Respuesta :
Answer:
[tex]a=66000 -0.842*21000=48318[/tex]
So the value of height that separates the bottom 20% of data from the top 80% is $484318. And this value represent the answer for our question.
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".
Solutio to the problem
Let X the random variable that represent the annual profits of a population, and for this case we know the distribution for X is given by:
[tex]X \sim N(66000,21000)[/tex]
Where [tex]\mu=66000[/tex] and [tex]\sigma=21000[/tex]
For this part we want to find a value a, such that we satisfy this condition:
[tex]P(X>a)=0.80[/tex] (a)
[tex]P(X<a)=0.20[/tex] (b)
Since we want the lowest 20% of profits
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 0.20 of the area on the left and 0.80 of the area on the right it's z=-0.842. On this case P(Z<-0.842)=0.2 and P(Z>-0.842)=0.80
If we use condition (b) from previous we have this:
[tex]P(X<a)=P(\frac{X-\mu}{\sigma}<\frac{a-\mu}{\sigma})=0.2[/tex]
[tex]P(z<\frac{a-\mu}{\sigma})=0.2[/tex]
But we know which value of z satisfy the previous equation so then we can do this:
[tex]z=-0.842<\frac{a-66000}{21000}[/tex]
And if we solve for a we got
[tex]a=66000 -0.842*21000=48318[/tex]
So the value of height that separates the bottom 20% of data from the top 80% is $484318. And this value represent the answer for our question.
