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Confidence intervals Several factors are involved in the creation of a confidence interval. Among them are the sample size, the level of confidence, and the margin of error. Which statements are true? a. For a given sample size, higher confidence means a smaller margin of error. b. For a specified confidence level, larger samples provide smaller margins of error. c. For a fixed margin of error, larger samples provide greater confidence. d. For a given confidence level, halving the margin of error requires a sample twice as large.

Confidence intervals, again Several factors are involved in the creation of a confidence interval. Among them are the sample size, the level of confidence, and the margin of error. Which statements are true? a. For a given sample size, reducing the margin of error will mean lower confidence. b. For a certain confidence level, you can get a smaller margin of error by selecting a bigger sample. c. For a fixed margin of error, smaller samples will mean lower confidence. d. For a given confidence level, a sample 9 times as large will make a margin of error one third as big.

30\. Parole A study of 902 decisions (to grant parole or not) made by the Nebraska Board of Parole produced the following computer output. Assuming these cases are representative of all cases that may come before the Board, what can you conclude? z-Interval for proportion With \(95.00 \%\) confidence, $$ 0.56100658<\mathrm{P}(\text { parole })<0.62524619 $$

31\. Mislabeled seafood In 2013 the environmental group Oceana (usa.oceana.org) analyzed 1215 samples of seafood purchased across the United States and genetically compared the pieces to standard gene fragments that can identify the species. Laboratory results indicated that \(33 \%\) of the seafood was mislabeled according to U.S. Food and Drug Administration guidelines. a. Construct a \(95 \%\) confidence interval for the proportion of all seafood sold in the United States that is mislabeled or misidentified. b. Explain what your confidence interval says about seafood sold in the United States. c. A 2009 report by the Government Accountability Office says that the Food and Drug Administration has spent very little time recently looking for seafood fraud. Suppose an official said, "That's only 1215 packages out of the billions of pieces of seafood sold in a year. With the small number tested, I don't know that one would want to change one's buying habits." (An official was quoted similarly in a different but similar context). Is this argument valid? Explain.

Baseball fans In a poll taken in December 2012, Gallup asked 1006 national adults whether they were baseball fans; \(48 \%\) said they were. Almost five years earlier, in February \(2008,\) only \(35 \%\) of a similar-size sample had reported being baseball fans. a. Find the margin of error for the 2012 poll if we want \(90 \%\) confidence in our estimate of the percent of national adults who are baseball fans. b. Explain what that margin of error means. c. If we wanted to be \(99 \%\) confident, would the margin of error be larger or smaller? Explain. d. Find that margin of error. e. In general, if all other aspects of the situation remain the same, will smaller margins of error produce greater or less confidence in the interval?

34\. Still living online The Pew Research poll described in Exercise 5 ? found that \(56 \%\) of a sample of 1060 teens go online several times a day. (Treat this as a Simple Random Sample.) a. Find the margin of error for this poll if we want \(95 \%\) confidence in our estimate of the percent of American mteens who go online several times a day. b. Explain what that margin of error means. c. If we only need to be \(90 \%\) confident, will the margin of error be larger or smaller? Explain. d. Find that margin of error. e. In general, if all other aspects of the situation remain the same, would smaller samples produce smaller or larger margins of error?

Contributions, please The Paralyzed Veterans of America is a philanthropic organization that relies on contributions. They send free mailing labels and greeting cards to potential donors on their list and ask for a voluntary contribution. To test a new campaign, they recently sent letters to a random sample of 100,000 potential donors and received 4781 donations. a. Give a \(95 \%\) confidence interval for the true proportion of their entire mailing list who may donate. b. A staff member thinks that the true rate is \(5 \%\). Given the confidence interval you found, do you find that percentage plausible?

Teenage drivers An insurance company checks police records on 582 accidents selected at random and notes that teenagers were at the wheel in 91 of them. a. Create a \(95 \%\) confidence interval for the percentage of all auto accidents that involve teenage drivers. b. Explain what your interval means. c. Explain what "95\% confidence" means. d. A politician urging tighter restrictions on drivers' licenses issued to teens says, "In one of every five auto accidents, a teenager is behind the wheel." Does your confidence interval support or contradict this statement? Explain.

Junk mail Direct mail advertisers send solicitations (a.k.a. "junk mail") to thousands of potential customers in the hope that some will buy the company's product. The acceptance rate is usually quite low. Suppose a company wants to test the response to a new flyer, and sends it to 1000 people randomly selected from their mailing list of over 200,000 people. They get orders from 123 of the recipients. a. Create a \(90 \%\) confidence interval for the percentage of people the company contacts who may buy something. b. Explain what this interval means. c. Explain what "90\% confidence" means. d. The company must decide whether to now do a mass mailing. The mailing won't be cost-effective unless it produces at least a \(5 \%\) return. What does your confidence interval suggest? Explain.

Death penalty, again In the survey on the death penalty you read about in the Step-by-Step Example, the Gallup Poll actually split the sample at random, asking 510 respondents the question quoted earlier, "Generally speaking, do you believe the death penalty is applied fairly or unfairly in this country today?" The other 510 were asked, "Generally speaking, do you believe the death penalty is applied unfairly or fairly in this country today?" Seems like the same question, but sometimes the order of the choices matters. Suppose that for the second way of phrasing it, \(64 \%\) said they thought the death penalty was fairly applied. (Recall that \(53 \%\) of the original 510 thought the same thing.) a. What kind of bias may be present here? b. If we combine them, considering the overall group to be one larger random sample of 1020 respondents, what is a \(95 \%\) confidence interval for the proportion of the general public that thinks the death penalty is being fairly applied? c. How does the margin of error based on this pooled sample compare with the margins of error from the separate groups? Why?