## [answered] Let be a random sample from a vs. distribution; with the te

please write the whole answer clearly. just need to solve problem 1 and 2!

Due Thursday, November 17th STAT 4202 HW 4 Problems 1 and 2 must be turned in; the rest are optional problems that may help you prepare for the

exam.

1. Let be a random sample from a

vs. distribution; with the test that rejects if that this test is not rejected iff is in the known. Consider testing

or if . Show CI. 2. An eighth grade reading test is constructed so that the mean should be 84.3. 45 randomly

selected eighth graders from certain school district averaged 87.8 with a standard deviation of

9.3.

a. Test to see if the students in this district are doing better than the purported average.

b. In the context of the problem, what is a type I error?

c. In the context of the problem, what is a type II error?

3. Suppose a test is rejected at

a.

?

b.

?

4. Consider testing vs. . Will it also be rejected at , where using the observed number of successes, , in is the success probability for a binomial, by

trials. a. Find an expression for the likelihood ratio statistic.

b. Show that the critical region of the likelihood ratio test can be written as

.

c. Show that the critical region can be written as . Explain why this form of the critical region makes sense.

5. Consider testing

vs.

, where is the success probability for a binomial,

by using the observed number of successes, , in 16 trials and rejecting if

. Complete the

following table (Hint:use http://stattrek.com/online-calculator/binomial.aspx): 0.95

0.90

0.85

0.80

0.75

0.70

0.65

0.60

0.55

0.50 6. Let

vs. be a random sample from a

. For a specified , a size

. If most powerful test at distribution;

known. Consider testing

test is any test satisfying , then the a test that rejects when is the ; call this Test 1. a. Now consider the test that rejects if ; call this Test 2. This is also a size test. Show that, for

,

where is the power function for Test .

Explain why this means there is no UMP test in this problem.

b. Now consider the test that rejects if

This is also a size test.

i. Show that for

ii. Show that for or if ; call this Test 3.

.

.

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