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Chapter 9: Problem 72

Will \(\hat{p}\) from a random sample from a population with \(60 \%\) successes tend to be closer to 0.6 for a sample size of \(n=400\) or a sample size of \(n=800 ?\) Provide an explanation for your choice.

Short Answer

Expert verified
The sample proportion \(\hat{p}\) will tend to be closer to 0.6 for a sample size of \(n=800\) in accordance with the principle of the Law of Large Numbers.

Step by step solution

01

Understand the Principle

Understand the concept of the Law of Large Numbers which implies that as the sample size increases, sample statistics like the sample mean or sample proportion tend to get closer to their corresponding population parameters.
02

Apply the Principle

Apply this principle to the given problem. Here, we're working with sample proportions - \(\hat{p}\) - and a given population proportion (0.6).
03

Compare the Sample Sizes

Compare the two specified sample sizes, n=400 and n=800. Given the law of large numbers, the sample proportion from the larger sample size, n=800, will tend to be closer to the population proportion.

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Most popular questions from this chapter

Consider taking a random sample from a population with \(p=0.40\) a. What is the standard error of \(\hat{p}\) for random samples of size \(100 ?\) b. Would the standard error of \(\hat{p}\) be larger for samples of size 100 or samples of size \(200 ?\) c. If the sample size were doubled from 100 to 200 , by what factor would the standard error of \(\hat{p}\) decrease?

A random sample will be selected from the population of all adult residents of a particular city. The sample proportion \(\hat{p}\) will be used to estimate \(p,\) the proportion of all adult residents who are registered to vote. For which of the following situations will the estimate tend to be closest to the actual value of \(p ?\) I. \(\quad n=1,000, p=0.5\) II. \(\quad n=200, p=0.6\) III. \(\quad n=100, p=0.7\)

The article "Viewers Speak Out Against Reality TV" (Associated Press, September 12,2005\()\) included the following statement: "Few people believe there's much reality in reality TV: a total of \(82 \%\) said the shows are either 'totally made up' or 'mostly distorted."" This statement was based on a survey of 1,002 randomly selected adults. Calculate and interpret the margin of error for the reported percentage.

For each of the following choices, explain which would result in a narrower large-sample confidence interval for \(p\) : a. \(95 \%\) confidence level or \(99 \%\) confidence level b. \(n=200\) or \(n=500\)

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 \(n=200 ?\)
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