Question

a set of data items is normally distributed with a mean of 200 and a standard deviation of 60. 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 ( 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 \(\mu = 200\), \(z = 2.5\), and \(\sigma=60\). Substitute into \(x=\mu+z\sigma\): \(x = 200+2.5\times60\).

Step4: Calculate the value of \(x\)

First, calculate \(2.5\times60 = 150\). Then \(x=200 + 150\).

Answer:

\(350\)