Respuesta :
Answer:
a) The area to the left of 18 is 0.5.
b) The area to the left of 14 is if 0.309.
c) The area to the right of 17 is of 0.55.
d) The area to the right of 19 is of 0.45.
e) The area between 14 and 30 is of 0.624.
Step-by-step explanation:
Normal Probability Distribution:
Problems of normal distributions can be solved using the z-score formula.
In a set 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]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X, which is the area to the left of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X, which is the area to the right of X.
A variable is normally distributed with mean 18 and standard deviation 8
This means that [tex]\mu = 18, \sigma = 8[/tex]
a) Find the area to the left of 18.
This is the pvalue of Z when X = 18.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{18 - 18}{8}[/tex]
[tex]Z = 0[/tex]
[tex]Z = 0[/tex] has a pvalue of 0.5.
So the area to the left of 18 is 0.5.
b) Find the area to the left of 14.
This is the pvalue of Z when X = 14. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{14 - 18}{8}[/tex]
[tex]Z = -0.5[/tex]
[tex]Z = -0.5[/tex] has a pvalue of 0.309
The area to the left of 14 is if 0.309.
c) Find the area to the right of 17.
This is 1 subtracted by the pvaleu of Z when X = 17. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{17 - 18}{8}[/tex]
[tex]Z = -0.125[/tex]
[tex]Z = -0.125[/tex] has a pvalue of 0.45
1 - 0.45 = 0.55
The area to the right of 17 is of 0.55.
d) Find the area to the right of 19.
This is 1 subtracted by the pvaleu of Z when X = 19. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{19 - 18}{8}[/tex]
[tex]Z = 0.125[/tex]
[tex]Z = 0.125[/tex] has a pvalue of 0.55
1 - 0.55 = 0.45
The area to the right of 19 is of 0.45.
e) Find the area between 14 and 30.
This is the pvalue of Z when X = 30 subtracted by the pvalue of Z when X = 14.
X = 30
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{30 - 18}{8}[/tex]
[tex]Z = 1.5[/tex]
[tex]Z = 1.5[/tex] has a pvalue of 0.933
From b), when X = 14, Z has a pvalue of 0.309.
0.933 - 0.309 = 0.624
The area between 14 and 30 is of 0.624.
The area to the left of 18 is 0.5, the area to the left of 14 is 0.309, the area to the right of 17 is 0.55, the area to the right of 19 is 0.45, and the area between 14 and 30 is 0.624.
Given :
A variable is normally distributed with a mean of 18 and a standard deviation of 8.
a) Use the z score formula in order to determine the area to the left of 18.
[tex]\rm Z = \dfrac{X-\mu}{\sigma}[/tex]
[tex]\rm Z = \dfrac{18-18}{8}=0[/tex]
For the z = 0 the p-value is 0.5.
Therefore, 0.5 is the area to the left of 18.
b) Use the z score formula in order to determine the area to the left of 14.
[tex]\rm Z = \dfrac{X-\mu}{\sigma}[/tex]
[tex]\rm Z = \dfrac{14-18}{8}=-0.5[/tex]
For the z = -0.5 the p-value is 0.309.
Therefore, 0.309 is the area to the left of 14.
c) Use the z score formula in order to determine the area to the right of 17.
[tex]\rm Z = \dfrac{X-\mu}{\sigma}[/tex]
[tex]\rm Z = \dfrac{17-18}{8}=-0.125[/tex]
For the z = -0.125 the p-value is 0.45.
Therefore, the area to the left of 17 is given by:
= 1 - 0.45 = 0.55
d) Use the z score formula in order to determine the area to the right of 19.
[tex]\rm Z = \dfrac{X-\mu}{\sigma}[/tex]
[tex]\rm Z = \dfrac{19-18}{8}=0.125[/tex]
For the z = 0.125 the p-value is 0.55.
Therefore, the area to the left of 19 is given by:
= 1 - 0.55 = 0.45
e) Use the z score formula in order to determine the area between 14 and 30.
At X = 30 the z score becomes:
[tex]\rm Z = \dfrac{X-\mu}{\sigma}[/tex]
[tex]\rm Z = \dfrac{30-18}{8}=1.5[/tex]
For the z = 1.5 the p-value is 0.933.
For the z = -0.5 the p-value is 0.309.
Therefore, the area to the left of 14 is given by:
= 0.933 - 0.309
= 0.624
For more information, refer to the link given below:
https://brainly.com/question/13299273
