Question

a set of data items is normally distributed with a mean of 100 and a standard deviation of 10. find the data item in this distribution that corresponds to the given z - score.
z = - 5
the data item that corresponds to z = - 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 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 \(\mu = 100\), \(z=- 5\), and \(\sigma = 10\). Substitute into \(x=\mu+z\sigma\): \(x = 100+(-5)\times10\).

Step4: Calculate the value of \(x\)

First, calculate \((-5)\times10=-50\). Then \(x = 100-50\).

Answer:

\(50\)