Respuesta :
Answer:
a) [tex]X \sim N(9.3,1)[/tex]
b) [tex]P(X>9.2)=P(\frac{X-\mu}{\sigma}>\frac{9.2-\mu}{\sigma})=P(Z>\frac{9.2-9.3}{1})=1-P(Z<-0.1)=1-0.4601=0.5398[/tex]
c) The value of height that separates the bottom 75% of data from the top 25% is 9.9745.
Step-by-step explanation:
1) 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".
2) Part a
Let X the random variable that represent the heights of a population, and for this case we know the distribution for X is given by:
[tex]X \sim N(9.3,1)[/tex]
Where [tex]\mu=9.3[/tex] and [tex]\sigma=1[/tex]
3) Part b
We are interested on this probability
[tex]P(X>9.2)[/tex]
And 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]
If we apply this formula to our probability we got this:
[tex]P(X>9.2)=P(\frac{X-\mu}{\sigma}>\frac{9.2-\mu}{\sigma})=P(Z>\frac{9.2-9.3}{1})=1-P(Z<-0.1)=1-0.4601=0.5398[/tex]
4) Part c
For this part we want to find a value a, such that we satisfy this condition:
[tex]P(X>a)=0.25[/tex] (a)
[tex]P(X<a)=0.75[/tex] (b)
Both conditions are equivalent on this case. We can use the z score again in order to find the value a.
As we can see on the figure attached the z value that satisfy the condition with 0.75 of the area on the left and 0.25 of the area on the right it's z=0.6745. On this case P(Z<0.6745)=0.75 and P(z>0.6745)=0.25
If we use condition (b) from previous we have this:
[tex]P(X<a)=P(\frac{X-\mu}{\sigma}<\frac{a-\mu}{\sigma})=0.75[/tex]
[tex]P(Z<\frac{a-\mu}{\sigma})=0.75[/tex]
But we know which value of z satisfy the previous equation so then we can do this:
[tex]z=0.6745<\frac{a-9.3}{1}[/tex]
And if we solve for a we got
[tex]a=9.3 +1*0.6745=9.9745[/tex]
So the value of height that separates the bottom 75% of data from the top 25% is 9.9745.
