Question

Bias and variability The figure below shows his programs of four sampling distributions of different statistics intended to estimate the same parameter.

(a) Which statistics are unbiased estimators? Justify your answer. (b) Which statistic does the best job of estimating the parameter? Explain.

Short answer

a). The statistic in the graph (b) and (c) seems to be an unbiased estimator.

b).The graph (b) is the best estimate.

Step-by-step solution

Part (a) Step 1: Given Information 

Four sampling distributions of different statistics intended to estimate the same parameter.

Part (a) Step 2: Explanation 

The statistic is an unbiased estimator if the value of the statistic is (about) equal to the mean.

The mean of the distribution is expected to lie at the highest bar in the histogram (if multiple highest bars, then take the middle highest bar).

We then note that the mean and the population parameter seem to fall together for graph (b) and (c) only.

Thus the statistic in graph (b) and (c) seems to be an unbiased estimator.

Part (b) Step 1: Given Information 

Four sampling distributions of different statistics intended to estimate the same parameter.

Part (b) Step 2: Explanation 

The graph corresponding to a statistic that does the best job of estimating the parameter, is the graph that has the bars centered about the population parameter with no gaps between the bars (because the gaps make it less likely to have an accurate estimate).

Thus graph (b) is then the best estimate.