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:

• 4
• 5

Section 9.2:

• 15
• 16
• 21 (for d, use \(\alpha = 0.01\))
• 24 (use R to compute sample statistics, use \(\alpha = 0.001\)
• 25 (use R to compute sample statistics)

Section 9.3:

• 36
• 37a
• 38

Additional Problem

Consider a random sample \(X_{1}, X_{2}, \ldots, X_{n} \stackrel{\text{iid}}{\sim} N(\mu, \sigma^{2})\) from a Gaussian population with known population standard deviation \(\sigma\). We wish to test the following hypothesis: \[ \begin{array}{ll} H_{0} : & \mu \leq \mu_{0} \\ H_{a} : & \mu > \mu_{0}\end{array}\] at significance level \(\alpha\).

1. Test Statistic Method:
   1. Determine an appropriate test statistic for this hypothesis test.
   2. Determine the rejection region for the appropriate test statistic for this hypothesis test.
2. Confidence Interval Method:
   1. Determine the confidence bound that can be used to test this hypothesis. Denote the true value of \(\mu\) by \(\mu_{0}\).
   2. Determine the condition for rejecting the null hypothesis using the confidence bound.
3. Show that the interval of values of \(\bar{x}\) 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 \(\mu_{0}\) and \(\bar{x}\). Then show that the two inequalities specify the same interval of values for \(\bar{x}\).