Question Details
[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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