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Chapter 9: Problem 10

A researcher wants to estimate the proportion of students enrolled at a university who are registered to vote. Would the standard error of the sample proportion \(\hat{p}\) be larger if the actual population proportion was \(p=0.4\) or \(p=0.8\) ?

Short Answer

Expert verified
The standard error of the sample proportion, \(\hat{p}\), would be larger if the actual population proportion was \(p=0.4\), compared to if it was \(p=0.8\).

Step by step solution

01

Understand the formula for calculating the standard error of a proportion

The formula used to calculate the standard error of a proportion is \(\sqrt{p(1-p)/n}\), where p is the population proportion and n is the sample size. This formula shows us that the standard error of a proportion is affected by the population proportion and the size of the sample.
02

Substitute the given values into the formula

First, substitute \(p=0.4\) into the formula. This will produce \(\sqrt{0.4(1-0.4)/n} = \sqrt{0.24/n}\). Now, substitute \(p=0.8\) into the formula. This will produce \(\sqrt{0.8(1-0.8)/n} = \sqrt{0.16/n}\).
03

Conclusion

Since we are not given the exact number of samples (n), we can still compare the two scenarios. Looking at the results, it is clear that the standard error would be larger if \(p=0.4\) than if \(p=0.8\). This is because the product of \(p(1-p)\) is larger when \(p=0.4\) than when \(p=0.8\).

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Key Concepts

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

Population Proportion
When researching statistical data, it's crucial to understand the concept of population proportion, often denoted as 'p'. It represents the fraction of members in a population that exhibit a particular attribute. For example, in a study of voters, 'p' could be the proportion of all students at a university who are registered to vote.

The value of 'p' plays a significant role in calculating various statistics, including the standard error of a proportion. It's interesting to note that as 'p' moves away from the extremes of 0 and 1, the product 'p(1-p)', which forms part of the standard error formula, increases. Consequently, the standard error is larger in scenarios where 'p' is closer to 0.5, since this is where the product 'p(1-p)' reaches its maximum value.
Sample Size
The term 'sample size', denoted as 'n', refers to the number of observations or measurements taken from a population to form a sample. This is a crucial concept because the size of the sample directly impacts the precision of statistical estimates.

A larger sample size generally leads to more precise estimates of population parameters, as it more closely reflects the true distribution of the population. This is why researchers aim for a larger 'n' in their studies. Within the standard error formula, \(\sqrt{p(1-p)/n}\), we see that the standard error is inversely proportional to the square root of the sample size. This means that as 'n' increases, the standard error decreases, leading to more accurate estimates of the population proportion.
Sampling Distribution
A sampling distribution is a probability distribution of a statistic that is formed by considering all possible samples of a given size from a population. It is a fundamental concept in statistics, as it allows us to understand the behavior of sample estimates and to make inferences about the population.

The standard error of a proportion is a measure of how much we expect the sample proportion (\(\hat{p}\)) to vary from the true population proportion ('p'). This variability is depicted in the sampling distribution of \(\hat{p}\). When the sampling distribution of \(\hat{p}\) is centered around 'p' with a smaller standard error, the spread is narrow, indicating that samples are likely to yield estimates close to the population proportion. Conversely, a larger standard error indicates a wider spread, suggesting a higher chance of obtaining sample proportions that differ from 'p'. Understanding this distribution helps researchers gauge the reliability of their sample estimates and the likelihood of drawing accurate conclusions about the population.

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Most popular questions from this chapter

The report "2005 Electronic Monitoring \& Surveillance Survey: Many Companies Monitoring, Recording, Videotapingand Firing-Employees" (American Management Association, 2005) summarized a survey of 526 U.S. businesses. The report stated that 137 of the 526 businesses had fired workers for misuse of the Internet, and 131 had fired workers for e-mail misuse. Assume that the sample is representative of businesses in the United States. a. Construct and interpret a \(95 \%\) confidence interval for the proportion of U.S. businesses that have fired workers for misuse of the Internet. b. What are two reasons why a \(90 \%\) confidence interval for the proportion of U.S. businesses that have fired workers for misuse of e-mail would be narrower than the \(95 \%\) confidence interval calculated in Part (a)?

High-profile legal cases have many people reevaluating the jury system. Many believe that juries in criminal trials should be able to convict on less than a unanimous vote. To assess support for this idea, investigators asked each individual in a random sample of Californians whether they favored allowing conviction by a \(10-2\) verdict in criminal cases not involving the death penalty. The Associated Press (San Luis ObispoTelegram-Tribune, September 13,1995 ) reported that \(71 \%\) favored conviction with a \(10-2\) verdict. Suppose that the sample size for this survey was \(n=900\). Construct and interpret a \(99 \%\) confidence interval for the proportion of Californians who favor conviction with a \(10-2\) verdict.

Suppose that 935 smokers each received a nicotine patch, which delivers nicotine to the bloodstream at a much slower rate than cigarettes do. Dosage was decreased to 0 over a 12 -week period. Of these 935 people, 245 were still not smoking 6 months after treatment. Assume this sample is representative of all smokers. a. Use the given information to estimate the proportion of all smokers who, when given this treatment, would refrain from smoking for at least 6 months. b. Verify that the conditions needed in order for the margin of error formula to be appropriate are met. c. Calculate the margin of error. d. Interpret the margin of error in the context of this problem.

The article "Consumers Show Increased Liking for Diesel Autos" (USA Today, January 29,2003 ) reported that \(27 \%\) of U.S. consumers would opt for a diesel car if it ran as cleanly and performed as well as a car with a gas engine. Suppose that you suspect that the proportion might be different in your area. You decide to conduct a survey to estimate this proportion for the adult residents of your city. What is the required sample size if you want to estimate this proportion with a margin of error of 0.05 ? Calculate the required sample size first using 0.27 as a preliminary estimate of \(p\) and then using the conservative value of \(0.5 .\) How do the two sample sizes compare? What sample size would you recommend for this study?

The Gallup Organization conducts an annual survey on crime. It was reported that \(25 \%\) of all households experienced some sort of crime during the past year. This estimate was based on a sample of 1,002 randomly selected adults. The report states, "One can say with \(95 \%\) confidence that the margin of sampling error is 卤3 percentage points." Explain how this statement can be justified.
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