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Homework 18
Problems
Instructions: Use the test statistic method to test all hypotheses in this homework. Follow the procedure outlined on page 31 of the notes from today's lecture.
Section 9.1:
Section 9.2:
- 15
- 16
- 21 (for d, use α=0.01)
- 24 (use R to compute sample statistics, use α=0.001
- 25 (use R to compute sample statistics)
Section 9.3:
Additional Problem
Consider a random sample X1,X2,…,Xniid∼N(μ,σ2) from a Gaussian population with known population standard deviation σ. We wish to test the following hypothesis:
H0:μ≤μ0Ha:μ>μ0
at significance level α.
- Test Statistic Method:
- Determine an appropriate test statistic for this hypothesis test.
- Determine the rejection region for the appropriate test statistic for this hypothesis test.
- Confidence Interval Method:
- Determine the confidence bound that can be used to test this hypothesis. Denote the true value of μ by μ0.
- Determine the condition for rejecting the null hypothesis using the confidence bound.
- Show that the interval of values of ˉx for which the Test Statistic Method rejects the null hypothesis is equivalent to the interval of values for which the Confidence Interval Method rejects the null hypothesis.
- Hint: For each of the two methods, write the interval of values for which the null hypothesis is rejected as an inequality involving μ0 and ˉx. Then show that the two inequalities specify the same interval of values for ˉx.