Chapter 6.3, Problem 32PS

Solution

a)

To find: The probability that the fire department takes at most 8 minutes.

The probability is 0.9772.

Given information:

  $\begin{array}{l}\text{Mean}\left(\mu \right)=6\\ \text{Standard deviation}\left(\sigma \right)=1\end{array}$

Formula used:

The formula of the z-score is:

  $\begin{array}{l}z=\frac{x-\mu }{\sigma }\\ \text{where}\\ x=\text{Row score}\\ \mu \text{=Population mean}\\ \sigma \text{=Population standard deviation}\end{array}$

Calculation:

Consider, X be the random variable that is distributed normally.

The probability is calculated as:

  $\begin{array}{c}P\left(x\le 8\right)=P\left(Z\le \frac{x-\mu }{\sigma }\right)\\ =P\left(Z\le \frac{8-6}{1}\right)\\ =P\left(Z\le 2\right)\\ =0.9772\end{array}$

The required probability is 0.9772.

b)

To find: The probability that the fire department takes between 4 and 7 minutes.

The probability is 0.8185.

Calculation:

Consider, X be the random variable that is distributed normally.

The probability is calculated as:

  $\begin{array}{c}P\left(4<x<7\right)=P\left(\frac{4-\mu }{\sigma }<Z<\frac{7-\mu }{\sigma }\right)\\ =P\left(\frac{4-6}{1}<Z<\frac{7-6}{1}\right)\\ =P\left(-2<Z<1\right)\\ =0.8185\end{array}$

The required probability is 0.8185.