Probability question involving the null hypothesis and type errors?

I am completely stuck on this problem here. I sat and stared at it for at least 20 minutes but I couldn't think of anything at all. If anyone could help me out with this problem I would surely appreciate it. Here it is:





The proportion of adults living in a small town who are college graduates is estimated to be 0.6. A random sample of 15 adults is selected. If the number of college graduates in our sample is anywhere from 6 to 12, we shall not reject the null hypothesis that the true proportion, p, is 0.6; otherwise, we shall conclude that p != 0.6.





a. Evaluate the probability of a type I error.


b. Evaluate the probability of a type II error for p = 0.5 and = p = 0.7.


c. Is this a good test procedure?





If anyone could please help me out here I would really appreciate anything. Thank you.

Probability question involving the null hypothesis and type errors?
Let's have X have a binomial distribution with number of trials n = 15 and probability of success p.





We will not reject the null hypothesis if 6 %26lt;= X %26lt;= 12, so we should reject the null if X %26lt; 6 or X %26gt; 12.





a. You commit a type I error if you reject the null hypothesis, however the null was true (meaning p = 0.6).


P(type I error) = P(Reject null | null is true)


= P(X %26lt; 6 or X %26gt; 12 | p = 0.6)


= 0.060947304. This probability was determined from a binomial distribution with n = 15 and p = 0.6. I used Excel to find this probability.





b. You commit a type II error if you do not reject the nul hypothesis, but the null is not true (meaning that p is something different from 0.6).





If p = 0.5 in reality, then the null is not true, so


P(type II error) = P(do not reject null | null is not true)


= P(6 %26lt;= X %26lt;= 12 | p = 0.5)


= 0.845428467 again using a binomial with n = 15 and p = 0.5. Again I used Excel.





If p = 0.7, then again the null will not be true, so


P(type II error) = P(do not reject null | null is not true)


= P(6 %26lt;= X %26lt;= 12 | p = 0.7)


= 0.869519764 using a binomial with n = 15 and p = 0.7.





c. The probability of a type I error is reasonable. The probabilities of a type II error is quite large, even though 0.5 and 0.7 is fairly far from 0.6, so I would say that this is not a good test procedure. I would suggest taking a larger sample size.
Reply:I'm not going to answer the question, but I'll try to walk you through it.





a. Type I error is when we reject the null hypothesis when it is true. You can calculate this probability using a binominal distribution where n =15, and p = 0.6. You will reject the null hypothesis if the number of college graduates in your sample of 15 is 0, 1, 2, 3, 4, 5, 13, 14, 15. Add up the probability of getting those number of college graduates in your sample from a binomial distribution with n = 15 and p = 0.6. That will be the probability of a Type I error.





b. Type II error is when we fail to reject the null hypothesis when it is false. If p = 0.5, what is the probability of getting 6 to 12 college graduates in sample of 15. It's a binomial probability problem, again, with n =15 and p = 0.5. Use the same approach with p = 0.7.





c. Look at the size of the Type I and Type II errors. If the errors are small, then this is a good test.