Respuesta :
Answer with Step-by-step explanation:
The Z score in a distribution is found as
[tex]Z=\frac{X-\overline{X}}{\sigma }[/tex]
where
X is the given value
[tex]\overline{X}[/tex] is the mean of the observations
[tex]\sigma [/tex] is the standard deviation of the data
The values of oach part is found as under:
Part a)
Since it is given that score is 20 points over mean thus we can write
[tex]X=20+\overline{X}\\\\X-\overline{X}=20[/tex]
Using the above relation we get
[tex]Z_a=\frac{20}{10}=2[/tex]
Part b)
Since it is given that score is 10 points below mean thus we can write
[tex]X=\overline{X}-10\\\\X-\overline{X}=-10[/tex]
Using the above relation we get
[tex]Z_b=\frac{-10}{10}=-1[/tex]
Part c)
Since it is given that score is 15 points over mean thus we can write
[tex]X=15+\overline{X}\\\\X-\overline{X}=15[/tex]
Using the above relation we get
[tex]Z_c=\frac{15}{10}=1.5[/tex]
Part d)
Since it is given that score is 30 points below mean thus we can write
[tex]X=\overline{X}-30\\\\X-\overline{X}=-30[/tex]
Using the above relation we get
[tex]Z_d=\frac{-30}{10}=-3[/tex]
Using the normal distribution, the corresponding z-scores are:
a) z = 2.
b) z = -1.
c) z = 1.5.
d) z = -3.
In a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
- It measures how many standard deviations the measure is from the mean.
In this problem, we have a standard deviation of [tex]\sigma = 10[/tex].
Item a:
- Score 20 points above the mean, thus [tex]X = \mu + 20[/tex]
The z-score is:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{\mu + 20 - \mu}{10}[/tex]
[tex]Z = 2[/tex]
Item b:
- Score 10 points below the mean, thus [tex]X = \mu - 10[/tex]
The z-score is:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{\mu - 10 - \mu}{10}[/tex]
[tex]Z = -1[/tex]
Item c:
- Score 15 points above the mean, thus [tex]X = \mu + 15[/tex]
The z-score is:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{\mu + 15 - \mu}{10}[/tex]
[tex]Z = 1.5[/tex]
Item d:
- Score 30 points below the mean, thus [tex]X = \mu - 30[/tex]
The z-score is:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{\mu - 30 - \mu}{10}[/tex]
[tex]Z = -3[/tex]
A similar problem is given at https://brainly.com/question/16645591
