Chapter 11, Problem 4CT

To determine

To calculate:

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

Between $67$ and $77$

Answer to Problem 4CT

The probability of $x$ between $67$ and $77$ is $68.26\%$ .

Explanation of Solution

Given information:

  $\begin{array}{l}\mu =72\\ \sigma =5\end{array}$

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 $67$ and $77$

To find the probability when $67\le x\le 77$ , use the standard normal distribution as shown below:

  

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

  $\begin{array}{l}P\left(67\le x\le 77\right)=P\left(\frac{67-72}{5}\le \frac{x-\mu }{\sigma }\le \frac{77-72}{5}\right)\\ P\left(67\le x\le 77\right)=P\left(-1\le \frac{x-\mu }{\sigma }\le 1\right)\end{array}$

Use the standard normal table:

  $P\left(67\le x\le 77\right)=0.6826=68.26\%$

The probability that $P\left(67\le x\le 77\right)=0.6826$ is equal to the shaded area under the curve as: