Answer:
a) She scored 74.46 on the exam.
b) 11% of the students scored better than Stephanie.
Step-by-step explanation:
The z-score measures how many standard deviation a score X is above or below the mean. It is given by the following formula:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
In which [tex]\mu[/tex] is the mean and [tex]\sigma[/tex] is the standard deviation.
In this problem, we have that:
[tex]\mu = 72, \sigma = 2[/tex]
a. What score did Stephanie get on the exam?
Stephanie scored 1.23 standard deviations above the mean. This means that her z-score is [tex]Z = 1.23[/tex]
We want to find X
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]1.23 = \frac{X - 72}{2}[/tex]
[tex]X - 72 = 2*1.23[/tex]
[tex]X = 74.46[/tex]
She scored 74.46 on the exam.
b. What percent of students scored better than Stephanie?
Each z-score has a pvalue, which is the percentile of the score. We look this pvalue at the z table.
[tex]Z = 1.23[/tex] has a pvalue of 0.89.
This means that Stephanie's score is in the 89th percentile, which means that she scored more than 89% of the students and scored less than 100-89 = 11% of the students.
So 11% of the students scored better than Stephanie.
