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
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
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
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
, 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
0.50 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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