Chapter 3.2, Problem 80E

To determine

To find:The highest gas mileage that fall in the top 10% of all vehicles

Answer to Problem 80E

The highest gas mileage that fall in the top 10% of all vehicles is 24.204mpg

Explanation of Solution

Given:

The mean gas mileage, $\mu =18.7\text{mpg}$

The standard deviation of gas mileage, $\sigma =4.3\text{mpg}$

Calculation:

If the random variable X denotes the number of vehicles that have gas mileage, then we are given that $X\sim N\left(\mu ,{\sigma }^2\right)$

Where $\mu =18.7\text{mpg}$ and $\sigma =4.3\text{mpg}$

It is supposed to find the highest gas mileage that falls in the top 10% of all vehicles

In other words, find z' that corresponds to the top 10%

That is,$\begin{array}{l}P\left(z>z^1\right)=10\%\\ \Rightarrow P\left(z>z^1\right)=0.90\end{array}$

From standard normal tables, the value of z corresponding to 0.90 cumulative area is 1.28.

That is,$\begin{array}{l}{z}^1=1.28\\ \Rightarrow \frac{x-\mu }{\sigma }=1.28\\ \Rightarrow \frac{x-18.7}{4.3}=1.28\\ \Rightarrow x=18.7+1.28\times 4.3\\ =24.204\end{array}$

Conclusion: Hence, the highest gas mileage highest gas mileage that fall in the top 10% of all vehicles is 24.204mpg