Let x1 and x2 be 2 random variables, iid,
xi ~ Exponential Law (lambda = 1 / theta) , in other words, E(x) = theta.
Test the following:
Ho: theta(0) = 2
H1: theta(1) = 4
We define the critical region C = {(x1,x2) | x1+x2 >= 9.5}
a) Find the confidence level of the test
I integrated over the region C... I found confidence level = 82.26% ... seem's kinda low
b) Find the power of the test.
In my notes it just says...
1-Beta = Power of the test = P(Accepting H1 is true | H1 is true)
so I'm trying to figure out a relationship between Beta and Alpha or Beta and the region C ... any ideas?
I looked in my book but... its Mathematical Statistics by John.A. Rice... worst book I've read in my life...You can only understand the book if your like a professor ... that must be why my professor is so in love with that book -_-' . And that book won awards???
Moving on... I found that the confidence level of the test is 82.26% so I used this formula that I found on this other book.
Beta(theta(1)) = P[Z <= Z_(alpha) - {|theta(0)-theta(1)| / sigma_(x bar)} ]
Z_(alpha) = Z_17.74% (Because the confidence level is 82.26% and the way the hypothesis test is set up were doing a one tailed test)
Z_17.74% = 0.925 (in table of Z).
The problem is that the standard deviation here is not a number... its actually in terms of theta...
So I replaced theta(0) = 2 found a value and then theta(1) = 4 in there and found another value and added that up to get the error of 2nd kind... is this right?
I found Beta to be 0.725 ... Kinda high for an error lol and the power of the test I found 1 - 0.725 = 0.275.
Is it possible that the error of 2nd kind is higher then the power of the test?