A coin has been bent in such a way that it comes up heads 75% of the time. This

Question

A coin has been bent in such a way that it comes up heads 75% of the time. This bent coin is placed in a box with 9 fair ones. A friend selects one coin randomly and tosses it 12 times. Consider the hypotheses

H0: It's a fair coin vs. Ha: It's the altered coin.

Use the following decision rule: Reject H0 if the coin comes up heads 11 or 12 times, and retain H0 otherwise.

(a) Specify the appropriate hypotheses in mathematical symbols.
(b) Find , the probability of a type I error.
(c) Find , the probability of a type II error.

Explanation / Answer

a=reject when true. So 11/12 heads for a fair coin. Ho: pi=.5 Ha: pi doesn't equal .5 Use the binomial theorem for b and c b. 12 choose 11 * (1/2)^(11)*(1/2)+12 choose 12*(1/2)^12=0.00317382812 or very low. c. 1-(12 choose 11 * (.75)^(11)*(1/2)+12 choose 12*(.75)^12)=0.714912832. So actually pretty high for beta.

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