Chapter 6: Q5 (page 293)

Unbiased Estimators
a. What is an unbiased estimator?
b. For the following statistics, identify those that are unbiased estimators: mean, median, range, variance, proportion.
c. Determine whether the following statement is true or false: “The sample standard deviation is a biased estimator, but the bias is relatively small in large samples, so is often used to estimate $σ$ ”

Short Answer

Expert verified

a. An unbiased estimator is a statistic whose sampling distribution has a mean value equal to the corresponding population parameter.

b. Out of the given list of estimators, mean, variance, and proportion are unbiased estimators.

c. The sample standard deviation is often used to estimate the population standard deviation $σ$ because the bias is small in large samples. Thus, this statement is true.

Step by step solution

Given Information

The definition of an unbiased estimator is considered. A set of estimators is provided.

Definition of unbiased estimator

A sample statistic is an unbiased estimator of the corresponding population parameter if the sample statistic's mean value (or expected value) over all possible samples of size is equal to the corresponding population parameter.

Identification of unbiased estimators

The following statistics are considered to be unbiased estimators of their respective population parameters:

Proportion
Mean
Variance

Thus, out of the given list of estimators, mean, variance, and proportion are unbiased estimators. On the other hand, median and range are biased estimators.

Sample standard deviation as an estimator of population standard deviation

The sample standard deviation $s$ is often used to estimate the population standard deviation $σ$ because the bias is small in large samples. Thus, this statement is true.