EXAMPLE 24 Constructing confidence intervals for the population variance and population standard deviation

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image electricmiles

The accompanying table shows the miles-per-gallon equivalent (MPGe) for fve electric cars, as reported by www.hybridcars.com in 2014. The normal probability plot in Figure 38 indicates that the data are normally distributed.

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Figure 8.38: FIGURE 38 Normal probability plot of miles-per-gallon equivalent for fve electric cars.
Electric Vehicle Mileage (MPGe)
Tesla Model S 89
Nissan Leaf 99
Ford Focus 105
Mitsubishi i-MiEV 112
Chevrolet Spark 119
  1. Find the critical values and for a confidence interval with a 95% confidence level.
  2. Construct and interpret a 95% confidence interval for the population variance of electric car MPG.
  3. Construct and interpret a 95% confidence interval for the population standard deviation of electric car MPG.

electricmiles

Solution

  1. There are electric cars in our sample, so the degrees of freedom equal .

    For a 95% confidence interval,

    From the table (Table E in the Appendix), therefore,

    Figures 39 through 41 show these results using Excel, Minitab, and JMP.

  2. Figure 42 shows the descriptive statistics for MPGe, as obtained by the TI-83/84. The sample standard deviation is .

    478

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    Figure 8.39: FIGURE 39 Excel results.
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    Figure 8.40: FIGURE 40 Minitab results.
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    Figure 8.41: FIGURE 41 JMP results.
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    Figure 8.42: FIGURE 42 TI-83/84 results.

    Thus, our 95% confidence interval for is given by

    We are 95% confident that the population variance lies between 48.17 and 1109.09 miles per gallon squared, that is, (MPG)2. (Recall that the variance is measured in units squared.) It is unclear what miles per gallon squared means, so we prefer to construct a confidence interval for the population standard deviation .

  3. Using the results from part (b),

We are 95% confident that the population standard deviation lies between 6.94 and 33.3 miles per gallon. Figure 43 shows the two confidence intervals obtained using Minitab.

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Figure 8.43: FIGURE 43 Minitab results showing the confidence intervals.

Figure 44 shows the confidence interval for obtained using JMP. We are interested in the bottom row, which has the confidence interval for the population standard deviation .

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Figure 8.44: FIGURE 44 JMP results showing the confidence interval for .

NOW YOU CAN DO

Exercises 17–32.