The method seems to work, finding beta error in t land is simple.
Convert alpha to x units, inches etc
Form the beta distribution in x units with x alpha.
Convert the beta x unit distribution to t units.
Solve for t </= t alpha beta.
Would someone look at the explanation? I'll email it.
Beta is the probability that you won't reject an incorrect H0.
You calculate the critical values based on the H0 distribution assumption, then you should calculate the probability to be in the acceptance area base on the H1 distribution assumption (if you use t-distribution it is a non-central t)
what t-test are you trying to calculate?
You should write what is the H0 distribution, whats is the critical value, what is the H1 distribution
You are doing the left tail. (one tail not two tails)
I prefer to calculate only the priori test power, in this case, you calculates the other avg(x) based on the "change" that you want the test to identify:
In the z-test the H1 also has the normal distribution, but in the t-test the H1 has the non-central t distribution,
"what t-test are you trying to calculate?" 2 sample right tail
What is H1; is it H beta? non central? does this mean mu 2 ain't = mu beta? Isn't that what TYPE II error is about? Do you have any questions about my explanation?
"In the t-test the H1 has the non-central t distribution"
I didn't explain clearly. My analysis is based on two t distributions with v1 = v2 and mu1 not equal to mu 2. I think of an x axis with a center at zero; and two identical t distributions, free to slide, sitting on the x axis. I can position each anywhere on the x axis. When they are exactly congruent = Ho. The x axis is graduated in inches or pounds or IQ or ????????
No non-central t distribution involved or required. t describes samples of x1-x2 taken from 1 = one distribution.
Does that help?
If you know the populations means: mu1 not equal to mu 2, you don't need to run a test
So I assume you know the averages, and try to understand if based on the samples you may conclude for the populations...
If you know the population standard deviation σ, you should use the z test.
If you don't know the population standard deviation you should estimate S and use a t-test.
In this case, the null assumption distributes non-central t.
Well, obh, I have failed to explain. You and I are talking about different things in different languages. I surrender. Back to the subject:
“Power” is the probability of rejecting Ho: µ 1 = µ 2 when Ho: is false.
The β error is the probability of accepting Ho when Ho is false.
Thus, power = 1 – β error.
In the example above, power = 1 - .2820 = .7180, 71.8%.
The non-central t distribution can be ignored.
I am beginning to suspect that the mathematical edifice constructed about t testing, β error and power was built on the premise that β error in t testing cannot be easily calculated; and that that premise is incorrect.