Question

a supervisor finds the mean number of miles that the employees in a department live from work. he finds $\bar{x}=29$ and $s = 3.6$. which statement must be true?
$z_{37}$ is within 1 standard deviation of the mean.
$z_{37}$ is between 1 and 2 standard deviations of the mean.
$z_{37}$ is between 2 and 3 standard deviations of the mean.
$z_{37}$ is more than 3 standard deviations of the mean.

Explanation:

Step1: Identify the z - score formula

$z=\frac{x-\bar{x}}{s}$

Step2: Substitute the given values

$z=\frac{37 - 29}{3.6}$

Step3: Calculate the z - score

$z=\frac{8}{3.6}\approx2.22$

Step4: Determine the range of the z - score

Since $2 < 2.22<3$, we conclude that $z_{37}$ is between 2 and 3 standard deviations of the mean.

Answer:

The formula for the z - score is $z=\frac{x-\bar{x}}{s}$. Here, we assume $x = 37$, $\bar{x}=29$ and $s = 3.6$.
First, calculate the z - score:
$z=\frac{37 - 29}{3.6}=\frac{8}{3.6}\approx2.22$

Since $2<2.22<3$, the statement " $z_{37}$ is between 2 and 3 standard deviations of the mean" is true. So the answer is: $z_{37}$ is between 2 and 3 standard deviations of the mean.