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 \(\mu = 100\) (mean) and \(\sigma=15\) (standard deviation).
For \(x = 90\):
\(z_1=\frac{90 - 100}{15}=\frac{- 10}{15}\approx - 0.67\)
For \(x = 120\):
\(z_2=\frac{120 - 100}{15}=\frac{20}{15}\approx1.33\)

Step2: Find the cumulative probabilities

Using the standard normal distribution table (z - table):
The cumulative probability for \(z=-0.67\) is \(P(Z < - 0.67)=0.2514\)
The cumulative probability for \(z = 1.33\) is \(P(Z < 1.33)=0.9082\)

Step3: Calculate the area between the two z - scores

The area between \(z_1\) and \(z_2\) is \(P(-0.67<Z<1.33)=P(Z < 1.33)-P(Z < - 0.67)\)
\(P(-0.67 < Z < 1.33)=0.9082-0.2514 = 0.6568\)

Answer:

\(0.6568\)