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Chapter 7: Problem 8

What is an estimator? When is an estimator unbiased? Is the sample mean, \(\bar{x}\), an unbiased estimator of \(\mu\) ? Explain.

Short Answer

Expert verified
An estimator is a statistic used to infer the value of an unknown parameter in a statistical model. It is unbiased if its expected value equals the true parameter. The sample mean ( \( \bar{x} \) ) is an unbiased estimator of the population mean ( \( \mu \) ), as its expected value equals the population mean.

Step by step solution

01

Define an Estimator

An estimator is a statistic (a function of the observed data) that is used to infer the value of an unknown parameter in a statistical model. For example, the sample mean is used as an estimator of the population mean.
02

Explain unbiased Estimator

An unbiased estimator is an estimator that on average correctly guesses the true parameter of the underlying population from which the data is sampled. Mathematically, an estimator \( \theta \) is unbiased for a parameter \( \mu \) if the expected value of \( \theta \) is equal to \( \mu \). In other words, \( E(\theta) = \mu \) .
03

Analyze Sample Mean as an Unbiased Estimator

The sample mean ( \( \bar{x} \) ) is indeed an unbiased estimator of the population mean ( \( \mu \) ). This can be proved by the law of large numbers which states that as the size of a sample becomes large, the sample mean will get closer and closer to the population mean. Hence, the expected value of the sample mean equals the population mean i.e., \( E(\bar{x}) = \mu \)

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