For any normally distributed random variable (X), the z-score corresponding to any particular value of the random variable is given by the formula,
[tex]z=\frac{x-\mu}{\sigma}[/tex]
The mean of the normal distribution is 425,
[tex]\mu=425[/tex]
The standard deviation of the normal distribution is 51,
[tex]\sigma=51[/tex]
Then the formula of z-score becomes,
[tex]z=\frac{x-425}{51}[/tex]
a)
Given the value of the random variable as 268,
[tex]x=268[/tex]
The corresponding z-score will be,
[tex]\begin{gathered} z=\frac{268-425}{51} \\ z\approx-3.08 \end{gathered}[/tex]
Thus, the required z-score is -3.08 approximately.
b)
Given the value of the random variable as 512,
[tex]x=512[/tex]
The corresponding z-score will be,
[tex]\begin{gathered} z=\frac{512-425}{51} \\ z\approx1.70 \end{gathered}[/tex]
Thus, the required z-score is 1.70 approximately.
c)
Given the value of the random variable is 450,
[tex]x=450[/tex]
The corresponding z-score will be,
[tex]\begin{gathered} z=\frac{450-425}{51} \\ z\approx0.49 \end{gathered}[/tex]
Thus, the required z-score is 0.49 approximately.
