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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
Yes, the estimator of the population proportion, \(\hat{p}\), is an unbiased estimator of the population parameter \(p\). This is because the expected value of \(\hat{p}\) equals \(p\).

Step by step solution

01

Definition of Population Proportion Estimator

The estimator of the population proportion, often denoted as \(\hat{p}\), is given by \( \hat{p} = \frac{X}{n} \) where \(X\) stands for the number of successes in a sample and \(n\) signifies the sample size.
02

Determine if the Estimator is Unbiased

To demonstrate whether an estimator is unbiased or not, the expected value, \(E(\hat{p})\), has to be computed. If \(E(\hat{p})\) equals the population parameter \(p\), the estimator is unbiased.
03

Calculate Expected Value of the Estimator

The expected value \(E(\hat{p})\) is calculated as follows: \(E(\hat{p}) = E(\frac{X}{n})\). Since \(E(X/n) = \frac{E(X)}{n}\), and the expected number of successes \(E(X)\) in a sample of size \(n\) with probability of success \(p\) is \(np\), then \(E(\hat{p}) = \frac{np}{n} = p\).
04

Conclude if the Estimator is Unbiased

From above, it's clear that the expected value \(E(\hat{p})\) equals the population parameter \(p\). Thus, the estimator \(\hat{p}\) is an unbiased estimator of \(p\).

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