Week 13 ‐ Discussion Questions STAT 400, Spring 2022, D. Unger Table of Contents Exercise 1 (Hypothesis Test for p) ............................................................................................................... 1 Exercise 2 (Hypothesis Test for p) ............................................................................................................... 1 Exercise 3 (Hypothesis Test for µ1 – µ2) .................................................................................................... 2 Exercise 4 (Hypothesis Test for µ1 – µ2) .................................................................................................... 2 Exercise 5 (Hypothesis Test for p1 – p2) .................................................................................................... 3 Exercise 6 (Hypothesis Test for p1 – p2) .................................................................................................... 3 Exercise 7 (Hypothesis Test for σ) ............................................................................................................... 4 Exercise 8 (Hypothesis Test for σ) ............................................................................................................... 4 Exercise 1 (Hypothesis Test for p) In a random sample of 160 students from Faber College, 56 believe that resume inflation (i.e., misrepresenting yourself on your resume) is unethical. (a) Find the p-value of the test H0: p = 0.40 versus H1: p < 0.40. (b) Is there significant evidence to suggest that fewer than 2 out of every 5 students think resume inflation is unethical at the 0.10 level of significance? Exercise 2 (Hypothesis Test for p) An economist states that 10% of Champaign-Urbana’s labor force is unemployed. A random sample of 400 people in the labor force is obtained, of whom 28 are unemployed. (a) Test whether Champaign-Urbana’s unemployment rate is as the economist claims or is different than the claim at a 1% level of significance using the critical region method for evidence. (b) Find the p-value of the test in part a. (c) Using the p-value from part b, state your decision at α = 0.05.Exercise 3 (Hypothesis Test for µ1 – µ2) A random sample of 9 adult white rhinos had the sample mean weight of 5,100 pounds and the sample standard deviation of 450 pounds. A random sample of 16 adult hippos had the sample mean weight of 3,300 pounds and the sample standard deviation of 400 pounds. Assume that the two populations are approximately normally distributed. (a) Construct a 95% confidence interval for the difference between their overall average weights of adult white rhinos and adult hippos. i. Assume the population standard deviations are equal. ii. Assume the population standard deviations are not equal, and use the conservative approach. iii. Assume the population standard deviations are not equal, and use Welch’s T. (b) It’s believed that on average an adult white rhino weighs 1,500 pounds more than an adult hippo. Test this claim against an alternative hypothesis that the average weight difference is greater than 1,500 pounds at an α = 0.05 level of significance. Assume that the population standard deviations are equal. Exercise 4 (Hypothesis Test for µ1 – µ2) A researcher wishes to determine whether the starting salaries of high-school math teachers in private schools are higher than those of high-school math teachers in public schools. She selects a sample of new math teachers from each type of school and calculates the sample means and sample standard deviations of their salaries. For private schools, the sample of size 10 yielded a sample mean of $36,800 and sample standard deviation of $600. For public schools, the sample of size 7 yielded a sample mean of $36,300 and sample standard deviation of $546. Assume that the populations are normally distributed and the population variances are equal. (a) Construct a 95% confidence interval for the difference in average starting salaries of high-school math teachers in private and public schools. (b) By the null hypothesis, there would be no difference between the salaries of the two groups. Test the researcher’s theory that the starting salaries of high-school math teachers in private schools are higher than those of high-school math teachers in public schools. Use a 1% level of significance and the critical region method. (c) Find the p-value of the test in part b.Exercise 5 (Hypothesis Test for p1 – p2) Students wishing to become actuaries must take a series of professional examinations. The first is called Exam P (Probability) and the second is called Exam FM (Financial Mathematics). In a random sample of 200 students who took Exam P last year, 74 successfully passed the exam. Suppose also that in a random sample of 150 students who took Exam FM, 45 successfully passed the exam (a) Construct a 95% confidence interval for the difference between the overall proportions of students who passed Exam P and Exam FM last year. (b) At a 10% level of significance, test whether Exam P had a higher success rate than Exam FM last year. Find the p-value of this test. Exercise 6 (Hypothesis Test for p1 – p2) In a comparative study of two new pain relief drugs, labeled A and B, 120 patients were treated with drug A and 150 patients with drug B, and the following results were obtained. Drug A Drug B Pain relief within 10 minutes 78 111 No pain relief within 10 minutes / pain relief after more than 10 minutes 42 39 Total 120 150 (a) Construct a 95% confidence interval for the difference in pain relief rates of the two drugs. (b) We wish to test whether drug B has a better pain relief rate than drug A. Find the p-value of the appropriate test and make a decision at the α = 0.05 level of significance. Exercise 7 (Hypothesis Test for σ) An examination of the records for a random sample of 16 motor vehicles in a large fleet resulted in the sample mean operating cost of 26.33 cents per mile and the sample standard deviation of 2.80 cents per mile. (Assume that operating costs are approximately normally distributed.) (a) Test whether the overall standard deviation of the operating costs is more than 2.30 cents per mile or not at a 5% significance level. (b) Using the Chi-Square Distribution table, what is the p-value of the test in part a? (You may give a range.) (c) Use a computer with R or an appropriate calculator to find the p-value of
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