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
120
The area of the shaded region is
(Round to four decimal places as needed.)

Answer 1

Answer:

The area of the shaded region is 0.9082.

Step-by-step explanation:

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value, which is the area of the shaded region, is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

$\mu = 100, \sigma = 15, X = 120$

Area of the shaded region:

pvalue of Z

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{120 - 100}{15}$

$Z = 1.33$

$Z = 1.33$ has a pvalue of 0.9082

So

The area of the shaded region is 0.9082.