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 - score

The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\), where \(x = 75\), \(\mu=100\), and \(\sigma = 15\).

$$z=\frac{75 - 100}{15}=\frac{-25}{15}\approx - 1.67$$

Step2: Find the area using the standard normal table

We want to find \(P(X<75)\), which is equivalent to \(P(Z<-1.67)\) in the standard normal distribution.
Looking up the value of \(z=-1.67\) in the standard normal table (the cumulative - distribution function for the standard normal distribution \(\varPhi(z)\)).
From the standard normal table, \(P(Z < - 1.67)=0.0475\)

Answer:

\(0.0475\)