Chapter 11, Problem 7CA

To determine

To find: The percent of the area and interval under the normal curve.

Answer to Problem 7CA

  $83.51\%$ of the area under normal curve lies in the shaded region .

Another interval with same percent of area is $43$ to $71$ .

Explanation of Solution

Given:

  $\mu =63and\sigma =8$

Calculation:

In this question, $\mu =63and\sigma =8.$

Find the percent of area under shaded region, which is the probability $p\left(55\le x\le 83\right).$

So, for that, find the $z-score$ corresponding to an $x-valueof55and83.$

  $\begin{array}{l}z=\frac{x-\mu }{\sigma }\\ =\frac{55-63}{8}\\ =-1\end{array}$

  $\begin{array}{l}z=\frac{x-\mu }{\sigma }\\ =\frac{83-63}{8}\\ =2.5\end{array}$

Thus the required probability is given by the value of $p\left(-1\le z\le 2.5\right)$

Now, use the standard normal table to find $p\left(z\le -1\right)andp\left(z\le 2.5\right).$

  

Thus,

  $\begin{array}{l}p\left(z\le -1\right)=0.1587\\ p\left(z\le 2.5\right)=0.9938\end{array}$

And, hence

  $\begin{array}{l}p\left(-1\le z\le 2.5\right)=p\left(z\le 2.5\right)-p\left(z\le -1\right)\\ =0.9938-0.1587\\ =0.8351\end{array}$

The percentage is found by turning the above decimal into percent.

Hence, $83.51\%$ of the area under normal curve lies in the shaded region .

Another interval with same percent of area is $43$ to $71$ .

Conclusion:

Hence, $83.51\%$ of the area under normal curve lies in the shaded region .

Another interval with same percent of area is $43$ to $71$ .