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A population of the amounts of caffeine in 3 different drink brands is provided.

Samples of size equal to 2 are extracted from this population with replacement.

a.

All possible samples of size 2 selected with replacement are tabulated below:

The range has the following formula:

$Range=\text{Maximum}\text{Value}-\text{Minimum}\text{Value}$

The following table shows all possible samples of size equal to 2 and the corresponding sample ranges:

Combining the values of ranges that are the same, the following probability values are obtained:

b.

The population range is computed as shown below:

$$\begin{gathered} \text{Population}\text{Range}=\text{Maximum}\text{Value}-\text{Minimum}\text{Value} \\ =51-34 \\ =17 \end{gathered}$$

Thus, the population range is equal to 17.

The mean of the sample ranges is computed below:

$$\begin{gathered} \text{Mean}\text{of}\text{Sample}\text{Ranges}=\frac{\text{Sample}{\text{Range}}_1+\text{Sample}{\text{Range}}_2+.....+\text{Sample}{\text{Range}}_{16}}{16} \\ =\frac{0+2+....+0}{16} \\ =7 \end{gathered}$$

Thus, the mean of the sampling distribution of the sample range is equal to 7.

Here, the population range (17) is not equal to the mean of the sampling distribution of the sample range (7).

c.

Since the population range is not equal to the mean of the sample ranges, it can be said that the sample rangesdo not target the value of the population range.

A good estimator is a sample statistic whose sampling distribution has a mean value equal to the population parameter.

The mean value of the sampling distribution of the sample range (7) is not equal to the population range (17).

Thus, the sample rangecannot be considered as a good estimator of the population range.