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 - score

The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\), where \(x = 97\), \(\mu=100\), and \(\sigma = 15\).

$$z=\frac{97 - 100}{15}=\frac{-3}{15}=- 0.2$$

Step2: Find the area using the standard normal distribution table

We want to find \(P(X>97)\), which is equivalent to \(P(Z>-0.2)\) in the standard normal distribution (\(Z\sim N(0,1)\)).
Since \(P(Z > z)=1 - P(Z\leq z)\), and from the standard normal distribution table, \(P(Z\leq - 0.2)=0.4207\)

$$P(Z>-0.2)=1 - 0.4207 = 0.5793$$

Answer:

\(0.5793\)