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

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)?

Solution

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