a) Percentage of scores that were between 59 and 67 = 95.45%
b) Percentage of scores above 69 = 0.135%
c) Percentage of scores below 59 = 2.2755%
d) Percentage of scores between 57 and 67 = 97.59%
Explanations:The mean, μ = 63
Standard deviation, σ = 2
When x = 59
[tex]\begin{gathered} z\text{ = }\frac{x-\mu}{\sigma} \\ z\text{ = }\frac{59-63}{2} \\ z\text{ = }-2 \end{gathered}[/tex]When x = 67
[tex]\begin{gathered} z\text{ = }\frac{x-\mu}{\sigma} \\ z\text{ = }\frac{67-63}{2} \\ \text{z = 2} \end{gathered}[/tex]P(59 < x < 67) = P(-2 < x < 2) = 0.9545
Probability that scores fall between 59 and 67 = 0.9545
Percentage of scores that were between 59 and 67 = 95.45%
b) above 69
P(x > 69)
[tex]\begin{gathered} z\text{ = }\frac{x-\mu}{\sigma} \\ z\text{ = }\frac{69-63}{2} \\ z\text{ = 3} \end{gathered}[/tex]P(x > 69) = P(z > 3) = 0.0013499
Percentage of scores above 69 = 0.135%
c) below 59
P(x < 59)
[tex]\begin{gathered} z\text{ = }\frac{59-63}{2} \\ z\text{ = -2} \end{gathered}[/tex]P(x < 59) = P(z < -2) = 0.02275
Percentage of scores below 59 = 2.2755%
d) between 57 and 67.
when x = 57
[tex]\begin{gathered} z\text{ = }\frac{57-63}{2} \\ z\text{ = -3} \end{gathered}[/tex]P(57 < x < 67) = P(-3 < x < 2) = 0.9759
Percentage of scores between 57 and 67 = 97.59%
