Question

the ages of the 5 officers for a school club are 18, 18, 17, 16, 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. using the values in the table, is the sample standard deviation an unbiased estimator? yes, 50% of the standard deviations are above 1.17, and 50% are below 1.17. yes, the mean of the sample standard deviations is 1.13, which is only 0.04 less than the population value, 1.17. no, the mean of the sample standard deviations is 1.13, which is not the same as 1.17. no, the standard deviation of the sample standard deviations is 0.68, which is not the same as 1.17.

Explanation:

Step1: Recall the definition of an unbiased estimator

An unbiased estimator is one where the mean of the sampling distribution of the estimator is equal to the population parameter.

Step2: Calculate the mean of the sample standard deviations

We have the sample standard deviations: \(0, 0.71, 0.71, 1.41, 1.41, 2.12, 2.12, 0.71, 1.41, 0.71\).
The sum of these values is \(0+0.71 + 0.71+1.41+1.41+2.12+2.12+0.71+1.41+0.71=11.3\).
Since there are \(n = 10\) samples, the mean \(\bar{x}=\frac{11.3}{10}=1.13\).

Step3: Compare the mean of the sample standard deviations with the population standard deviation

The population standard deviation is \(1.17\). Since \(1.13
eq1.17\), the sample standard deviation is not an unbiased estimator.

Answer:

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