Question

a set of data items is normally distributed with a mean of 600 and a standard deviation of 40. find the data item in this distribution that corresponds to the given z - score. $z = 6$. the data item that corresponds to $z = 6$ is $\square$ (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 item, \(\mu\) is the mean, and \(\sigma\) is the standard deviation. We need to solve for \(x\).

Step2: Rearrange the formula to solve 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 \(\mu = 600\), \(z = 6\), and \(\sigma=40\). Substitute into \(x=\mu+z\sigma\): \(x = 600+6\times40\).

Step4: Calculate the value of \(x\)

First, calculate \(6\times40 = 240\). Then \(x=600 + 240\).

Answer:

\(840\)