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Chapter 9: Problem 15

A researcher wants to estimate the proportion of city residents who favor spending city funds to promote tourism. Would the standard error of the sample proportion \(\hat{p}\) be smaller for random samples of size \(n=100\) or random samples of size \(n=200 ?\)

Short Answer

Expert verified
The standard error of the sample proportion \(\hat{p}\) would be smaller for random samples of size \(n=200\) compared to random samples of size \(n=100\). This means that using a larger sample size will yield more accurate estimates of the true proportion of city residents who favor spending city funds on promoting tourism.

Step by step solution

01

Understand the formula for the standard error of the sample proportion

The formula for the standard error of the sample proportion \(\hat{p}\) is given by: \[\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}\] where \(p\) is the population proportion, and \(n\) is the sample size.
02

Find the standard error for a sample size of \(n=100\)

We can plug in \(n=100\) into the standard error formula. Since we don't know the value of \(p\), we can't compute the exact standard error, but we can write the general expression: \[\sigma_{\hat{p}} (\text{for } n=100) = \sqrt{\frac{p(1-p)}{100}}\]
03

Find the standard error for a sample size of \(n=200\)

Similarly, plug in \(n=200\) into the standard error formula: \[\sigma_{\hat{p}} (\text{for } n=200) = \sqrt{\frac{p(1-p)}{200}}\]
04

Compare the standard errors

Now we can see that both standard errors have the same structure, but with different denominators, which are the sample sizes: \[\sigma_{\hat{p}} (\text{for } n=100) = \sqrt{\frac{p(1-p)}{100}}\] \[\sigma_{\hat{p}} (\text{for } n=200) = \sqrt{\frac{p(1-p)}{200}}\] Notice that the standard error formula for \(n=200\) has a larger denominator than the standard error formula for \(n=100\). This means that the standard error for the larger sample size will be smaller: \[\sigma_{\hat{p}} (\text{for } n=200) < \sigma_{\hat{p}} (\text{for } n=100)\]
05

Conclusion

The standard error of the sample proportion \(\hat{p}\) would be smaller for random samples of size \(n=200\) compared to random samples of size \(n=100\). This means that using a larger sample size will yield more accurate estimates of the true proportion of city residents who favor spending city funds on promoting tourism.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sample Proportion
When conducting surveys or experiments, researchers often need to estimate the proportion of a population that exhibits a certain characteristic. This proportion found in the sample is known as the sample proportion, denoted by \( \hat{p} \). With this, researchers can make inferences about the entire population.
  • The sample proportion is basically the fraction of sampled individuals with the desired characteristic.
  • If you're surveying 100 people and 60 of them favor a policy, the sample proportion (\( \hat{p} \)) is \( \frac{60}{100} = 0.6 \).
The sample proportion provides an estimate for the population proportion but will vary from sample to sample. This variability is captured by a concept called the standard error, which assesses how much \( \hat{p} \) would tend to differ from the true population proportion value if we drew multiple samples.
Sample Size
Sample size, denoted as \( n \), refers to the number of individuals or observations used in a sample. It's a crucial factor because it directly influences the reliability of the estimates obtained from the sample.
  • Larger sample sizes provide more accurate estimates.
  • They reduce the "noise" or variability in estimates.
When the sample size increases, the standard error decreases, leading to more confidence in the insights derived from the data. For example, if you have a larger sample size, the estimate of a population proportion will be more precise. This is why, in our scenario, the standard error of \( n=200 \) is less than that of \( n=100 \), even without knowing the exact population proportion \( p \). Hence, a larger sample size contributes to a smaller standard error, enhancing the accuracy of assessing the population proportion.
Population Proportion
The population proportion, symbolized by \( p \), signifies the fraction of the entire population that shares a specific attribute. This is what researchers seek to estimate using sample data.
  • A true value of \( p \) is usually unknown.
  • Sample proportions \( \hat{p} \) are used to approach it.
In practice, while the size and accuracy of the sample play a significant role, the inherent variability or diversity within the population influences \( p \). Understanding the population proportion is paramount as it provides details about preferences, behaviors, or characteristics of the broad group of interest. In our example, the population proportion would tell us the exact percentage of all city residents who support tourism funding, a statistic researchers gather via sample surveys.

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Most popular questions from this chapter

If two statistics are available for estimating a population characteristic, under what circumstances might you choose a biased statistic over an unbiased statistic?

In a survey on supernatural experiences, 722 of 4013 adult Americans reported that they had seen a ghost ("What Supernatural Experiences We've Had," USA TODAY, February 8,2010 ). Assume that this sample is representative of the population of adult Americans. a. Use the given information to estimate the proportion of adult Americans who would say they have seen a ghost. b. Verify that the conditions for use of the margin of error formula to be appropriate are met. c. Calculate the margin of error. d. Interpret the margin of error in context. e. Construct and interpret a \(90 \%\) confidence interval for the proportion of all adult Americans who would say they have seen a ghost. f. Would a \(99 \%\) confidence interval be narrower or wider than the interval calculated in Part (e)? Justify your answer.

The report "The Politics of Climate" (Pew Research Center, October \(4,2016,\) www.pewinternet.org \(/ 2016 / 10 / 04\) /the-politics-of-climate, retrieved May 6,2017 ) summarized data from a survey on public opinion of renewable and other energy sources. It was reported that \(52 \%\) of the people in a sample from western states said that they have considered installing solar panels on their homes. This percentage was based on a representative sample of 369 homeowners in the western United States. Use the given information to construct and interpret a \(90 \%\) confidence interval for the proportion of all homeowners in western states who have considered installing solar panels.

The USA Snapshot titled "Social Media Jeopardizing Your Job?" (USA TODAY, November 12,2014\()\) summarized data from a survey of 1855 recruiters and human resource professionals. The Snapshot indicted that \(53 \%\) of the people surveyed had reconsidered a job candidate based on his or her social media profile. Assume that the sample is representative of the population of recruiters and human resource professionals in the United States. a. Use the given information to estimate the proportion of recruiters and human resource professionals who have reconsidered a job candidate based on his or her social media profile using a \(95 \%\) confidence interval. Give an interpretation of the interval in context and an interpretation of the confidence level of \(95 \%\). b. Would a \(90 \%\) confidence interval be wider or narrower than the \(95 \%\) confidence interval from Part (a)?

Use the formula for the standard error of \(\hat{p}\) to explain why a. the standard error is greater when the value of the population proportion \(p\) is near 0.5 than when it is near 1 . b. the standard error of \(\hat{p}\) is the same when the value of the population proportion is \(p=0.2\) as it is when \(p=0.8\).
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