Chapter 13.2, Problem 55E

To determine

To find: The probability for a randomly selected $x$ value from the distribution.

Answer to Problem 55E

The probability is approximately $0\cdot 657$ .

Explanation of Solution

Given:

The mean of a normal distribution is $64$ and standard deviation is $7$

Calculation:

From the given data, $\mu =64$ and $\sigma =7$ .

To find the probability for randomly selected $x$ value from the distribution in the given interval, first, the z statistic for $x=60$ and $x=75$ needs to be determined as shown below:

  $\begin{array}{c}z=\frac{x-\mu }{\sigma }\\ =\frac{60-64}{7}\\ =-0\cdot 57\end{array}$

Further simplified as:

  $\begin{array}{c}z=\frac{x-\mu }{\sigma }\\ =\frac{75-64}{7}\\ =1\cdot 57\end{array}$

So, $P\left(60\le x\le 75\right)=P\left(-0\cdot 57\le z\le 1\cdot 57\right)$ . Solve using the standard normal table as shown below:

  $\begin{array}{c}P\left(60\le x\le 75\right)=P\left(-0\cdot 57\le z\le 1\cdot 57\right)\\ =P\left(z\le 1\cdot 57\right)-P\left(z\le -0\cdot 57\right)\\ \approx 0\cdot 941-0\cdot 284\left[\text{From the table}\right]\\ =0\cdot 657\end{array}$

So, the probability is approximately $0\cdot 657$ .