Question

act scores are normally distributed with a mean of 20 and standard deviation 5. find the data item in this distribution that corresponds to a z - score of 1.5.

Explanation:

Step1: Recall the z - score formula

The z - score formula is \(z=\frac{x-\mu}{\sigma}\), where \(x\) is the data item, \(\mu\) is the mean, and \(\sigma\) is the standard deviation. We need to solve for \(x\).

Step2: Rearrange the formula for \(x\)

Multiply both sides of \(z = \frac{x-\mu}{\sigma}\) by \(\sigma\): \(z\sigma=x - \mu\). Then add \(\mu\) to both sides: \(x=\mu+z\sigma\).

Step3: Substitute the given values

Given \(\mu = 20\), \(z = 1.5\), and \(\sigma=5\). Substitute into \(x=\mu+z\sigma\): \(x = 20+1.5\times5\).

Step4: Calculate the value of \(x\)

First, calculate \(1.5\times5 = 7.5\). Then \(x=20 + 7.5=27.5\).

Answer:

\(27.5\)