91Ó°ÊÓ

According to data released in 2016 , \(69 \%\) of students in the United States enroll in college directly after high school graduation. Suppose a sample of 200 recent high school graduates is randomly selected. After verifying the conditions for the Central Limit Theorem are met, find the probability that at most \(65 \%\) enrolled in college directly after high school graduation. (Source: nces.ed.gov)

Human blood is divided into 8 possible blood types. The rarest blood type is AB negative. Only \(1 \%\) of the population has this blood type. Suppose a random sample of 50 people is selected. Can we find the probability that more than \(3 \%\) of the sample has AB negative blood? If so, find the probability. If not, explain why this probability cannot be calculated.

According to a 2017 Gallup Poll, 617 out of 1028 randomly selected adults living in the United States felt the laws covering the sale of firearms should be more strict. a. What is the value of \(\hat{p}\), the sample proportion who favor stricter gun laws? b. Check the conditions to determine whether the CLT can be used to find a confidence interval. c. Find a \(95 \%\) confidence interval for the population proportion who favor stricter gun laws. d. Based on your confidence interval, do a majority of Americans favor stricter gun laws?

A random sample of likely voters showed that \(55 \%\) planned to vote for Candidate \(\mathrm{X}\), with a margin of error of 2 percentage points and with a \(95 \%\) confidence level. a. Use a carefully worded sentence to report the \(95 \%\) confidence interval for the percentage of voters who plan to vote for Candidate \(\mathrm{X}\). b. Is there evidence that Candidate \(\mathrm{X}\) could lose. c. Suppose the survey was taken on the streets of New York City and the candidate was running for U.S. president. Explain how that would affect your conclusion.

A random sample of likely voters showed that \(49 \%\) planned to support Measure \(\mathrm{X}\). The margin of error is 3 percentage points with a \(95 \%\) confidence level. a. Using a carefully worded sentence, report the \(95 \%\) confidence interval for the percentage of voters who plan to support Measure \(X\). b. Is there evidence that Measure X will fail? c. Suppose the survey was taken on the streets of Miami and the measure was a Florida statewide measure. Explain how that would affect your conclusion.

In a simple random sample of 1200 Americans age 20 and over, the proportion with diabetes was found to be \(0.115\) (or \(11.5 \%)\). a. What is the standard error for the estimate of the proportion of all Americans age 20 and over with diabetes? b. Find the margin of error, using a \(95 \%\) confidence level, for estimating this proportion. c. Report the \(95 \%\) confidence interval for the proportion of all Americans age 20 and over with diabetes. d. According to the Centers for Disease Control and Prevention, nationally, \(10.7 \%\) of all Americans age 20 or over have diabetes. Does the confidence interval you found in part c support or refute this claim? Explain.

The Gallup poll reported that \(45 \%\) of Americans have tried marijuana. This was based on a survey of 1021 Americans and had a margin of error of plus or minus 5 percentage points with a \(95 \%\) level of confidence. a. State the survey results in confidence interval form and interpret the interval. b. If the Gallup Poll was to conduct 100 such surveys of 1021 Americans, how many of them would result in confidence intervals that did not include the true population proportion? c. Suppose a student wrote this interpretation of the interval: "We are \(95 \%\) confident that the percentage of Americans who have tried marijuana is between \(40 \%\) and \(50 \% .\) " What, if anything, is incorrect in this interpretation?

In the 1960 presidential election, \(34,226,731\) people voted for Kennedy, \(34,108,157\) for Nixon, and 197,029 for third-party candidates (Source: www.uselectionatlas.org). a. What percentage of voters chose Kennedy? b. Would it be appropriate to find a confidence interval for the proportion of voters choosing Kennedy? Why or why not?

In a 2017 Harris poll conducted for Uber Eats, 438 of 1019 U.S. adults polled said they were "picky eaters." a. What proportion of the respondents said they were picky eaters? b. Find a \(95 \%\) confidence interval for the population proportion of U.S. adults who say they are picky eaters. c. Would a \(90 \%\) confidence interval based on this sample be wider or narrower than the \(95 \%\) interval? Give a reason for your answer. d. Construct the \(90 \%\) confidence interval. Was your conclusion in part c correct?

Of 1019 U.S. adults responding to a 2017 Harris poll, \(47 \%\) said they always or often read nutrition labels when grocery shopping. a. Construct a \(95 \%\) confidence interval for the population proportion of U.S. adults who always or often read nutrition labels when grocery shopping. b. What is the width of the \(95 \%\) confidence interval? c. Name a confidence level that would produce an interval wider than the 95\% confidence interval. Explain why you think this interval would be wider than a \(95 \%\) confidence interval. d. Construct the interval using the confidence level you proposed in part c and find the width of the interval. Is this interval wider than the \(95 \%\) confidence interval?