To convert a value from a normally distributed data set to a [tex]$z$[/tex]-score, you can use the following formula:
[tex]\[
z = \frac{{x - \mu}}{{\sigma}}
\][/tex]
where:
- [tex]\( x \)[/tex] is the value you want to convert
- [tex]\( \mu \)[/tex] is the mean of the data set
- [tex]\( \sigma \)[/tex] is the standard deviation of the data set
Given:
- Mean ([tex]\( \mu \)[/tex]) = 40
- Standard deviation ([tex]\( \sigma \)[/tex]) = 9
- Value to convert ([tex]\( x \)[/tex]) = 22
We can plug these values into the formula:
[tex]\[
z = \frac{{22 - 40}}{{9}}
\][/tex]
First, calculate the numerator:
[tex]\[
22 - 40 = -18
\][/tex]
Next, divide the result by the standard deviation:
[tex]\[
z = \frac{{-18}}{{9}} = -2.0
\][/tex]
So, the [tex]$z$[/tex]-score for the value 22 is:
[tex]\[
z_{22} = -2.0
\][/tex]
