Question

the average miles per gallon of a particular automobile model are approximately normally distributed with a given mean $mu = 43.8$ miles per gallon and standard deviation $sigma = 5.1$ miles per gallon. what percentage of the automobiles have an average miles per gallon between 38.7 miles per gallon and 48.9 miles per gallon?
68%
75%
95%
100%

Explanation:

Step1: Calculate z - scores

The z - score formula is $z=\frac{x-\mu}{\sigma}$.
For $x = 38.7$, $z_1=\frac{38.7 - 43.8}{5.1}=\frac{-5.1}{5.1}=- 1$.
For $x = 48.9$, $z_2=\frac{48.9 - 43.8}{5.1}=\frac{5.1}{5.1}=1$.

Step2: Use the empirical rule

The empirical rule for a normal distribution states that approximately 68% of the data lies within 1 standard - deviation of the mean, that is between $z=-1$ and $z = 1$.

Answer:

68%