Answer:
b. [278.90, 288.55]
See the explanation below.
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".
Solution to the problem
Let X the random variable that represent the amounts of energy of a population, and for this case we know the distribution for X is given by:
Where
and ![\sigma=9.1](https://tex.z-dn.net/?f=%5Csigma%3D9.1)
LOWER LIMIT
For this part we want to find a value a, such that we satisfy this condition:
(a)
(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 0.25 of the area on the left and 0.75 of the area on the right it's z=-0.674. On this case P(Z<-0.674)=0.25 and P(z>-0.674)=0.75
If we use condition (b) from previous we have this:
![P(z](https://tex.z-dn.net/?f=P%28z%3C%5Cfrac%7Ba-%5Cmu%7D%7B%5Csigma%7D%29%3D0.25)
But we know which value of z satisfy the previous equation so then we can do this:
![z=-0.674](https://tex.z-dn.net/?f=z%3D-0.674%3C%5Cfrac%7Ba-285%7D%7B9.1%7D)
And if we solve for a we got
![a=285 -0.674*9.1=278.90](https://tex.z-dn.net/?f=a%3D285%20-0.674%2A9.1%3D278.90)
So the value of height that separates the bottom 25% of data from the top 75% is 278.90.
UPPER LIMIT
For this part we want to find a value a, such that we satisfy this condition:
(a)
(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 0.7 of the area on the left and 0.3 of the area on the right it's z=0.524. On this case P(Z<0.524)=0.7 and P(z>0.524)=0.3
If we use condition (b) from previous we have this:
![P(z](https://tex.z-dn.net/?f=P%28z%3C%5Cfrac%7Ba-%5Cmu%7D%7B%5Csigma%7D%29%3D0.7)
But we know which value of z satisfy the previous equation so then we can do this:
![z=0.524](https://tex.z-dn.net/?f=z%3D0.524%3C%5Cfrac%7Ba-285%7D%7B9.1%7D)
And if we solve for a we got
![a=285 +0.524*9.1=289.78](https://tex.z-dn.net/?f=a%3D285%20%2B0.524%2A9.1%3D289.78)
So the value of height that separates the bottom 70% of data from the top 30% is 289.78.
With the procedure we got for the limits [278.9 , 289.78]
And since any of the options are on the list we take th most similar for this case:
b. [278.90, 288.55]