Question

Environmental Protection Agency (EPA) fuel economy estimates for automobile models tested recently predicted a mean of \(24.8 \mathrm{mpg}\) and a standard deviation of \(6.2 \mathrm{mpg}\) for highway driving. Assume that a Normal model can be applied. a) Draw the model for auto fuel economy. Clearly label it, showing what the \(68-95-99.7\) Rule predicts. b) In what interval would you expect the central \(68 \%\) of autos to be found? c) About what percent of autos should get more than \(31 \mathrm{mpg}\) ? d) About what percent of cars should get between 31 and \(37.2 \mathrm{mpg}\) ? e) Describe the gas mileage of the worst \(2.5 \%\) of all cars.

Short answer

a) Normal model, mean 24.8 mpg, sd 6.2 mpg. b) [18.6, 31.0] mpg. c) 16%. d) 13.5%. e) Below 12.4 mpg.

Step-by-step solution

Identifying Mean and Standard Deviation

The problem gives us a mean (\(\mu\)) of \(24.8\) mpg and a standard deviation (\(\sigma\)) of \(6.2\) mpg for highway driving.

Applying the 68-95-99.7 Rule

This rule tells us that for a normal distribution: \(68\%\) of data falls within one standard deviation of the mean, \(95\%\) within two standard deviations, and \(99.7\%\) within three standard deviations. We use this information to draw the distribution.

Calculating Central 68% Interval

The central \(68\%\) of autos are expected to be within one standard deviation of the mean. Thus, calculate \([\mu - \sigma, \mu + \sigma]\): \(24.8 - 6.2 = 18.6\) and \(24.8 + 6.2 = 31.0\). Therefore, the interval is \([18.6, 31.0]\) mpg.

Calculating the Percentage Greater than 31 mpg

31 mpg represents the upper bound of one standard deviation above the mean. Since \(68\%\) is within one standard deviation, \(32\%\) is outside, split evenly as \(16\%\) is above and below. Therefore, about \(16\%\) of cars should get more than \(31\) mpg.

Calculating Percentage between 31 and 37.2 mpg

Calculate the range from one to two standard deviations above the mean. One standard deviation (31 mpg) is the beginning and two standard deviations (\(24.8 + 2\times 6.2 = 37.2\) mpg) is the end. With \(68\%\) within one standard deviation and \(95\%\) within two, \(27\%\) lies between one and two standard deviations: \((95\% - 68\%)/2 = 13.5\%\). Therefore, \(13.5\%\) of cars should get between 31 and 37.2 mpg.

Determining Gas Mileage at Worst 2.5%

For the worst \(2.5\%\), we look beyond two standard deviations below the mean, since \(95\%\) of data is within two deviations. Thus, the worst \(2.5\%\) begins below \(24.8 - 2\times 6.2 = 12.4\) mpg.

Key concepts

68-95-99.7 Rule

The 68-95-99.7 Rule, also known as the Empirical Rule, is a fundamental concept used in statistics to understand data that follows a Normal Distribution. When data is Normally distributed, like the EPA fuel economy estimates for highway driving, the rule describes how data is distributed around the mean:

• 68% of the data falls within one standard deviation ( \( \pm 1\sigma \) ) of the mean.
• 95% of the data is within two standard deviations ( \( \pm 2\sigma \) ) of the mean.
• 99.7% of the data falls within three standard deviations ( \( \pm 3\sigma \) ) of the mean.

This rule helps in visualizing the distribution. For the EPA’s car models:

• 68% of the cars should have fuel efficiency between \( 18.6 \, \text{mpg} \) and \( 31.0 \, \text{mpg} \) .
• 95% should be between \( 12.4 \, \text{mpg} \) and \( 37.2 \, \text{mpg} \) .
• Nearly all cars, 99.7%, should fall within the range of \( 6.2 \, \text{mpg} \) to \( 43.4 \, \text{mpg} \) .

This rule provides a quick way to determine where a particular segment of data, like fuel economy, may lie in a given distribution.

Mean and Standard Deviation

The mean and standard deviation are key parameters in statistics that describe a normal distribution. In the context of fuel economy statistics from the EPA for highway driving, these parameters are insightful:

• The mean ( \( \mu \) ) is the average score or central point of a dataset, which in this case is \( 24.8 \, \text{mpg} \) . It summarizes the central tendency of the data, providing a location around which the rest of the data is distributed.
• The standard deviation ( \( \sigma \) ) is \( 6.2 \, \text{mpg} \) . It measures the amount of variation or dispersion in a set of values. A smaller standard deviation indicates that the values tend to be close to the mean, whereas a larger standard deviation indicates that the values are more spread out.

Understanding these can help in predicting and interpreting the variation in car fuel economy statistics, making it easier for users to estimate how widely each car model’s fuel efficiency may vary from the average.

Probability Intervals

Probability intervals describe the range of values in a dataset where certain proportions of the data lie. This is crucial when using the 68-95-99.7 Rule to address questions about the probabilities:

• The central 68% probability interval for car fuel efficiencies is within the mean plus or minus one standard deviation. So, for cars, this interval is between \( 18.6 \, \text{mpg} \) and \( 31.0 \, \text{mpg} \) .
• To find the percentage of cars exceeding a certain mpg, like \( 31 \, \text{mpg} \) , subtract the interval from 100%. Since 68% of cars lie below 31 mpg (upper bound of one standard deviation), 16% exceed this mark.
• For cars with mileage between \( 31 \, \text{mpg} \) and \( 37.2 \, \text{mpg} \) , which corresponds to 1 to 2 standard deviations above the mean, \( 13.5\% \) of cars fall into this bracket.
Understanding probability intervals empowers you to predict and interpret car fuel economy based on statistical data models.

Fuel Economy Statistics

Fuel economy statistics are important for understanding the performance and efficiency of automobile models. These statistics generally follow a normal distribution, as indicated by the EPA's estimates. The given mean of \( 24.8 \, \text{mpg} \) and the standard deviation of \( 6.2 \, \text{mpg} \) provide insights into the expected performance of cars:

• On average, cars are expected to deliver around \( 24.8 \, \text{mpg} \) on highways.
• The lower 2.5% of performers, essentially the least efficient cars, have fuel economies below \( 12.4 \, \text{mpg} \) .
• These statistics help manufacturers improve fuel efficiency and guide consumers in understanding what "standard" fuel economy looks like.

Grasping fuel economy statistics through normal distribution principles aids individuals in making informed decisions about vehicle purchases or fuel economy improvements.