Environmental Protection Agency (EPA) fuel economy estimates...

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 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.

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 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.

Key Concepts

Probability Intervals and Percentiles

This concept involves using the normal distribution and Z-scores to find the probability that a value falls within a specified interval or exceeds a certain threshold. It is essential for interpreting the likelihood of experiencing a value that is significantly above or below the mean.

Empirical Rule (68-95-99.7 Rule)

This rule states that in a normal distribution approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. This concept helps quickly estimate the proportion of observations within certain intervals.

Z-Score

A Z-score measures how many standard deviations an observation is from the mean. It is used to standardize values from any normal distribution so that probabilities and percentiles can be computed using the standard normal distribution.

Mean and Standard Deviation

The mean provides the central value around which data is centered, while the standard deviation measures the spread or dispersion of the data points around the mean. These parameters are crucial in determining the intervals where most data values lie within a normal distribution.

Normal Distribution

This is a continuous probability distribution that is symmetric about its mean, resembling a bell-shaped curve. It is completely defined by its mean and standard deviation, and many real-world phenomena, like fuel economy in automobile models, are approximated using this model.

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