Chapter 11.1, Problem 13E

To determine

To calculate:

The probability value of $x$ for the following distribution:

Between $29$ and $37$

Answer to Problem 13E

The probability of $x$ between $29$ and $37$ is $0.6826$ .

Explanation of Solution

Given information:

  $\begin{array}{l}\mu =33\\ \sigma =4\end{array}$

Formula used:

  $Z-$ Score is given by the following expression:

  $Z=\frac{x-\mu }{\sigma }$

Where,

Standard deviation: $\sigma$

Mean: $\mu$

Calculation:

To determine the probability of a randomly selected $x$ -value from the standard normal distribution using the given information for $x$ in between $29$ and $37$

To find the probability when $29\le x\le 37$ , use the standard normal distribution as shown below:

  

Here, need to use the $Z-$ score formula to get probability value:

  $\begin{array}{l}P\left(29\le x\le 37\right)=P\left(\frac{29-33}{4}\le \frac{x-\mu }{\sigma }\le \frac{37-33}{4}\right)\\ P\left(29\le x\le 37\right)=P\left(-1\le \frac{x-\mu }{\sigma }\le 1\right)\end{array}$

Use the standard normal table:

  $P\left(29\le x\le 37\right)=0.6826$

The probability that $P\left(29\le x\le 37\right)=0.6826$ is equal to the shaded area under the curve as: