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matheagle · #2 June 20th, 2009, 06:39 PM

This a binomial test and I can give you an $\alpha$ if you give me a decision rule/critical region.
Similarly if you give me an $\alpha$ I can then give you the critical region.
Clearly we reject the null hypothesis and accept the alternative for small X,
since p under the alternative is smaller than p under the null.

If we make the critical region X=0,

then $\alpha=P(X=0)={4 \choose 0} (3/5)^0(2/5)^4$

and $\beta=P(X>0)$ when p=2/5.

If we make the critical region X=0 or X=1, then

$\alpha=P(X=0)+P(X=1)={4 \choose 0} (3/5)^0(2/5)^4+{4 \choose 1} (3/5)^1(2/5)^3$

and $\beta=P(X>1)$ when p=2/5.

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