Respuesta :
Answer:
a) Using the z score formula we got:
[tex] z = \frac{\mu +5 -\mu}{\sigma}= \frac{5}{10}= 0.5[/tex]
b) Using the z score formula we got:
[tex] z = \frac{\mu +2 -\mu}{\sigma}= \frac{2}{10}= 0.2[/tex]
c) Using the z score formula we got:
[tex] z = \frac{\mu -20 -\mu}{\sigma}= -\frac{20}{10}= -2[/tex]
d) Using the z score formula we got:
[tex] z = \frac{\mu -15 -\mu}{\sigma}= -\frac{15}{10}= -1.5[/tex]
Step-by-step explanation:
Previous concepts
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".
Solution to the problem
Let X the random variable that represent the variable os interest of a population, and for this case we know the distribution for X is given by:
[tex]X \sim N(\mu,10)[/tex]
Where [tex]\mu[/tex] and [tex]\sigma=10[/tex]
The best way to solve this problem is using the normal standard distribution and the z score given by:
[tex]z=\frac{x-\mu}{\sigma}[/tex]
Part a
[tex] X = \mu +5[/tex]
Using the z score formula we got:
[tex] z = \frac{\mu +5 -\mu}{\sigma}= \frac{5}{10}= 0.5[/tex]
Part b
[tex] X = \mu +2[/tex]
Using the z score formula we got:
[tex] z = \frac{\mu +2 -\mu}{\sigma}= \frac{2}{10}= 0.2[/tex]
Part c
[tex] X = \mu -20[/tex]
Using the z score formula we got:
[tex] z = \frac{\mu -20 -\mu}{\sigma}= -\frac{20}{10}= -2[/tex]
Part d
[tex] X = \mu -15[/tex]
Using the z score formula we got:
[tex] z = \frac{\mu -15 -\mu}{\sigma}= -\frac{15}{10}= -1.5[/tex]
a. The value of z-score above the mean by 5 points is 0.5.
b. The value of z-score above the mean by 5 points is 0.2.
c. The value of z-score below the mean by 20 points is -2.
d. The value of z-score below the mean by 15 points is -1.5.
z-score:
The normal standard distribution and the z score given by:
[tex]z=\frac{X-\mu}{\sigma}[/tex]
Where [tex]X[/tex] is random variable, [tex]\mu[/tex] is mean and [tex]\sigma[/tex] is standard deviation.
Given that, [tex]\sigma=10[/tex]
The value of z-score above the mean by 5 points.
Substitute [tex]X=\mu+5[/tex] in above equation.
[tex]z=\frac{\mu+5-\mu}{10}=0.5[/tex]
The value of z-score above the mean by 2 points.
[tex]z=\frac{\mu+2-\mu}{10}=0.2[/tex]
The value of z-score below the mean by 20 points.
[tex]z=\frac{\mu-20-\mu}{10}=-2[/tex]
The value of z-score below the mean by 15 points.
[tex]z=\frac{\mu-15-\mu}{10}=-1.5[/tex]
Learn more about the z-score here:
https://brainly.com/question/25638875
