Question

find the area of the shaded region. the graph to the right depicts iq scores of adults, and those scores are normally distributed with a mean of 100 and a standard deviation of 15. click to view page 1 of the table. click to view page 2 of the table. the area of the shaded region is (round to four decimal places as needed.)

Explanation:

Step1: Calculate the z - scores

The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\), where \(\mu = 100\), \(\sigma=15\).
For \(x = 80\):
\(z_1=\frac{80 - 100}{15}=\frac{- 20}{15}\approx - 1.33\)
For \(x = 110\):
\(z_2=\frac{110 - 100}{15}=\frac{10}{15}\approx0.67\)

Step2: Use the standard normal table

The area to the left of \(z_1=-1.33\) is \(P(Z\lt - 1.33)=0.0918\) (from the standard normal table).
The area to the left of \(z_2 = 0.67\) is \(P(Z\lt0.67)=0.7486\) (from the standard normal table).

Step3: Calculate the area of the shaded region

The area of the shaded region is \(P(-1.33\lt Z\lt0.67)=P(Z\lt0.67)-P(Z\lt - 1.33)\)
\(=0.7486-0.0918 = 0.6568\)

Answer:

\(0.6568\)