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Gambling A city ballot includes a local initiative that would legalize gambling. The issue is hotly contested, and two groups decide to conduct polls to predict the outcome. The local newspaper finds that \(53 \%\) of 1200 randomly selected voters plan to vote "yes," while a college statistics class finds \(54 \%\) of 450 randomly selected voters in support. Both groups will create \(95 \%\) confidence intervals. a. Without finding the confidence intervals, explain which one will have the larger margin of error. b. Find both confidence intervals. c. Which group concludes that the outcome is too close to call? Why?

Back to campus ACT, Inc. reported that \(74 \%\) of 1644 randomly selected college freshmen returned to college the next year. The study was stratified by type of college-public or private. The retention rates were \(71.9 \%\) among 505 students enrolled in public colleges and \(74.9 \%\) among 1139 students enrolled in private colleges. a. Will the \(95 \%\) confidence interval for the true national retention rate in private colleges be wider or narrower than the \(95 \%\) confidence interval for the retention rate in public colleges? Explain. b. Do you expect the margin of error for the overall retention rate to be larger or smaller? Explain.

Deer ticks Wildlife biologists inspect 153 deer taken by hunters and find 32 of them carrying ticks that test positive for Lyme disease. a. Create a \(90 \%\) confidence interval for the percentage of deer that may carry such ticks. b. If the scientists want to cut the margin of error in half, how many deer must they inspect? c. What concerns do you have about this sample?

Pilot study A state's environmental agency worries that many cars may be violating clean air emissions standards. The agency hopes to check a sample of vehicles in order to estimate that percentage with a margin of error of \(3 \%\) and \(90 \%\) confidence. To gauge the size of the problem, the agency first picks 60 cars and finds 9 with faulty emissions systems. How many should be sampled for a full investigation?

Another pilot study During routine screening, a doctor notices that \(22 \%\) of her adult patients show higher than normal levels of glucose in their blood-a possible warning signal for diabetes. Hearing this, some medical researchers decide to conduct a large-scale study, hoping to estimate the proportion to within \(4 \%\) with \(98 \%\) confidence. How many randomly selected adults must they test?