Chapter 6, Problem 25E

a)

To determine

To draw the model which shows 68-95-99.7

Explanation of Solution

Given:

  $\mu$ = 24.8

  $\sigma$ = 6.2

Following is the model which shows 68-95-99.7:

  

b)

To determine

To find the interval in which 68% of autos to be found.

Answer to Problem 25E

The central 68% of autos can be found between 18.6 mpg and 31.0 mpg

Explanation of Solution

Given:

  $\mu$ = 24.8

  $\sigma$ = 6.2

According to rule of 68-95-99.7, the 68% of the values are within 1 standard deviation of the mean.

Therefore,

  $\begin{array}{l}\mu -\sigma =24.7-6.2=18.6\\ \mu +\sigma =24.7+6.2=31\end{array}$

Hence, the central 68% of autos can be found between 18.6 mpg and 31.0 mpg.

c)

To determine

To find the percent of autos should get more than 31 mpg.

Answer to Problem 25E

The 16% of autos should get more than 31 mpg.

Explanation of Solution

Given:

  $\mu$ = 24.8

  $\sigma$ = 6.2

The central 68% of autos can be found between 18.6 mpg and 31.0 mpg.

According to rule, 31 is 1 standard deviation above the mean. Therefore, when data is in total 100%, 32% of the data is then more than 1 standard deviation from the mean. We know, the normal curve is symmetric about the mean then 16% is more than 1 standard deviation below the mean and 16% is more than 1 standard deviation above the mean. So, we can say, 16% of autos should get more than 31 mpg.

d)

To determine

To find the percent of cars should bet between 31 and 37.2 mpg

Answer to Problem 25E

The 13.5% of cars should get between 31 and 37.2 mpg.

Explanation of Solution

Given:

  $\mu$ = 24.8

  $\sigma$ = 6.2

  

According to rule, 31 is 1 standard deviation above the mean. As per graph,37.2 is 2 standard deviation above the mean. Therefore, 95%-68% = 27% which is more than 1 standard deviation and less than 2 standard deviation of the mean. We know, the normal curve is symmetric about the mean then 13.5% is more than 1 standard deviation but less than 2 standard deviation below the mean and 13.5% is more than 1 standard deviation but less than 2 standard deviationabove the mean. So, we can say, 13.5% of cars should get between 31 and 37.2 mpg.

e)

To determine

To explain the gas mileage of the worst 2.5% of all cars.

Answer to Problem 25E

The worst 2.5% have a gas mileage less than 12.4 mpg.

Explanation of Solution

Given:

  $\mu$ = 24.8

  $\sigma$ = 6.2

  

According to rule, 95% of the data lies between 2 standard deviations from the mean. Since, total data is 100%, so 5% of the data is the more than 2 standard deviations from the mean. Therefore, as per symmetry, 2.5% is more than 2 standard deviations below the mean and 2.5% is more than 2 standard deviations above the mean.

  $\mu -2\sigma =24.7-2\times 6.2=12.4$

Therefore, the worst 2.5% have a gas mileage less than 12.4 mpg.