Let's solve the problem step-by-step.
### Part (a): Finding the [tex]$z$[/tex]-score
Given the information:
- Mean score ([tex]\(\mu\)[/tex]) = 77
- Standard deviation ([tex]\(\sigma\)[/tex]) = 9
- Nicole's score ([tex]\(X\)[/tex]) = 70
The formula to calculate the [tex]$z$[/tex]-score is:
[tex]\[ z = \frac{X - \mu}{\sigma} \][/tex]
Substituting the values:
[tex]\[ z = \frac{70 - 77}{9} \][/tex]
[tex]\[ z = \frac{-7}{9} \][/tex]
[tex]\[ z = -0.78 \][/tex]
So the [tex]$z$[/tex]-score of Nicole's exam score is:
[tex]\[ z = -0.78 \][/tex]
### Part (b): Interpreting the [tex]$z$[/tex]-score
A [tex]$z$[/tex]-score tells us how many standard deviations a particular score is from the mean. The [tex]$z$[/tex]-score in this case is [tex]\(-0.78\)[/tex], which indicates that Nicole's score is below the mean. To express this in terms of a positive number of standard deviations:
- Take the absolute value of the [tex]$z$[/tex]-score: [tex]\(|-0.78| = 0.78\)[/tex]
- Nicole's score is 0.78 standard deviations below the mean.
The interpretation will be:
Nicole's exam score was [tex]\(0.78\)[/tex] standard deviations [tex]\(\text{below}\)[/tex] the mean exam score among all students in the course.
