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Chapter 7: Problem 68

Is the sample proportion a consistent estimator of the population proportion? Explain why or why not.

Short Answer

Expert verified
Yes, the sample proportion is a consistent estimator of the population proportion. This is due to the Law of Large Numbers, which states that as the sample size increases, the sample proportion converges to the population proportion.

Step by step solution

01

Understanding the Key Concepts

Key concepts to understand in this problem are 'sample proportion' and 'consistent estimator'. The sample proportion is a statistical estimate of a certain characteristic in a sample, like the proportion of students who are male in a randomly selected class. A 'consistent estimator' is an estimate that gets closer and closer to the true value of the parameter as the size of the sample increases.
02

The Law of Large Numbers

The Law of Large Numbers states that as the size of a sample increases, the sample mean will get closer and closer to the population mean. This law also applies to proportions. That is, as the size of a sample increases, the sample proportion will get closer and closer to the population proportion.
03

Consistency of the Sample Proportion

According to the definition of consistent estimators and the Law of Large Numbers, the sample proportion is indeed a consistent estimator for the population proportion. As the sample size increases, the sample proportion converges to the population proportion. This is because larger samples provide better approximations of the population, reducing the sampling error.

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Most popular questions from this chapter

The standard deviation of the 2011 gross sales of all corporations is known to be $$\$ 139.50$$ million. Let \(\bar{x}\) be the mean of the 2011 gross sales of a sample of corporations. What sample size will produce the standard deviation of \(\bar{x}\) equal to $$\$ 15.50$$ million? Assume \(n / N \leq .05\).

Consider a large population with \(\mu=60\) and \(\sigma=10\). Assuming \(n / N \leq .05\), find the mean and standard deviation of the sample mean, \(\bar{x}\), for a sample size of a. \(18 \quad\) b. 90

In a population of 5000 subjects, 600 possess a certain characteristic. In a sample of 120 subjects selected from this population, 18 possess the same characteristic. What are the values of the population and sample proportions?

Explain the central limit theorem.

If all possible samples of the same (large) size are selected from a population, what percentage of all the sample means will be within \(1.5\) standard deviations \(\left(\sigma_{\bar{x}}\right)\) of the population mean?
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