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Chapter 8: Problem 54

A \(U S A\) Today/CNN/Gallup survey of 369 working parents found 200 who said they spend too little time with their children because of work commitments. a. What is the point estimate of the proportion of the population of working parents who feel they spend too little time with their children because of work commitments? b. At \(95 \%\) confidence, what is the margin of error? c. What is the \(95 \%\) confidence interval estimate of the population proportion of working parents who feel they spend too little time with their children because of work commitments?

Short Answer

Expert verified
In summary, the point estimate for the proportion of working parents who feel they spend too little time with their children due to work commitments is approximately 0.5420. The margin of error at a 95% confidence level is approximately 0.0509. Therefore, the 95% confidence interval estimate of the population proportion is between 0.4911 and 0.5929, or 49.11% to 59.29%.

Step by step solution

01

Calculate the point estimate for the proportion of working parents who spend too little time with their children due to work commitments

First, we'll find the point estimate for the proportion of working parents who feel this way. To do this, we'll use the first formula mentioned in the analysis. In this case, the successful outcomes are 200 parents, and the total number of outcomes is 369 parents. Point estimate: \(p = \frac{200}{369} \approx 0.5420\)
02

Determine the Margin of Error

Now, we'll determine the margin of error at a 95% confidence level. This corresponds to a critical value of \(Z_{\frac{\alpha}{2}} = 1.96\). We'll use the second formula mentioned in the analysis using the point estimate and sample size. Margin of error: \(ME = 1.96 * \sqrt{\frac{0.5420(1 - 0.5420)}{369}} \approx 1.96 * 0.0260 \approx 0.0509\)
03

Calculate the 95% Confidence Interval

Finally, we'll calculate the 95% confidence interval estimate of the population proportion. We'll use the third formula mentioned in the analysis using the point estimate and margin of error. Confidence interval: (\(0.5420 - 0.0509, 0.5420 + 0.0509\)) = (\(0.4911, 0.5929\)) So, based on the given sample of working parents, we can be 95% confident that the proportion of the population of working parents who feel they spend too little time with their children because of work commitments is between 49.11% and 59.29%.

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Most popular questions from this chapter

A Phoenix Wealth Management/Harris Interactive survey of 1500 individuals with net worth of \(\$ 1\) million or more provided a variety of statistics on wealthy people (Business Week, September 22,2003 ). The previous three-year period had been bad for the stock market, which motivated some of the questions asked. a. The survey reported that \(53 \%\) of the respondents lost \(25 \%\) or more of their portfolio value over the past three years. Develop a \(95 \%\) confidence interval for the proportion of wealthy people who lost \(25 \%\) or more of their portfolio value over the past three years. b. The survey reported that \(31 \%\) of the respondents feel they have to save more for retirement to make up for what they lost. Develop a \(95 \%\) confidence interval for the population proportion. c. Five percent of the respondents gave \(\$ 25,000\) or more to charity over the previous year. Develop a \(95 \%\) confidence interval for the proportion who gave \(\$ 25,000\) or more to charity. d. Compare the margin of error for the interval estimates in parts (a), (b), and (c). How is the margin of error related to \(\bar{p} ?\) When the same sample is being used to estimate a variety of proportions, which of the proportions should be used to choose the planning value \(p^{*} ?\) Why do you think \(p^{*}=.50\) is often used in these cases?

The travel-to-work time for residents of the 15 largest cities in the United States is reported in the 2003 Information Please Almanac. Suppose that a preliminary simple random sample of residents of San Francisco is used to develop a planning value of 6.25 minutes for the population standard deviation. a. If we want to estimate the population mean travel-to-work time for San Francisco residents with a margin of error of 2 minutes, what sample size should be used? Assume \(95 \%\) confidence b. If we want to estimate the population mean travel-to-work time for San Francisco residents with a margin of error of 1 minute, what sample size should be used? Assume \(95 \%\) confidence

Playbill magazine reported that the mean annual household income of its readers is \(\$ 119,155(\text {Playbill}, \text { January } 2006) .\) Assume this estimate of the mean annual household income is based on a sample of 80 households, and, based on past studies, the population standard deviation is known to be \(\sigma=\$ 30,000\) a. Develop a \(90 \%\) confidence interval estimate of the population mean. b. Develop a \(95 \%\) confidence interval estimate of the population mean. c. Develop a \(99 \%\) confidence interval estimate of the population mean. d. Discuss what happens to the width of the confidence interval as the confidence level is increased. Does this result seem reasonable? Explain.

A well-known bank credit card firm wishes to estimate the proportion of credit card holders who carry a nonzero balance at the end of the month and incur an interest charge. Assume that the desired margin of error is .03 at \(98 \%\) confidence. a. How large a sample should be selected if it is anticipated that roughly \(70 \%\) of the firm's card holders carry a nonzero balance at the end of the month? b. How large a sample should be selected if no planning value for the proportion could be specified?

A poll for the presidential campaign sampled 491 potential voters in June. A primary purpose of the poll was to obtain an estimate of the proportion of potential voters who favored each candidate. Assume a planning value of \(p^{*}=.50\) and a \(95 \%\) confidence level. a. For \(p^{*}=.50,\) what was the planned margin of error for the June poll? b. Closer to the November election, better precision and smaller margins of error are desired. Assume the following margins of error are requested for surveys to be conducted during the presidential campaign. Compute the recommended sample size for each survey.
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