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I have got one problem as following.

There are two coins in a box (that look very much the same). For one coin, the probability of heads is $0.7$ and for the other, the probability of heads is $0.1$. You will randomly select one of the coins and toss it once. Based on the result of this toss, you will decide which coin you have tossed. Suppose that your null hypothesis is that you have selected the coin with a 70% chance of heads.

(1) What is the alternative hypothesis?

(2) If the results of the toss is heads, will you “accept” or reject the null hypothesis? Explain your reasoning.

(3) What is $\alpha$ for this test?

(4) What is the power for this test if the alternative hypothesis is true? (Note that there is only a single possible alternative hypothesis.)

I have finished the first two questions. For (1), I think the alternative hypothesis is "I have selected the coin with a $10\%$ chance of heads".

For (2), I used the the basic probability theory to compute both probabilities of getting each coins ($0.875$ v.s. $0.175$), and concluded that I will "accept" my null hypothesis.

However, I have no ideas how to approach (3) and (4). I think $\alpha$, the probability of type 1 error, should be set before the test. We can interpret it as a threshold of our test, then why and how we can compute $\alpha$?

For (4), I don't really understand why the power would depend on the alternative hypothesis? Shouldn't it be all the same? I also have no ideas how to find the power in this problem.

Any hints or suggestions would be appreciate!

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  • $\begingroup$ I am still thinking about this problem. Can I calculate $\alpha, \beta$ using conditional probability? Like $\alpha = \mathbb{P}(reject\mid H_o)$? $\endgroup$
    – Jay Wang
    Nov 4, 2016 at 1:13

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