Question

a set of data items is normally distributed with a mean of 300 and a standard deviation of 30. find the data item in this distribution that corresponds to the given z - score. z = 2.5. the data item that corresponds to z = 2.5 is □ (type an integer or a decimal)

Explanation:

Step1: Recall the z - score formula

The z - score formula is \(z=\frac{x - \mu}{\sigma}\), where \(x\) is the data value, \(\mu\) is the mean, and \(\sigma\) is the standard deviation. We want to solve for \(x\) when \(z = 2.5\), \(\mu=300\), and \(\sigma = 30\).

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

From \(z=\frac{x - \mu}{\sigma}\), we can multiply both sides by \(\sigma\) and then add \(\mu\) to get \(x=\mu+z\sigma\).

Step3: Substitute the given values into the formula

Substitute \(\mu = 300\), \(z = 2.5\), and \(\sigma=30\) into \(x=\mu + z\sigma\). So \(x=300+2.5\times30\).
First, calculate \(2.5\times30=75\). Then, \(x=300 + 75\).

Answer:

\(375\)