be 31.5 mpg. Since this estimate exceeds the EPA standard of 31 mpg, we might also conclude that the automaker qualifies for the tax credit. However, the estimate is intuitive, so we do not have any information about its reliability. In Chapter 2 we will study more precise ways to both define and estimate a “typical” population value. Then in Chapters 3 through 7 we will study tools for assessing the reliability of estimation procedures and for estimating “with confidence.”




FIGURE 1.6  MegaStat Output of a Runs Plot of the 49 Mileages

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CONCEPTS

1.12	Define a process. Then give an example of a process you might study when you start your career after graduating from college.	<a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=jpg::::/sites/dl/free/0072977477/hm_logo_cmyk.jpg','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif"> (K)</a>
1.13	Explain what it means to say that a process is in statistical control.
1.14	What is a runs plot? What does a runs plot look like when we sample and plot a process that is in statistical control?
METHODS AND APPLICATIONS
1.15	The data below give 18 measurements of a critical dimension for an automobile part (measurements in inches). Here one part has been randomly selected each hour from the previous hour’s production, and the measurements are given in time order.   <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=jpg::::/sites/dl/free/0072977477/339821/cdicon_small.jpg','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif"> (11.0K)</a>  AutoPart1 <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=gif::::/sites/dl/free/0072977477/0073191914_001_041.gif','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif"> (K)</a> Construct a runs plot and determine if the process appears to be in statistical control.
1.16	Table 1.9 presents the time (in days) needed to settle the 67 homeowners’ insurance claims handled by an Indiana insurance agent over a year. The claims are given in time order by loss date.   <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=jpg::::/sites/dl/free/0072977477/339821/cdicon_small.jpg','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif"> (11.0K)</a>  ClaimSet TABLE 1.9  Number of Days Required to Settle Homeowners’ Insurance Claims (Claims Made from July 2, 2004 to June 25, 2005)   <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=jpg::::/sites/dl/free/0072977477/339821/cdicon_small.jpg','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif"> (11.0K)</a>  ClaimSet <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=gif::::/sites/dl/free/0072977477/339821/table1_9.gif','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif"> (51.0K)</a> Figure 1.7 shows a MINITAB runs plot of the claims data in Table 1.9. Does the claims-handling process seem to be in statistical control? Why or why not? FIGURE 1.7  MINITAB Runs Plot of the Insurance Claims Data for Exercise 1.16 <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=jpg::::/sites/dl/free/0072977477/bow77477_0107.jpg','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif"> (K)</a> In March of 2005, Indiana was hit by a widespread ice storm that caused heavy damage to homes in the area. Did this ice storm have a significant impact on the time needed to settle homeowners’ claims? Should the agent consider improving procedures for handling claims in emergency situations? Why or why not?
1.17	In the article “Accelerating Improvement” published in Quality Progress (October 1991), Gaudard, Coates, and Freeman describe a restaurant that caters to business travelers and has a self-service breakfast buffet. Interested in customer satisfaction, the manager conducts a survey over a three-week period and finds that the main customer complaint is having to wait too long to be seated. On each day from September 11, 1989, to October 1, 1989, a problem-solving team records the percentage of patrons who must wait more than one minute to be seated. A runs plot of the daily percentages is shown in Figure 1.8.12 What does the runs plot suggest? FIGURE 1.8  Runs Plot of Daily Percentages of Customers Waiting More Than One Minute to Be Seated (for Exercise 1.17) <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=jpg::::/sites/dl/free/0072977477/bow77477_0108.jpg','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif"> (K)</a>
1.18	THE TRASH BAG CASE13   <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=jpg::::/sites/dl/free/0072977477/339821/cdicon_small.jpg','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif"> (11.0K)</a>  TrashBag A company that produces and markets trash bags has developed an improved 30-gallon bag. The new bag is produced using a specially formulated plastic that is both stronger and more biodegradable than previously used plastics, and the company wishes to evaluate the strength of this bag. The breaking strength of a trash bag is considered to be the amount (in pounds) of a representative trash mix that when loaded into a bag suspended in the air will cause the bag to sustain significant damage (such as ripping or tearing). The company has decided to carry out a 40-hour pilot production run of the new bags. Each hour, at a randomly selected time during the hour, a bag is taken off the production line. The bag is then subjected to a breaking strength test. The 40 breaking strengths obtained during the pilot production run are given in Table 1.10, and an Excel runs plot of these breaking strengths is given in Figure 1.9. TABLE 1.10  Trash Bag Breaking Strengths   <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=jpg::::/sites/dl/free/0072977477/339821/cdicon_small.jpg','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif"> (11.0K)</a>  TrashBag <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=gif::::/sites/dl/free/0072977477/0073191914_001_045.gif','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif"> (K)</a> FIGURE 1.9  Excel Runs Plot of Breaking Strengths for Exercise 1.18 <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=jpg::::/sites/dl/free/0072977477/bow77477_0109.jpg','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif"> (K)</a> Do the 40 breaking strengths appear to be in statistical control? Explain. Estimate limits between which most of the breaking strengths of all trash bags would fall.
1.19	THE BANK CUSTOMER WAITING TIME CASE   <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=jpg::::/sites/dl/free/0072977477/339821/cdicon_small.jpg','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif"> (11.0K)</a>  WaitTime Recall that every 150th customer arriving during peak business hours was sampled until a systematic sample of 100 customers was obtained. This systematic sampling procedure is equivalent to sampling from a process. Figure 1.10 shows a MegaStat runs plot of the 100 waiting times in TABLE 1.6. Does the process appear to be in statistical control? Explain. FIGURE 1.10  MegaStat Runs Plot of Waiting Times for Exercise 1.19 <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=jpg::::/sites/dl/free/0072977477/bow77477_0110.jpg','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif"> (K)</a>

Exercises  1.12, 1.13, 1.14, 1.15, 1.16, 1.17, 1.18, 1.19