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Stats 250 W15 Exam 2 SOLUTIONS 1. Protecting the Environment Part 1 ~ Some people consider themselves “green” meaning they are supportive when it comes to environmental issues. But how do people... really act? American per capita use of energy is roughly double that of Western Europeans. Would people be willing to pay more for gas to fund environmental projects? A research team at Michigan State University selected a random sample of 1000 Michigan adults and asked each if they would support an additional 2% tax on gasoline in order to fund various environmental projects. For this sample, 450 of the 1000 adults reported a willingness to do so. Create a 90% confidence interval to estimate the population proportion for all Michigan adults who would support such a gasoline tax. [3] Note that a general (not conservative) interval was requested here and z* of 1.64 or 1.65 is ok. ( ) ˆ ˆ (1 ) (0.45)(1 0.45) ˆ 0.45 1.645 1000 0.45 1.645 0.0157 0.45 0.0259 p p p z n ∗ − − ± →± ± →± Final Answer: ____0.4241______ to ____0.4759______ 2. Protecting the Environment Part 2 ~ A researcher at the University of Minnesota wanted to conduct a similar study as did Michigan State University to estimate the population proportion of Minnesota adults who would support an additional 2% tax on gasoline to fund environmental projects. a. The researcher would like to have a 95% confidence interval estimate with a width of (at most) 8%. Determine the minimum sample size that would be required. [3] Note that a width of 0.08 is the same as 2m (with a margin of error of 0.04). z* = 1.96  n = � ??∗ ????� ?? =� ??.???? ??.????� ?? = (24.5) 2 = 600.25 and ALWAYS ROUND UP If use z* of 2, then n = 625 is the final answer. Final answer: _____ 601 ________ The resulting confidence interval from the Minnesota study was reported as (0.40, 0.48). For part (b) and (c) indicate if the statement is correct or incorrect. [2] b. We are 95% confident that the population proportion of all Minnesota adults who would support an additional 2% tax on gasoline to fund environmental projects is in the interval 0.40 to 0.48. Correct Incorrect [2] c. Based on the same survey results, the width for a 90% confidence interval for the population proportion all Minnesota adults who would support an additional 2% tax on gasoline to fund environmental projects would be more than 8%. Correct Incorrect [2] d. Consider now using these same Minnesota survey results to test, at a significance level of 0.05, the null hypothesis H0: p = 0.50 against the majority hypothesis of Ha: p > 0.50. Then the resulting p-value will be (circle one) greater than less than 0.05, so that H0 (circle one) would be would not be rejected. Note: if selected “LESS THAN” as answer at start of sentence, then must circle “WOULD BE” to earn 1 of 2 points; if “LESS THAN” and “WOULD NOT BE” both points are lost as not consistent. Stats 250 W15 Exam 2 SOLUTIONS ~ Page 1 of 8 Stats 250 W15 Exam 2 SOLUTIONS 3. Poor Roads in Cleveland ~ A recent UM study reported that 25% of Michigan’s top elected or appointed state government officials rated the roads in their area as poor. The Director of Transportation and Roadways in Cleveland read this article and decides to repeat the study in Cleveland. She wishes to assess if the population proportion of all Cleveland adults rating their roads as poor is less than 25%. She approves the survey design which will take a random sample 300 Cleveland adults and use the results to test H0: p = 0.25 vs Ha: p < 0.25 at the 10% significance level. a. Given budgetary issues in Cleveland, the director recommends that the survey be piloted by initially taking just a random sample of 20 Cleveland adults and analyzing the results. Then the administration could re-evaluate whether it is reasonable to follow with a larger scale study. What distribution would be used for this pilot analysis in order to determine a p-value? Be complete. [2] Since the sample size is too small (20(0.25) = 5 which is less than 10), a small sample binomial test would need to be conducted. The test statistic would be the count statistic X and the distribution to use to find the p-value is the Bin(n=20, p=0.25). Final answer: _____ Binomial (n = 20, p = 0.25) _________ [Show More]

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