Consider again the conditions of Exercise 1. Determine the

Suppose that X1,...,Xn form a random sample from the normal distribution with unknown mean and known variance 1, and it is desired to test the following hypotheses for a given number 0: H0: = 0, H1: =

Consider again the conditions of Exercise 1, and suppose that c1 = 0 1.96n1/2. Determine

Consider again the conditions of Exercise 1 and also the test procedure described in that exercise. Determine the smallest value of n for which (0|) = 0.10 and (0 + 1|) = (0 1|)

Suppose that X1,...,Xn form a random sample from the normal distribution with unknown mean and known variance 1, an

Consider again the conditions of Exercise 4, and suppose also that n = 25. Determine the values of the constants

Suppose that X1,...,Xn form a random sample from the uniform distribution on the interval [0, ], where the value of is unknown, and it is desired to test the following hypotheses: H0: 3, H1: > 3. a. Show that for each level of significance 0 (0 0 < 1), there exists a UMP test that specifies that H0 should be rejected if max{X1,...,Xn} c. b. Det

For a given sample size n and a given value of 0, sketch the power function of the UMP test found in E

Suppose that X1,...,Xn form a random sample from the uniform distribution described in Exercise 6, but suppose now that it is desired to test the following hypotheses: H0:

For a given sample size n and a given value of 0, sketch the power function o

Suppose that X1,...,Xn form a random sample from the uniform distribution described in Exercise 6, but suppose now that it is desired to test the following hypotheses: H0: = 3, H1: = 3. (9.4.15) Consider a test procedure such that the hypothesis H0 is rejected if either max{X1,...,Xn} c1 or max{X1,..., Xn} c2, and let (|) denote the power function of . a. Determine the values of the constants c1 and c2 such that (3|) = 0.05 and is unbiased. b. Prove that the test found in part (a) is UMP of level 0.05 for testing the hypotheses in (9.4.15). Hint: Compare this test to the UM

Consider again the conditions of Exercise 1. Determine the values of the constants c1 and c2 such that (0|) = 0.10 and is unb

. Let X have the exponential distribution with parameter . Suppose that we wish to test the hypotheses H

Prove Eq. (9.4.14) in Example 9.4.6.Hint: Both parts of the integrand in Eq. (9.4.13) differ from gamma distribution p.d.f.s by some factor that does not depend on t.