Testing of a hypothesis based on a single sample

The 3 problems are from the topic "Test of Hypothesis Based on a Single Sample" of Probability and Statistic. I need assistance with these problems. Need to find out the correct solution and answers.

9. Two different companies have applied to provide cable television service in a certain region. Let p denote the proportion of all potential subscribers who favor the first company over the second. Consider testing H_0:p=.5 versus H_a:p≠.5 based on a random sample of 25 individuals. Let X denote the number in the sample who favor the first company and x represent the observed value of X.

a) Which of the following rejection regions is most appropriate and why?
R_1= {x: x ≤ 7 or x ≥ 18}, R_2= {x: x ≤ 8},
R_3 = {x: x ≥ 17}

b) In the context of this problem situation, describe what type I and type II errors are.

c) What is the probability distribution of the test statistic X when H_0 is true? Use it to compute the probability of a type I error.

d) Compute the probability of a type II error for the selected region when p = .3, again when p = .4, and also for both p = .6 and p = .7.

e) Using the selected region, what would you conclude if 6 of the 25 queried favored company 1?

10. A mixture of pulverized fuel ash and Portland cement to be used for grouting should have a compressive strength of more than 1300 KN/m^2. The mixture will not be used unless experimental evidence indicates conclusively that the strength specification has been met. Suppose compressive strength for specimens of this mixture is normally distributed with o = 60. Let u denote the true average compression strength.

a) What are the appropriate null and alternative hypotheses?

b) Let xbar denote the sample average compressive strength for n = 20 randomly selected specimens. Consider the test procedure with test statistic xbar and rejection region xbar ≥ 1331.26. What is the probability distribution of the test statistic when H_0 is true? What is the probability of a type I error for the test procedure?

c) What is the probability distribution of the test statistic when u = 1350? Using the test procedure of part (b), what is the probability that the mixture will be judged unsatisfactory when in fact u = 1350 (a type II error)?

d) How would you change the test procedure of part (b) to obtain a test with a significance level of .05? What impact would this change have on the error probability of part (c)?

e) Consider the standardized test statistic Z = (Xbar - 1300)/(o/√n) = (Xbar) - 1300)/13.42. What are the values of Z corresponding to the rejection region of part (b)?

13. Let X_1,...,X_n denote a random sample from a normal population distribution with a known value of o.

a) For testing the hypothesis H_0: u = u_0 versus H_a: u > u_0 (where u_0 is a fixed number), show that the test with test statistic Xbar and rejection region of xbar ≥ u_0 + 2.33o/√n has significance level .01.

b) Suppose the procedure of part (a) is used to test H_0: u ≤ u_0 versus H_a: u > u_0. If u_0 = 100, n = 25, and o = 5, what is the probability of committing a type I error when u = 99? When u = 98? In general, what can be said about the probability of a type I error when the actual value of u is less than u_0? Verify your assertion.