The mean and standard deviation of the scores are given below:
• Mean = 75
,• Standard deviation = 4
We make use of the z-score formula below:
[tex]z-\text{score}=\frac{X-\mu}{\sigma}\text{ where}\begin{cases}\mu=\text{Mean} \\ \sigma=\text{Standard Deviation}\end{cases}[/tex]Part A (between 63 and 87)
First, we determine the z-scores.
[tex]\begin{gathered} z-score=\frac{63-75}{4}=-\frac{12}{4}=-3 \\ z-score=\frac{87-75}{4}=\frac{12}{4}=3 \\ \text{From the z-score table: }P(-3Therefore, the percentage of scores that were between 63 and 87 is 99.73%.Part B (Above 83)
[tex]\begin{gathered} z-score=\frac{83-75}{4}=\frac{8}{4}=2 \\ \text{From the z-score table: }P(x>2)=0.02275 \end{gathered}[/tex]Therefore, the percentage of scores that were above 83 is 2.28%.
Part C (below 71)
[tex]\begin{gathered} z-\text{score}=\frac{71-75}{4}=-\frac{4}{4}=-1 \\ \text{From the z-score table: }P(x<-1)=0.15866 \end{gathered}[/tex]Therefore, the percentage of scores that were below 71 is 15.87%.
Part D (between 67 and 79)
[tex]\begin{gathered} z-score=\frac{67-75}{4}=\frac{-8}{4}=-2 \\ z-score=\frac{79-75}{4}=\frac{4}{4}=1 \\ \text{From the z-score table: }P(-2Therefore, the percentage of scores that were between 67 and 79 is 81.86%.