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 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 who support 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 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 C P 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 C P 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 C P versus H1 C P Assume that the two population variances are equal d State the critical region for the test in part c and your statistical decision using 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 Explain
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