a) We have a distribution with mean 50 and standard deviation of 20.
We can find the z-score for X = 40 as:
[tex]z=\frac{X-\mu}{\sigma}=\frac{40-50}{20}=\frac{-10}{20}=-0.5[/tex]
We can find the X given that z = 0.25 as:
[tex]X=\mu+z\sigma=50+0.25\cdot20=50+5=55[/tex]
b) For a distribution with mean 50 and standard deviation of 4, the z-score for X = 40 is:
[tex]z=\frac{X-\mu}{\sigma}=\frac{40-50}{4}=\frac{-10}{4}=-2.5[/tex]
We can calculate X for z = 0.25 as:
[tex]X=\mu+z\sigma=50+0.25\cdot4=50+1=51[/tex]
c) For a distribution with mean 30 and standard deviation of 8, the z-score for X = 40 is:
[tex]z=\frac{X-\mu}{\sigma}=\frac{40-30}{8}=\frac{10}{8}=1.25[/tex]
We can calculate X for z = 0.25 as:
[tex]X=\mu+z\sigma=30+0.25\cdot8=30+2=32[/tex]
d) For a distribution with mean 30 and standard deviation of 4, the z-score for X = 40 is:
[tex]z=\frac{X-\mu}{\sigma}=\frac{40-30}{4}=\frac{10}{4}=2.5[/tex]
We can calculate X for z = 0.25 as:
[tex]X=\mu+z\sigma=30+0.25\cdot4=30+1=31[/tex]
Answer:
a) z = -0.5, X = 55
b) z = -2.5, X = 51
c) z = 1.25, X = 32
d) z = 2.5, X = 31
