Example from class I don't understand ...

#1
The Prof did an example in class which I don't understand.. Here it is:

Consider a Bernoulli population with parameter p which is the probability of success. A random sample of size 3 is chosen.

H0: 0 <= p <= .5
HA: .5 <= p <= 1

Critical region1 is 3 successes
Critical region2 is 2 or 3 successes

We can show power function for CR1 is P1(p)=p^3
Also power function for CR2 is P2(p)=p^3 + 3p^2(1-p)

Anyways the prof draws the 2 power functions on the same graph.

We see that P2(p) is always above P1(p).

This is the part I don't get:
Prof says that
Test 2 is better than test 1 if p falls in HA range
Test 1 is better than test 2 if p falls in H0 range
 
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#2
Never mind I found out what is going on here... Basically if p is in H0 then only the alpha comparisons matter, as beta is undefined since p is not in HA. In this case P1 is better.

Now if p is in HA then alpha errors are undefined, only betas matter and here P2 has better beta.

So depends on the value of p which is the better test.