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Chapter 8: Problem 8

Explain what the term sampling variability means in the context of using a sample proportion to estimate a population proportion.

Short Answer

Expert verified
Sampling variability in the context of using a sample proportion to estimate a population proportion refers to the variation that occurs in estimates when different samples are taken from the same population. It means that different samples may yield different sample proportions, thus the estimates of the population proportion could vary.

Step by step solution

01

Understanding Sample Proportion

A sample proportion is the ratio of members of a subgroup or the instances of an attribute within that sample to the total sample size. So for a sample 's' from a population 'p', if 'x' members of the sample exhibit a certain attribute, the sample proportion, denoted \( \hat{p} \), is calculated as \( \hat{p} = x/s \).
02

Understanding Sampling Variability

Sampling variability is a concept in statistics which implies that different samples drawn from the same population tend to produce slightly different estimates. It tells us how much our statistics vary from sample to sample. It accounts for the fact that different samples from the same population may not give identical results because each sample is subject to its own unique errors.
03

Connection Between Sample Proportion and Sampling Variability

The term sampling variability in the context of using a sample proportion is related to estimation of population proportions. When we derive multiple samples from a population, the sample proportion varies. That variation in sample proportion for different samples is sampling variability. This affects the estimation of population proportions because the greater the variability, the less precise our estimates are.

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Most popular questions from this chapter

The article "Career Expert Provides DOs and DON'Ts for Job Seekers on Social Networking" (CareerBuilder.com, August 19,2009 ) included data from a survey of 2,667 hiring managers and human resource professionals. The article noted that many employers are using social networks to screen job applicants and that this practice is becoming more common. Of the 2,667 people who participated in the survey, 1,200 indicated that they use social networking sites (such as Facebook, MySpace, and LinkedIn) to research job applicants. For the purposes of this exercise, assume that the sample can be regarded as a random sample of hiring managers and human resource professionals. a. Suppose you are interested in learning about the value of \(p,\) the proportion of all hiring managers and human resource managers who use social networking sites to research job applicants. This proportion can be estimated using the sample proportion, \(p .\) What is the value of \(p\) for this sample? b. Based on what you know about the sampling distribution of \(p,\) is it reasonable to think that this estimate is within 0.02 of the actual value of the population proportion? Explain why or why not.

Suppose that a particular candidate for public office is favored by \(48 \%\) of all registered voters in the district. A polling organization will take a random sample of 500 of these voters and will use \(\hat{p}\), the sample proportion, to estimate \(p\). a. Show that \(\sigma_{p}\), the standard deviation of \(\hat{p}\), is equal to \(0.0223 .\) b. If for a different sample size, \(\sigma_{p}=0.0500\), would you expect more or less sample-to-sample variability in the sample proportions than when \(n=500 ?\) c. Is the sample size that resulted in \(\sigma_{\rho}=0.0500\) larger than 500 or smaller than \(500 ?\) Explain your reasoning.

The article "Thrillers" (Newsweek, April 22,1985 ) stated, "Surveys tell us that more than half of America's college graduates are avid readers of mystery novels." Let \(p\) denote the actual proportion of college graduates who are avid readers of mystery novels. Consider a sample proportion \(\hat{p}\) that is based on a random sample of 225 college graduates. If \(p=0.5,\) what are the mean value and standard deviation of the sampling distribution of \(\hat{p}\) ? Answer this question for \(p=0.6 .\) Is the sampling distribution of \(\hat{p}\) approximately normal in both cases? Explain.

Consider the following statement: The proportion of all students enrolled at a particular university during 2012 who lived on campus was \(\mathbf{0 . 2 1}\). a. Is the number that appears in boldface in this statement a sample proportion or a population proportion? b. Which of the following use of notation is correct, \(p=0.21\) or \(\hat{p}=0.21 ?\)

The article referenced in the previous exercise also reported that \(38 \%\) of the 1,200 social network users surveyed said it was OK to ignore a coworker's friend request. If \(p=0.38\) is used as an estimate of the proportion of all social network users who believe this, is it likely that this estimate is within 0.05 of the actual population proportion? Use what you know about the sampling distribution of \(\hat{p}\) to support your answer.
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