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The ages of the 5 officers for a school club are [tex]$18, 18, 17, 16$[/tex], and 15. The standard deviation of the distribution of ages is 1.17. The table displays all possible samples of size 2 and the corresponding ranges for each sample.

\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
Sample [tex]$n=2$[/tex] & 18,18 & 18,17 & 18,17 & 18,16 & 18,16 & 18,15 & 18,15 & 17,16 & 17,15 & 16,15 \\
\hline
\begin{tabular}{c}
Sample Standard \\
Deviation
\end{tabular} & 0 & 0.71 & 0.71 & 1.41 & 1.41 & 2.12 & 2.12 & 0.71 & 1.41 & 0.71 \\
\hline
\end{tabular}

Using the values in the table, is the sample standard deviation an unbiased estimator?

A. Yes, [tex]$50\%$[/tex] of the standard deviations are above 1.17, and [tex]$50\%$[/tex] are below 1.17.

B. Yes, the mean of the sample standard deviations is 1.13, which is only 0.04 less than the population value, 1.17.

C. No, the mean of the sample standard deviations is 1.13, which is not the same as 1.17.

D. No, the standard deviation of the sample standard deviations is 0.68, which is not the same as 1.17.



Answer :

GinnyAnswer
To determine whether the sample standard deviation is an unbiased estimator of the population standard deviation, we need to compare the mean of the sample standard deviations to the population standard deviation. Here's a step-by-step approach for evaluating this:

1. List the Sample Standard Deviations:
From the table provided, the sample standard deviations are:
0, 0.71, 0.71, 1.41, 1.41, 2.12, 2.12, 0.71, 1.41, 0.71.

2. Calculate the Mean of the Sample Standard Deviations:
To find the mean of these sample standard deviations, we sum them all up and then divide by the number of samples (which is 10 in this case):

[tex]\[ \text{Mean} = \frac{0 + 0.71 + 0.71 + 1.41 + 1.41 + 2.12 + 2.12 + 0.71 + 1.41 + 0.71}{10} \approx 1.131 \][/tex]

3. Compare the Mean Sample Standard Deviation to Population Standard Deviation:
The population standard deviation is given as 1.17.

- The calculated mean of the sample standard deviations is 1.131.
- 1.131 is quite close to 1.17, with a difference of 0.039.

4. Consider the Standard Deviation of the Sample Standard Deviations:
The standard deviation of the sample standard deviations is calculated (given in the problem) as approximately 0.646.

5. Assess Unbiasedness:

An unbiased estimator has a mean that is equal to the population parameter. In this case, the mean sample standard deviation (1.131) is very close to the population standard deviation (1.17), which suggests that it is quite close to being an unbiased estimator.

Given these details and focusing on how close the mean sample standard deviation is to the population standard deviation, we conclude:

- Yes, the mean of the sample standard deviations is 1.131, which is only 0.039 less than the population value of 1.17.

Thus, based on the closeness of the values, the correct answer is:

Yes, the mean of the sample standard deviations is 1.131, which is only 0.039 less than the population value, 1.17.

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