Consider the confidence interval found in Exercise 5 in

. Let X have the exponential distribution with parameter . Suppose that we wish to test the hypotheses H0 : 1 versus H1 : < 1. Consider the test procedure that rejects H0 if X 1. a. Dete

Suppose that X1,...,Xn form a random sample from the uniform distribution on the interval [0, ], and that the following hyp

Suppose that the proportion p of defective items in a large population of items is unknown, and that it is desired to test the following hypotheses: H0: p = 0.2, H1: p = 0.2. Suppose also that a rando

Suppose that X1,...,Xn form a random sample from the normal distribution with unknown mean and known variance 1. Suppose also that 0 is a certain specified number, and that the following hypotheses are to be tested: H0: = 0, H1: = 0. Finally, suppose that the samp

Suppose that X1,...,Xn form a random sample from the normal distribution with unknown mean and unknown variance 2. Classify each of the following hypotheses as either simple or composite: a. H0: = 0 and = 1 b. H0: > 3 an

. Suppose that a single observation X is to be taken from the uniform distribution on the interval # 1 2 , + 1 2 $ , and suppose that the following hypotheses are to be tested: H0: 3, H1:

Return to the situation described in Example 9.1.7. Consider a different test that rejects H0 if Yn 2.9 or Yn 4.5. Let be the test described in Example 9.1.7. a. Prove that (|) = (|) for all 4. b. Prove that (|) < (|) for all > 4. c. Which of the two test

Assume that X1,...,Xn are i.i.d. with the normal distribution that has mean and variance 1. Suppose that we wish to test the hypotheses H0: 0, H1:

Assume that X1,...,Xn are i.i.d. with the normal distribution that has mean and variance 1. Suppose that we wish to test the hypotheses H0: 0, H1: <0. Find a test statistic T such that, for every c, the test c that rejects H0 when T c

In Exercise 8, assume that Z = z is observed. Find a formula for the p-value

Assume that X1,...,X9 are i.i.d. having the Bernoulli distribution with parameter p. Suppose that we wish to test the hypotheses H0: p

Consider a single observation X from a Cauchy distribution centered at . That is, the p.d.f. of X is f (x| ) = 1 [1 + (x )2] , for

3. Let X have the Poisson distribution with mean . Suppose that we wish to test the hypotheses H0: 1.0, H1: > 1.0.

Let X1,...,Xn be i.i.d. with the exponential distribution with parameter . Suppose that we wish to test the hypotheses H0: 0, H1: <0. Let X = n i=1 Xi. Let c be the test that rejects H0 if X c. a. Show that (|c) is a d

Let X have the uniform distribution on the interval [0, ], and suppose that we wish to test the hypotheses H0: 1, H1:

Consider the confidence interval found in Exercise 5 in Sec. 8.5. Find the collection of hypothesis tests that are equivalent to this interval. That is, for each c > 0, find a test c of the null hypothesis H0,c : 2 = c versus some alternative such that c rejects H0,c if and only if c is not in the interval. Write the test in terms of a test statistic T = r(X) bei

Let X1,...,Xn be i.i.d. with a Bernoulli distribution that has parameter p. Let Y = n i=1 Xi. We wish to find a coefficient confidence interval for p of the form (q(y), 1). Prove that, if Y = y is observed, then q(y) should be chosen to be the smallest value p0 such that Pr(Y y|p = p0) 1

Consider the situation described immediately before Eq. (9.1.12). Prove that the expression (9.1.12) equals the smallest 0 such that we would reject H0 at level of significance 0.

Return to the situation described in Example 9.1.17. Suppose that we wish to test the hypotheses H0: 0, H1: <0 (9.1.27) at level 0. It makes sense to reject H0 if Xn is small. Construct a one-sided coefficient 1 0 confidence interval for such that we can reject H0 if 0 is not in the interval. Make sur

Prove Theorem 9.1.3.

Return to the situations described in Example 9.1.17 and Exercise 19. We wish to compare what might happen if we switch the null and alternative hypotheses. That is, we want to compare the results of testing the hypotheses in (9.1.22) at level 0 to the results of testing the hypotheses in (9.1.27) at level 0. a. Let 0 < 0.5. Prove that there are no possible data sets such that we would reject both of the null hypotheses simultaneously. That is, for every possible Xn and