Question

environmental protection agency (epa) fuel economy estimates for automobile models tested recently predicted a mean of 24.81 mpg and a standard deviation of 6.22 mpg for highway driving. assume that a normal model can be applied. use the 68 - 95 - 99.7 rule to complete parts a through e.
a) draw the model for auto fuel economy. clearly label it, showing what the 68 - 95 - 99.7 rule predicts.

Explanation:

Step1: Recall the 68 - 95 - 99.7 Rule

The 68 - 95 - 99.7 Rule for a normal distribution states that about 68% of the data lies within 1 standard - deviation of the mean, about 95% lies within 2 standard - deviations of the mean, and about 99.7% lies within 3 standard - deviations of the mean. Given mean $\mu = 24.81$ mpg and standard deviation $\sigma=6.22$ mpg.

Step2: Calculate the values for 1, 2 and 3 standard - deviations from the mean

For 1 standard - deviation:
Lower limit: $\mu-\sigma=24.81 - 6.22=18.59$
Upper limit: $\mu+\sigma=24.81 + 6.22=31.03$
For 2 standard - deviations:
Lower limit: $\mu - 2\sigma=24.81-2\times6.22=24.81 - 12.44 = 12.37$
Upper limit: $\mu + 2\sigma=24.81+2\times6.22=24.81 + 12.44 = 37.25$
For 3 standard - deviations:
Lower limit: $\mu-3\sigma=24.81-3\times6.22=24.81 - 18.66 = 6.15$
Upper limit: $\mu + 3\sigma=24.81+3\times6.22=24.81+18.66 = 43.47$

Answer:

The correct normal - distribution model should have the mean labeled as 24.81, and the intervals for 68% (from 18.59 to 31.03), 95% (from 12.37 to 37.25) and 99.7% (from 6.15 to 43.47) labeled correctly. Without seeing the exact details of each option, but based on the calculations above, the model with these correct values and the 68 - 95 - 99.7 labels in the correct places is the right one. If one of the options has the mean at 24.81 and the lower and upper bounds for 1, 2 and 3 standard - deviations as calculated above, that is the correct option.