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. 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 \(\mu = 100\) (mean) and \(\sigma=15\) (standard deviation).
For \(x = 104\):
\(z_1=\frac{104 - 100}{15}=\frac{4}{15}\approx0.27\)
For \(x = 126\):
\(z_2=\frac{126-100}{15}=\frac{26}{15}\approx1.73\)

Step2: Find the area using the standard normal table

The area to the left of \(z_1 = 0.27\) is \(P(Z<0.27)=0.6064\) (from the standard normal table).
The area to the left of \(z_2 = 1.73\) is \(P(Z < 1.73)=0.9582\) (from the standard normal table).

Step3: Calculate the area of the shaded region

The area of the shaded region is \(P(0.27<Z<1.73)=P(Z < 1.73)-P(Z < 0.27)\)
\(P(0.27<Z<1.73)=0.9582 - 0.6064=0.3518\)

Answer:

\(0.3518\)