To find the z-score of a company employee who makes an annual salary of [tex]$54,000, we need to use the formula for calculating the z-score in a normal distribution:
\[ z = \frac{X - \mu}{\sigma} \]
Where:
- \( X \) is the value of the data point (the employee's salary, in this case, $[/tex]54,000).
- [tex]\( \mu \)[/tex] is the mean of the data (the average salary, which is [tex]$34,000 in this case).
- \( \sigma \) is the standard deviation of the data (the standard deviation of the salaries, given as $[/tex]4,000).
Let's plug these values into the formula:
[tex]\[ z = \frac{54000 - 34000}{4000} \][/tex]
First, compute the difference in the numerator:
[tex]\[ 54000 - 34000 = 20000 \][/tex]
Next, divide the result by the standard deviation:
[tex]\[ z = \frac{20000}{4000} \][/tex]
[tex]\[ z = 5 \][/tex]
Therefore, the z-score of a company employee who makes an annual salary of $54,000 is [tex]\( 5 \)[/tex]. This means that the employee's salary is 5 standard deviations above the mean salary.
