Answer:
[tex]\textsf{If }\: X \sim\textsf{N}(\mu,\sigma^2)\:\textsf{ then }\: \dfrac{X-\mu}{\sigma}=Z, \quad \textsf{where }\: Z \sim \textsf{N}(0,1)[/tex]
Math
[tex]\textsf{If }\: X \sim\textsf{N}(71,6.5^2)\:\textsf{ then }\: \dfrac{X-71}{6.5}=Z, \quad \textsf{where }\: Z \sim \textsf{N}(0,1)[/tex]
Science
[tex]\textsf{If }\: X \sim\textsf{N}(62,8.0^2)\:\textsf{ then }\: \dfrac{X-62}{8.0}=Z, \quad \textsf{where }\: Z \sim \textsf{N}(0,1)[/tex]
z-score Information
- Positive z-score = data point is above average
- Negative z-score = data point is below average
- z-score close to zero = data point is close to average
- If a data point's z-score is below -3 or above 3 (more than 3 standard deviations from the mean) it can be considered unusual.
Question 3
Math
[tex]X=75 \implies Z=\dfrac{75-71}{6.5}\right) \implies Z=0.6154[/tex]
Science
[tex]X=71 \implies Z= \dfrac{71-62}{8.0} \implies Z=1.125[/tex]
The student's math score is below average, whereas their science score is above average. Therefore, the student did better on the science test.
Question 4
Math
[tex]X=91 \implies Z=\dfrac{91-71}{6.5} \implies Z=3.0769[/tex]
Science
[tex]X=85 \implies Z=\dfrac{85-62}{8.0} \implies Z=2.875[/tex]
The student's math and science scores are both above average, but the math score is further away from the mean. Therefore, the student did better on the math test.
Question 5
Math
[tex]X=65 \implies Z=\dfrac{65-71}{6.5} \implies Z=-0.9231[/tex]
Science
[tex]X=54 \implies Z=\dfrac{54-62}{8.0} \implies Z=-1[/tex]
The student's math and science scores are both below average, but the science score is further away from the mean. Therefore, the student did better on the math test.
