Respuesta :
Answer:
a) [tex] z_{\alpha}= -1.036[/tex]
b) [tex] z_{\alpha}= -0.385[/tex]
c) [tex] z_{\alpha}= -1.282[/tex]
d) [tex] z_{\alpha}= 1.645[/tex]
e) [tex] z_{\alpha/2}= -1.960 , z_{\alpha/2}= 1.960[/tex]
f) [tex] z_{\alpha/2}= -2.576 , z_{\alpha/2}= 2.576[/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".
Part a
Let Z the random variable that represent the variable of a population, and for this case we know the distribution for Z is given by:
[tex]Z \sim N(0,1)[/tex]
Where [tex]\mu=0[/tex] and [tex]\sigma=1[/tex]
We want 15% of the area in the left so we can use z table or the following code in excel in order to find the value required:
"=NORM.INV(0.15,0,1)"
And we got [tex] z_{\alpha}= -1.036[/tex]
Part b
We want 65% of the area in the right and that implies 100-65=35% of the area in the left so we can use z table or the following code in excel in order to find the value required:
"=NORM.INV(0.35,0,1)"
And we got [tex] z_{\alpha}= -0.385[/tex]
Part c
We want 10% of the area in the left so we can use z table or the following code in excel in order to find the value required:
"=NORM.INV(0.10,0,1)"
And we got [tex] z_{\alpha}= -1.282[/tex]
Part d
We want 5% of the area in the right and that implies 100-5=95% of the area in the left so we can use z table or the following code in excel in order to find the value required:
"=NORM.INV(0.95,0,1)"
And we got [tex] z_{\alpha}= 1.645[/tex]
Part e
For this case we waant two values that accumulates 95% of the area in the middle. That means we need to have 100-95 % = 5% of the area in the tails and since the distribution is symmetric we need to have 5/2 = 2.5% of the area on each tail. So we can use the following codes to find the two values:
"=NORM.INV(0.025,0,1)"
"=NORM.INV(1-0.025,0,1)"
[tex] z_{\alpha/2}= -1.960 , z_{\alpha/2}= 1.960[/tex]
Part f
For this case we waant two values that accumulates 99% of the area in the middle. That means we need to have 100-99 % = 1% of the area in the tails and since the distribution is symmetric we need to have 1/2 = 0.5% of the area on each tail. So we can use the following codes to find the two values:
"=NORM.INV(0.005,0,1)"
"=NORM.INV(1-0.005,0,1)"
[tex] z_{\alpha/2}= -2.576 , z_{\alpha/2}= 2.576[/tex]
Using the normal distribution, we have that:
a) z = -1.037.
b) z = -0.385.
c) z = -1.28.
d) z = 1.645.
e) z = 1.96.
f) z = 2.575.
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.
- After finding the z-score, we look at the z-score table and find the p-value associated with this z-score, which is the percentile of X, which is the area to the left of Z.
- The area to the right of Z is 1 subtracted by it's p-value.
Item a:
This is z with a p-value of 0.15, hence z = -1.037.
Item b:
This is z with a p-value of 1 - 0.65 = 0.35, hence z = -0.385.
Item c:
This is z with a p-value of 0.1, hence z = -1.28.
Item d:
This is z with a p-value of 1 - 0.05 = 0.95, hence z = 1.645.
Item e:
This is z with a p-value of [tex]\frac{1 + 0.95}{2} = 0.975[/tex], hence z = 1.96.
Item f:
This is z with a p-value of [tex]\frac{1 + 0.99}{2} = 0.995[/tex], hence z = 2.575.
A similar problem is given at https://brainly.com/question/16036056
