Question

find the indicated iq score. 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 indicated iq score, x, is (round to one decimal place as needed.)

Explanation:

Step1: Find the z - score

Since the area to the left of \(x\) is \(0.55\), we use the standard normal distribution table (or a calculator with a normal - distribution function). The \(z\) - score corresponding to an area of \(A = 0.55\) is \(z\approx0.13\) (using a standard normal table or a calculator like the invNorm function on a TI - 84: invNorm\((0.55,0,1)\)).

Step2: Use the z - score formula

The z - score formula is \(z=\frac{x-\mu}{\sigma}\), where \(\mu = 100\) (mean) and \(\sigma=15\) (standard deviation).
We want to solve for \(x\). Rearranging the formula gives \(x=\mu + z\sigma\).
Substitute \(\mu = 100\), \(z = 0.13\), and \(\sigma = 15\) into the formula:
\(x=100+0.13\times15\)

Step3: Calculate the value of \(x\)

First, calculate \(0.13\times15 = 1.95\).
Then, \(x=100 + 1.95=101.95\approx102.0\)

Answer:

\(102.0\)