Here is dependence between scores and x-values:
[tex]Z_i=\dfrac{X_i-\mu}{\sigma},[/tex]
where [tex]\mu[/tex] is the mean, [tex]\sigma[/tex] is standard deviation and i changes from 1 to 2.
1. When i=1, [tex]Z_1=-0.25,\ X_1=57,[/tex] then
[tex]-0.25=\dfrac{57-\mu}{\sigma}.[/tex]
2. When i=2, [tex]Z_2=1.25,\ X_2=87,[/tex] then
[tex]1.25=\dfrac{87-\mu}{\sigma}.[/tex]
Now solve the system of equations:
[tex]\left\{\begin{array}{l}-0.25=\dfrac{57-\mu}{\sigma}\\ \\1.25=\dfrac{87-\mu}{\sigma}.\end{array}\right.[/tex]
[tex]\left\{\begin{array}{l}-0.25\sigma=57-\mu\\ \\1.25\sigma=87-\mu.\end{array}\right.[/tex]
Subtract first equation from the second:
[tex]1.25\sigma-(-0.25\sigma)=87-57,\\ \\1.5\sigma=30,\\ \\\sigma=20.[/tex]
Then
[tex]1.25=\dfrac{87-\mu}{20},\\ \\87-\mu=25,\\ \\\mu=87-25=62.[/tex]
Answer: the mean is 62, the standard deviation is 20.
Answer: The mean is 62 and the standard deviation is 20 for the population.
Step-by-step explanation:
Let [tex]\mu[/tex] be the mean and [tex]\sigma[/tex] be the standard deviation .
Formula to calculate z-score corresponds to random variable x on normal curve.
[tex]Z=\dfrac{X-\mu}{\sigma},[/tex]
Given : In a population distribution, a score of x=57 corresponds to z=-0.25 and a score of x=87 corresponds to z=1.25.
[tex]-0.25=\dfrac{57-\mu}{\sigma}\\\\-0.25\sigma=57-\mu---------(1)[/tex]
[tex]1.25=\dfrac{87-\mu}{\sigma}\\\\1.25\sigma=87-\mu----------(2)[/tex]
Eliminate equation(1) from equation(2), we get
[tex]1.50\sigma=30\\\\\Rightarrow\ \sigma=\dfrac{30}{1.5}=20[/tex]
Put value of [tex]\sigma=20[/tex] in (1)
[tex]1.25(20)=87-\mu\\\\87-\mu=25,\\\\\Rightarrow\mu=87-25=62.[/tex]
Hence, the mean is 62 and the standard deviation is 20 for the population.
