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Homework 13 Due Tuesday, April 26, 11:59pm STAT 400, Spring 2022, D. Unger * Late Homework deadline is Wednesday, April 27 at 12:00pm, noon, so that solutions may be made available prior to Thursday’s exam. Each Exercise or lettered part of an Exercise is worth 5 points. Homework assignments are worth 50 points. Exercise 1 Several years ago, ballots in Champaign-Urbana contained the following question to assess public opinion on an issue: “Should the State of Illinois legalize and regulate the sale and use of marijuana in a similar fashion as the State of Colorado?” Suppose that we would like to understand Champaign-Urbana’s opinion on marijuana legalization. Your friend claims that among people she talks to, at least three out of every five people support legalization. To satisfy your curiosity, we obtain a random sample of 120 Champaign-Urbana residents and find that 87 support marijuana legalization. (a) State the null and alternative hypotheses, calculate the test statistic, and provide the p-value. (b) Using a level of significance of α = 0.01, make a decision about the null hypothesis and provide a concluding statement in the context of this situation. (c) In Homework 10, you constructed a 99% (two-sided) confidence interval for the true proportion of Champaign-Urbana residents who support legalizing marijuana. Using that as evidence for the hypothesis test in part a, do you come to the same conclusion that you did in part b? Explain. Exercise 2 Several years ago, ballots in Champaign-Urbana contained the following question to assess public opinion on an issue: “Should the State of Illinois legalize and regulate the sale and use of marijuana in a similar fashion as the State of Colorado?” Suppose we obtain a random sample of 80 Champaign voters, of which 55 support marijuana legalization. We also obtain a random sample of 100 Urbana voters, of which 75 support marijuana legalization. Let pC be the true proportion of Champaign voters whosupport marijuana legalization, and let pU be the true proportion of Urbana voters who support marijuana legalization. (a) Calculate a 95% confidence interval for pU – pC. (b) Calculate the p-value for the test H0: pU = pC versus H1: pU ≠ pC. Using a level of significance of α = 0.05, make a decision about the null hypothesis and provide a concluding statement in the context of this situation. Exercise 3 A behavioral scientist is interested in the effect of caffeine on the typing speed of students. The scientist obtains a random sample of 8 students who are given 400 mg of caffeine then given a typing test. They type an average of 51.4 words per minute (wpm), with a sample standard deviation of 12.3 wpm. He also obtains a random sample of 13 students who are given a placebo before the typing test. The placebo group types an average of 43.9 wpm, with a sample standard deviation of 15.1 wpm. Assume typing speeds follow a normal distribution in both groups. (a) Construct a 99% confidence interval for μC − μP, the true difference in average typing speed between the caffeine and placebo groups. (Assume that the two population variances are equal.) (b) Construct a 99% confidence interval for μC − μP, the true difference in average typing speed between the caffeine and placebo groups. (Do not assume that the two population variances are equal. Use Welch’s T.) (c) Calculate the value of the test statistic for testing H0: μC = μP versus H1: μC ≠ μP. (Assume that the two population variances are equal.) (d) State the critical region for the test in part c, and your statistical decision using α=0.05. Provide a concluding statement in the context of this situation. (e) Calculate the p-value for the test in part c. Does a decision based on p-value agree with your result from part d?

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