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[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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