Please show full work so I can understand, thank you so much. 1. We are interest

Question

Please show full work so I can understand, thank you so much.

1. We are interested in testing whether a coin is fair or not, based on the number of heads Y on 36 tosses of the coin. (a) (3 points) If we use the rejection region |y - 18| >= 4, what is the level of significance of the test (i.e. the probability of Type I error)? (b) (2 points) If the observed number of heads is 24, what is the p-value of the test? (c) (2 points) Compare your p-value at the level of significance computed in part (a), and write your conclusions. (d) (3 points) Suppose now we want to test H0 : p = 0.5 versus H1 : p = 0.7. What is the power of the test based on the same rejection region as in part (a)?

Explanation / Answer

a) Level of significance is decided by the experiementer. It is the probability that null hypothesis will be falsely rejected. That depends on the investigator that how much chance s/he is willing to allow for this error to creep in. Usually it is fixed at .05 or .01

b) p-value can be computed based on whether the test is left tailed right tailed or both tailed.

Assuming this is a right tailed test p-value= P(Y=>24|H_0), Y~Binomial(n=36,p=.5) under H_0. The probability I Mentioned can be computed in binomial probability calculator available online. The value obtained is .032

For left tailed test p-value= P(Y<=24|H_0)=.986

For both tailed test p-value= 2*min{P(Y=>24,|H_0), P(Y<=24|H_0))=2*min{.032,.986}=2*.032=.064

c) for the right tailed test p-value=.032<.05 => H_0 is rejected at 5% level for the right tailed test p-value=.986>.05 => The H_0 is accpeted for both tailed test p-value=.064>.05 => H_0 is accepted at 5% level.

d) The power is P(|y-18|=>4|H_1)=.6844

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