Chapter 10, Problem 4PT3

To determine

To find: The interval that is expected to cover 81.5% of all GPAs for all the college students

Answer to Problem 4PT3

The interval is (1.9, 3.7)

Explanation of Solution

Given:

  $\begin{array}{l}\text{Population mean}\left(\mu \right)=2.5\\ \text{Population standard deviation}\left(\sigma \right)=0.6\end{array}$

Formula used:

The formula to compute the z score is:

  $\begin{array}{l}z=\frac{x-\mu }{\sigma }\\ x=\text{ Score}\end{array}$

Calculation:

The probability is calculated as:

  $\begin{array}{c}P\left(1.9<X<3.7\right)=P\left(\frac{1.9-\mu }{\sigma }<\frac{x-\mu }{\sigma }<\frac{3.7-\mu }{\sigma }\right)\\ =P\left(\frac{1.9-2.5}{0.6}<\frac{x-\mu }{\sigma }<\frac{3.7-2.5}{0.6}\right)\\ =P\left(-1<Z<2\right)\\ =0.8185\end{array}$

Thus, the interval (1.9, 3.7) has the probability is 0.8185.

Hence, the correct option is (c).