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Chapter 7: Problem 67

What is the estimator of the population proportion? Is this estimator an unbiased estimator of \(p ?\) Explain why or why not.

Short Answer

Expert verified
The estimator of the population proportion is the sample proportion, denoted by \( \hat{p} \). Yes, this estimator is unbiased. The reason for this estimation being unbiased is that its expected value is equal to the population proportion, which is the parameter being estimated.

Step by step solution

01

Understand the Terms

An 'estimator' is a statistic or a rule for calculating an attribute of a population from a sample. 'Population proportion' (\(p\)) is the ratio in the population that has a particular characteristic. Finally, an 'unbiased estimator' is a statistic that has an expected value equal to the population parameter being estimated.
02

Identify the Estimator for Population Proportion

The estimator for population proportion is the sample proportion (\(\hat{p}\)). Falling under point estimation, it is calculated as \( \hat{p} = x/n \), where \(x\) is the number of successes in the sample and \(n\) is the sample size.
03

Determine If the Estimator is Unbiased

For an estimator to be unbiased, the expected value of the estimator should be equal to the parameter being estimated. The expected value of the sample proportion is \(E(\hat{p}) = p \), which is the population proportion. Thus, sample proportion (\( \hat{p} \)) is an unbiased estimator of the population proportion (\( p \)).

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