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Chapter 9: Problem 10

For each of the following choices, explain which would result in a wider large-sample confidence interval for \(p\) : a. \(90 \%\) confidence level or \(95 \%\) confidence level b. \(n=100\) or \(n=400\)

Short Answer

Expert verified
For choice a, the \(95\%\) confidence level results in a wider confidence interval. For choice b, \(n=100\) results in a wider confidence interval.

Step by step solution

01

Analyze Choice A

The wider confidence interval will correspond to the higher confidence level. This is due to the fact that the higher the confidence level, the greater the standard error, which results in a wider confidence interval. So, choice a. \(95\%\) confidence level would result in a wider large-sample confidence interval for \(p\).
02

Analyze Choice B

The size of the confidence interval is inversely proportional to the square root of the sample size. Hence, increasing the sample size decreases the standard error, which in turn makes the confidence interval narrower. So, in choice b, \(n=100\) would result in a wider large-sample confidence interval for \(p\).

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Most popular questions from this chapter

USA Today (October 14, 2002) reported that \(36 \%\) of adult drivers admit that they often or sometimes talk on a cell phone when driving. This estimate was based on data from a sample of 1004 adult drivers, and a bound on the error of estimation of \(3.1 \%\) was reported. Assuming a \(95 \%\) confidence level, do you agree with the reported bound on the error? Explain.

The paper "The Curious Promiscuity of Queen Honey Bees (Apis mellifera): Evolutionary and Behavioral Mechanisms" (Annals of Zoology Fennici [2001]:255- 265) describes a study of the mating behavior of queen honeybees. The following quote is from the paper: Queens flew for an average of \(24.2 \pm 9.21\) minutes on their mating flights, which is consistent with previous findings. On those flights, queens effectively mated with \(4.6 \pm 3.47\) males (mean \(\pm \mathrm{SD})\). a. The intervals reported in the quote from the paper were based on data from the mating flights of \(n=\) 30 queen honeybees. One of the two intervals reported is stated to be a confidence interval for a population mean. Which interval is this? Justify your choice. b. Use the given information to construct a \(95 \%\) confidence interval for the mean number of partners on a mating flight for queen honeybees. For purposes of this exercise, assume that it is reasonable to consider these 30 queen honeybees as representative of the population of queen honeybees.

In spite of the potential safety hazards, some people would like to have an Internet connection in their car. A preliminary survey of adult Americans has estimated this proportion to be somewhere around. 30 (USA Today. May 1,2009 ). a. Use the given preliminary estimate to determine the sample size required to estimate the proportion of adult Americans who would like an Internet connection in their car to within \(.02\) with \(95 \%\) confidence. b. The formula for determining sample size given in this section corresponds to a confidence level of \(95 \%\). How would you modify this formula if a \(99 \%\) confidence level was desired? c. Use the given preliminary estimate to determine the sample size required to estimate the proportion of adult Americans who would like an Internet connection in their car to within \(.02\) with \(99 \%\) confidence.

An article in the Chicago Tribune (August \(29 .\) 1999) reported that in a poll of residents of the Chicago suburbs, \(43 \%\) felt that their financial situation had improved during the past year. The following statement is from the article: "The findings of this Tribune poll are based on interviews with 930 randomly selected suburban residents. The sample included suburban Cook County plus DuPage, Kane, Lake, McHenry, and Will Counties. In a sample of this size, one can say with \(95 \%\) certainty that results will differ by no more than \(3 \% \mathrm{t}\) from results obtained if all residents had been included in the poll." Comment on this statement. Give a statistical argument to justify the claim that the estimate of \(43 \%\) is within \(3 \%\) of the true proportion of residents who feel that their financial situation has improved.

In a study of 1710 schoolchildren in Australia (Herald Sun, October 27,1994 ), 1060 children indicated that they normally watch TV before school in the morning. (Interestingly, only \(35 \%\) of the parents said their children watched TV before school!) Construct a \(95 \%\) confidence interval for the true proportion of Australian children who say they watch TV before school. What assumption about the sample must be true for the method used to construct the interval to be valid?
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