Question

fuel economy estimates for automobiles built one year predicted a mean of 25.8 mpg and a standard deviation of 6.2 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
b
c
b) in what interval would you expect the central 99.7% of autos to be found?
using the 68 - 95 - 99.7 rule, the central 99.7% of autos can be expected to be found in the interval from 7.2 to 44.4 mpg.
(do not round. type integers or decimals.)
c) about what percent of autos should get more than 32 mpg?
using the 68 - 95 - 99.7 rule, about 16% of autos should get more than 32 mpg.
(do not round. type an integer or a decimal.)
d) about what percent of autos should get between 32 and 38.2 mpg?
using the 68 - 95 - 99.7 rule, about □% of autos should get between 32 and 38.2 mpg.
(do not round. type an integer or a decimal.)

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 \( \mu\pm\sigma \), 95% within \( \mu\pm2\sigma \), and 99.7% within \( \mu\pm3\sigma \), where \( \mu \) is the mean and \( \sigma \) is the standard deviation.

Step2: Calculate the values for each part

• Part (b):

The central 99.7% of a normal distribution lies within \( \mu - 3\sigma \) and \( \mu+3\sigma \). Given \( \mu = 25.8\) and \( \sigma=6.2\), then \( \mu - 3\sigma=25.8-3\times6.2 = 25.8 - 18.6=7.2\) and \( \mu + 3\sigma=25.8+3\times6.2=25.8 + 18.6 = 44.4\).

• Part (c):

First, find the \( z - \)score. The formula for the \( z - \)score is \( z=\frac{x-\mu}{\sigma}\). For \( x = 32\), \( z=\frac{32 - 25.8}{6.2}=\frac{6.2}{6.2}=1\).
Using the standard normal distribution table or the 68 - 95 - 99.7 rule (since the total area under the normal curve is 1), the area to the right of \( z = 1\) is \(\frac{1 - 0.68}{2}=0.16\) (because 68% of the data is within \( z=- 1\) and \( z = 1\)). So about 16% of autos should get more than 32 mpg.

• Part (d):

For \( x_1 = 32\) and \( x_2 = 38.2\), \( z_1=\frac{32 - 25.8}{6.2}=1\) and \( z_2=\frac{38.2 - 25.8}{6.2}=\frac{12.4}{6.2}=2\).
The area between \( z = 1\) and \( z = 2\) is \(\frac{0.95 - 0.68}{2}=0.135\). So about 13.5% of autos should get between 32 and 38.2 mpg.

Answer:

b) The interval is from \(7.2\) to \(44.4\) mpg.
c) About \(16\%\) of autos should get more than 32 mpg.
d) About \(13.5\%\) of autos should get between 32 and 38.2 mpg.