Question

Bias and variability The figure shows approximate sampling distributions of $4$different

statistics intended to estimate the same parameter.

1. 
   
2. 
   
3. 
   
4. 

a. Which statistics are unbiased estimators? Justify your answer.

b. Which statistic does the best job of estimating the parameter? Explain your answer.

Short answer

Our required answers are:

1. Graph (ii) and (iii) are unbiased estimators.
2. Graph (ii) is the best estimate.

Step-by-step solution

Part (a) Step 1: Given Information

We are given some approximate sampling distributions of $4$different statistics intended to estimate the same parameter. Different statistics figures are:

1. 
2. 
3. 
4. 

We need to explain which statistic is an unbiased estimator.

Part (a) Step 2: Explanation

If the statistic's value is equal to the mean, it is an unbiased estimator.

The highest bar in the histogram should represent the distribution's mean.

The mean and population parameter thus appear to come together only in graphs (ii) and (iii).

As a result, it appears that the statistic in graphs (ii) and (iii) is an unbiased estimator.

Part (b) Step 1: Given Information

We need to explain which statistic performs the best in terms of parameter estimation.

Statistics given are:

1. 
2. 
3. 
4. 

Part (b) Step 2: Explanation

A statistic that does the best job of estimating the parameter is the graph (ii) with the bars centered around the population parameter and no gaps between them.