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 \(\mu = 100\), \(\sigma=15\).
For \(x = 108\), \(z_1=\frac{108 - 100}{15}=\frac{8}{15}\approx0.53\).
For \(x = 126\), \(z_2=\frac{126-100}{15}=\frac{26}{15}\approx1.73\).

Step2: Use the standard normal distribution table

We know that the area under the standard normal curve \(P(Z < z_1)\) and \(P(Z < z_2)\) can be found from the standard - normal table.
From the standard - normal table, \(P(Z<0.53)=0.7019\) and \(P(Z < 1.73)=0.9582\).

Step3: Calculate the area of the shaded region

The area of the shaded region is \(P(0.53<Z<1.73)=P(Z < 1.73)-P(Z < 0.53)\).
Substitute the values: \(0.9582 - 0.7019=0.2563\).

Answer:

\(0.2563\)