# Question

Let X have a Bernoulli distribution with pmf

We would like to test the null hypothesis H0: p ≤ 0.4 against the alternative hypothesis H1: p > 0.4. For the test statistic, use

is a random sample of size n from this Bernoulli distribution. Let the critical region be of the form C = {y: y ≥ c}.

(a) Let n = 100. On the same set of axes, sketch the graphs of the power functions corresponding to the three critical regions, C1 = {y : y ≥ 40}, C2 = {y : y ≥ 50}, and C3 = {y : y ≥ 60}. Use the normal approximation to compute the probabilities.

(b) Let C = {y : y ≥ 0.45n}. On the same set of axes, sketch the graphs of the power functions corresponding to the three samples of sizes 10, 100, and 1000.

We would like to test the null hypothesis H0: p ≤ 0.4 against the alternative hypothesis H1: p > 0.4. For the test statistic, use

is a random sample of size n from this Bernoulli distribution. Let the critical region be of the form C = {y: y ≥ c}.

(a) Let n = 100. On the same set of axes, sketch the graphs of the power functions corresponding to the three critical regions, C1 = {y : y ≥ 40}, C2 = {y : y ≥ 50}, and C3 = {y : y ≥ 60}. Use the normal approximation to compute the probabilities.

(b) Let C = {y : y ≥ 0.45n}. On the same set of axes, sketch the graphs of the power functions corresponding to the three samples of sizes 10, 100, and 1000.

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