Define the element for the example? What are the variables?

Based on the table(s) or figure(S) from the example, please determine whether the data set is quantitative data or qualitative (i.e. categorical) data. Please further define if it is ratio, interval, ordinal, or nominative variable.

Based on the table(s) or figure(S) from the example, please determine whether the data set is cross-sectional data or time-series data.

Based on the example, please determine where the data comes from (i.e. “existing sources” or “experimental and observational studies”).

Based on the example, please determine if the example is a census study. If not, what are the population and sample size for the example?

Based on the example, is it the case of descriptive statistics or a study of statistical inference?

EXAMPLE 1.4 The Car Mileage Case: Estimating Mileage

Part 1: Auto Fuel Economy Personal budgets, national energy security, and the global environment are all affected by our gasoline consumption. Hybrid and electric cars are a vital part of a long-term strategy to reduce our nation’s gasoline consumption. However, until use of these cars is more widespread and affordable, the most effective way to conserve gasoline is to design gasoline powered cars that are more fuel efficient. In the short term, “that will give you the biggest bang for your buck,” says David Friedman, research director of the Union of Concerned Scientists’ Clean Vehicle Program.

In this case study we consider a tax credit offered by the federal government to automakers for improving the fuel economy of gasoline-powered midsize cars. According to The Fuel Economy Guide—2015 Model Year, virtually every gasoline-powered midsize car equipped with an automatic transmission and a six-cylinder engine has an EPA combined city and 15

TABLE 1.7    A Sample of 50 Mileages  GasMiles

30.8	30.8	32.1	32.3	32.7
31.7	30.4	31.4	32.7	31.4
30.1	32.5	30.8	31.2	31.8
31.6	30.3	32.8	30.7	31.9
32.1	31.3	31.9	31.7	33.0
33.3	32.1	31.4	31.4	31.5
31.3	32.5	32.4	32.2	31.6
31.0	31.8	31.0	31.5	30.6
32.0	30.5	29.8	31.7	32.3
32.4	30.5	31.1	30.7	31.4
Note: Time order is given by reading down the columns from left to right.

FIGURE 1.5   A Time Series Plot of the 50 Mileages

highway mileage estimate of 26 miles per gallon (mpg) of less.⁷ As a matter of fact, when this book was written, the mileage leader in this category was the Honda Accord, which registered a combined city and highway mileage of 26 mpg. While fuel economy has seen improvement in almost all car categories, the EPA has concluded that an additional 5 mpg increase in fuel economy is significant and feasible.⁸ Therefore, suppose that the government has decided to offer the tax credit to any automaker selling a midsize model with an automatic transmission and a six-cylinder engine that achieves an EPA combined city and highway mileage estimate of at least 31 mpg.

Part 2: Sampling a Process Consider an automaker that has recently introduced a new midsize model with an automatic transmission and a six-cylinder engine and wishes to demonstrate that this new model qualifies for the tax credit. In order to study the population of all cars of this type that will or could potentially be produced, the automaker will choose a sample of 50 of these cars. The manufacturer’s production operation runs 8-hour shifts, with 100 midsize cars produced on each shift. When the production process has been fine-tuned and all start-up problems have been identified and corrected, the automaker will select one car at random from each of 50 consecutive production shifts. Once selected, each car is to be subjected to an EPA test that determines the EPA combined city and highway mileage of the car.

To randomly select a car from a particular production shift, we number the 100 cars produced on the shift from 00 to 99 and use a random number table or a computer software package to obtain a random number between 00 and 99. For example, starting in the upper left-hand corner of Table 1.4(a) and proceeding down the two leftmost columns, we see that the first three random numbers between 00 and 99 are 33, 3, and 92. This implies that we would select car 33 from the first production shift, car 3 from the second production shift, car 92 from the third production shift, and so forth. Moreover, because a new group of 100 cars is produced on each production shift, repeated random numbers would not be discarded. For example, if the 15th and 29th random numbers are both 7, we would select the 7th car from the 15th production shift and the 7th car from the 29th production shift.

Part 3: The Sample and Inference Suppose that when the 50 cars are selected and tested, the sample of 50 EPA combined mileages shown in Table 1.7 is obtained. A time series plot of the mileages is given in Figure 1.5. Examining this plot, we see that, although the mileages vary over time, they do not seem to vary in any unusual way. For example, the mileages do not tend to either decrease or increase (as did the basic cable rates in Figure 1.3) over time. This intuitively verifies that the midsize car manufacturing process is producing consistent car mileages over time, and thus we can regard the 50 mileages as an approximately random sample that can be used to make statistical inferences about the population of all 16 possible midsize car mileages.⁹ Therefore, because the 50 mileages vary from a minimum of 29.8 mpg to a maximum of 33.3 mpg, we might conclude that most midsize cars produced by the manufacturing process will obtain between 29.8 mpg and 33.3 mpg.

We next suppose that in order to offer its tax credit, the federal government has decided to define the “typical” EPA combined city and highway mileage for a car model as the mean of the population of EPA combined mileages that would be obtained by all cars of this type. Therefore, the government will offer its tax credit to any automaker selling a midsize model equipped with an automatic transmission and a six-cylinder engine that achieves a mean EPA combined mileage of at least 31 mpg. As we will see in Chapter 3, the mean of a population of measurements is the average of the population of measurements. More precisely, the population mean is calculated by adding together the population measurements and then dividing the resulting sum by the number of population measurements. Because it is not feasible to test every new midsize car that will or could potentially be produced, we cannot obtain an EPA combined mileage for every car and thus we cannot calculate the population mean mileage. However, we can estimate the population mean mileage by using the sample mean mileage. To calculate the mean of the sample of 50 EPA combined mileages in Table 1.7, we add together the 50 mileages in Table 1.7 and divide the resulting sum by 50. The sum of the 50 mileages can be calculated to be

30.8 + 31.7 + . . . + 31.4 = 1578

and thus the sample mean mileage is 1578/50 5 31.56. This sample mean mileage says that we estimate that the mean mileage that would be obtained by all of the new midsize cars that will or could potentially be produced this year is 31.56 mpg. Unless we are extremely lucky, however, there will be sampling error. That is, the point estimate of 31.56 mpg, which is the average of the sample of 50 randomly selected mileages, will probably not exactly equal the population mean, which is the average mileage that would be obtained by all cars. Therefore, although the estimate 31.56 provides some evidence that the population mean is at least 31 and thus that the automaker should get the tax credit, it does not provide definitive evidence. To obtain more definitive evidence, we employ what is called statistical modeling. We introduce this concept in the next subsection.

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