Problem 2: Monte Carlo Simulation In previous labs, we have generated lots of samples and their corresponding confidence intervals for a parameter , then checked to see the proportion of the intervals that captured - the coverage. We saw coverage was 1 . Now conduct a similar task, but this time generate the test statistic associated with each sample, then, see how often the test statistic falls in the rejection region. Consider samples of size n = 30 from N( = 8, = 1). We wish to test H0 : = 8 vs. HA : < 8 at significance level = 0.05. a) Assuming the variance is unknown, what is the test statistic you would use for this hypothesis? b) What is the sampling distribution of this test statistic? Use this value along with = 0.05 to compute the rejection region.

c) Generate a large number of samples. Compute and store the test statistic for each sample. For each sample, generate the data from N( = 8, = 1), but compute your test statistic under the assumption that is unknown. d) Check to see how often your test statistics fell in the rejection region. Explain your answer and how it compares to the idea of confidence interval coverage. e) Plot the sampling distribution of your test statistics to verify it resembles the distribution you described in part b. Problem 3: Another Simulation Repeat problem 2 under the following circumstances. Consider samples of size n = 19 from N( = 8, = 1). We wish to test H0 : 2 = 1 vs. HA : > 1 at significance level = 0.01.