Bias and Unbiased Estimators

The following graphs show the sampling distributions for two different point estimators, R and W, of the same population parameter.

Which of the following statements is true?

• Both R and W are unbiased.

• R is biased, and W is unbiased

• Both R and W are biased.

• R is unbiased, and W is biased.

• The assessment of bias is not possible, because the sampling distributions display too much variability.

A statistic is biased if...

• the value of the statistic from any sample is NOT equal to the parameter

• The value of the statistic from any sample IS equal to the parameter

• The mean of the sampling distribution is NOT equal to the true value of the parameter

• The mean of the sampling distribution IS equal to the value of the parameter

Calculate the bias of an estimator if the true value is 10 and the estimator gives an average value of 9.5 based on 5 samples.

• -0.5

• 2.5

• 1.5

• 0.5

Apply the concept of unbiased estimators to calculate the mean of a sample with values 8, 10, 12, 9, and 11.

• 13

• 10

• 11

• 8

A statistic used to estimate a parameter is an ____________ if the mean of its sampling distribution is equal to the value of the parameter being estimated.

• unbiased estiamtor

• biased estimator

• highly variable

• less variable

Which best describes the sampling distribution shown below.

• Unbiased Estimator

• Biased Estimator

A certain statistic d-hat is being used to estimate a population parameter D. The expected value of d-hat is not equal to D. What property does d-hat exhibit?

• The sampling distribution of d-hat is Normal.

• The sampling distribution of d-hat is Binomial.

• The sampling distribution of d-hat is uniform.

• d-hat is unbiased

• d-hat is biased

Answer explanation

A biased estimator is one in which its value, on average, is not equal to the value of the parameter it is intended to estimate.

In a national study on transportation patterns, 1,000 randomly selected adults will be asked the question: How many trips per week do you make to the grocery store? The sample mean will be computed. Let µ denote the population mean response to the question if everyone in the population is to be asked the question. Is the sample mean x unbiased for estimating µ?

• Yes, because for random samples the mean (expected value) of the sample mean x is equal to the population mean µ.

• Yes, because with a sample size of 1,000 the standard deviation of the sample mean x is small.

• Yes, because the wording of the question is not biased.

• No, because the sample mean x does not always equal the population mean µ.

• No, because number of trips to the grocery story is not normally distributed so the mean (expected value) of the sample mean x does not equal the population mean µ.

Tags

(SR.DSR.4.3

Which best describes the sampling distribution shown below.

• Unbiased Estimator

• Biased Estimator

Calculate the bias of an estimator if the true value is 30 and the estimator gives an average value of 28 based on 10 samples.

• 5

• 20

• -2

• 12