Question: In one faculty the distribution of the heights of the students is N(180,25) In the second faculty it's N(160,20)
We take some students from both faculties and put them together in a class. There are 2 hypothesis regarding how they are distributed:
$H_0$: $\frac 14$ of them are from faculty I, and $\frac 34$ are from faculty II.
$H_1$: $\frac 14$ of them are in faculty II, and $\frac 34$ are from faculty I.
If we look at a random student- If his height is bigger than 168 we reject $H_0$.
What is the rejection zone?
What I did: To try to formalize it I rephrased this in math terms:
X is a Bernulli RV representing a single random student, s.t. p is the probability of a student to be in faculty I.
Therefore $H_0: p=0.75$
I think the statistic S(X) is the height of a single random student, so I want to find out the prob. under $H_0$ that his height is bigger than 168. I did it this way : $0.75P(\frac {S(X)-180}{25}>\frac {168-180}{25})+0.25 P(\frac{S(X)-160}{20}<\frac{168-160}{20})=0.5994$ (using R) Is this result the alpha used to calculate the rejection zone $C_a$?
(We've been told that $P_{H_0} (S(X)\in C_a)=a$ is how to do a statistic test)
I would love some thorough explanation (maybe some good links too) for this(not only hints) as I feel my understanding of the whole subject is still lacking. Thanks a lot