Chapter 6.3, Problem 17E

Solution

To find: the probability that a randomly selected x-value from the distribution is between 74 and 94.

  $C.\text{ 0}\text{.95}$

Given information:

A normal distribution has a mean of 84 and a standard deviation of 5.

The given options are:

1. 0.475
2. 0.68
3. 0.95
4. 0.997

Concept Used:

Areas under a Normal Curve

A normal distribution with mean $\overline{x}$ and standard deviation $\sigma$ has the following properties:

• The total area under the related normal curve is 1.
• About 68% of the area lies within 1 standard deviation of the mean.
• About 95% of the area lies within 2 standard deviation of the mean.
• About 99.7% of the area lies within 3 standard deviation of the mean.

  

Explanation:

The normal distribution curve for the given distribution with mean $\overline{x}=84$ and standard deviation $\sigma =5$ is:

  

The required interval lies between 74 and 94 represents the shaded area under the normal curve (it is the area lies between $\overline{x}-2\sigma$ and $\overline{x}+2\sigma$ ) as shown below:

  

So, the probability that a randomly selected x-value from the distribution is in the given interval is:

  $\begin{array}{c}P\left(74\le x\le 94\right)=0.135+0.34+0.34+0.135\\ =0.95\end{array}$

Correct answer is C.