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 \(z\) is the z - score, \(x\) is the data item, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.

Step2: Substitute the given values into the formula

We are given that \(z = 1.5\), \(\mu=20\), and \(\sigma = 5\). Substituting these into the formula \(1.5=\frac{x - 20}{5}\).

Step3: Solve for \(x\)

Multiply both sides of the equation by \(5\): \(1.5\times5=x - 20\). So, \(7.5=x - 20\).
Then add \(20\) to both sides: \(x=20 + 7.5\).

Answer:

\(x = 27.5\)