Question

a set of data items is normally distributed with a mean of 300 and a standard deviation of 20. find the data item in this distribution that corresponds to the given z - score
z = - 1
the data item that corresponds to z = - 1 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 are given that \(\mu = 300\), \(\sigma=20\), and \(z=-1\).

Step2: Solve the formula for \(x\)

Starting with \(z=\frac{x - \mu}{\sigma}\), we can multiply both sides by \(\sigma\) to get \(z\sigma=x-\mu\). Then add \(\mu\) to both sides: \(x=\mu+z\sigma\).
Substitute the given values: \(x = 300+(-1)\times20\).

Step3: Calculate the value of \(x\)

First, calculate \((-1)\times20=-20\). Then \(x=300 - 20\).

Answer:

\(280\)