Question

z - score of 2.5 on an iq test is considered a genius (mean = 100, sd = 16). what score makes someone a genius? hint: solve for x

Explanation:

Step1: Recall the z - score formula

The z - score formula is \(z=\frac{x - \mu}{\sigma}\), where \(z\) is the z - score, \(x\) is the raw score, \(\mu\) is the mean, and \(\sigma\) is the standard deviation. We are given \(z = 2.5\), \(\mu=100\), and \(\sigma = 16\), and we need to solve for \(x\).

Step2: Rearrange the formula to solve for \(x\)

Starting with \(z=\frac{x - \mu}{\sigma}\), we can multiply both sides by \(\sigma\) to get \(z\sigma=x-\mu\). Then, add \(\mu\) to both sides: \(x=\mu + z\sigma\).

Step3: Substitute the given values into the formula

Substitute \(\mu = 100\), \(z = 2.5\), and \(\sigma=16\) into \(x=\mu + z\sigma\). So \(x=100+2.5\times16\).
First, calculate \(2.5\times16 = 40\). Then, \(x=100 + 40\).

Answer:

\(140\)