To determine the [tex]$z$[/tex]-score of a given employee's salary, we use the following formula for the [tex]$z$[/tex]-score calculation in a normal distribution:
[tex]$ z = \frac{X - \mu}{\sigma} $[/tex]
Where:
- [tex]\( X \)[/tex] is the value for which we are calculating the [tex]$z$[/tex]-score (the individual's salary).
- [tex]\( \mu \)[/tex] is the mean of the distribution (the average salary).
- [tex]\( \sigma \)[/tex] is the standard deviation of the distribution (the standard deviation of salaries).
In this situation:
- The mean salary, [tex]\(\mu\)[/tex], is \[tex]$34,000.
- The standard deviation, \(\sigma\), is \$[/tex]4,000.
- The employee's salary, [tex]\(X\)[/tex], is \[tex]$28,000.
Plugging these values into the formula, we get:
$[/tex][tex]$ z = \frac{28000 - 34000}{4000} $[/tex][tex]$
Simplify the numerator:
$[/tex][tex]$ z = \frac{-6000}{4000} $[/tex][tex]$
Perform the division:
$[/tex][tex]$ z = -1.5 $[/tex][tex]$
Thus, the $[/tex]z[tex]$-score for an employee with a salary of \$[/tex]28,000 is [tex]\(-1.5\)[/tex].
Therefore, the correct answer is:
[tex]\[
-1.5
\][/tex]
