Question

the ages of the 5 officers for a school club are 18, 18, 17, 16, and 15. the proportion of officers who are younger than 18 is 0.6. the table displays all possible samples of size 2 and the corresponding proportion for each sample. using the proportions in the table, is the sample proportion an unbiased estimator?
yes, the sample proportions are calculated using samples from the population.
yes, the mean of the sample proportions is 0.6, which is the same as the population proportion.
no, 0.6 is not one of the possible sample proportions.
no, 70% of the sample proportions are less than or equal to 0.5.

Explanation:

Step1: Calculate the mean of sample proportions

The sample proportions are \(0, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 1, 1, 1\).
The formula for the mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\). Here \(n = 10\) and \(\sum_{i=1}^{10}x_{i}=0+(0.5\times6)+(1\times3)=0 + 3+3=6\).
So \(\bar{x}=\frac{6}{10}=0.6\).

Step2: Compare with population proportion

The population proportion is \(0.6\). An estimator is unbiased if \(E(\hat{p})=p\) (where \(E(\hat{p})\) is the expected value of the sample proportion and \(p\) is the population proportion). Since the mean of the sample proportions (\(E(\hat{p}) = 0.6\)) is equal to the population proportion (\(p = 0.6\)).

Answer:

Yes, the mean of the sample proportions is \(0.6\), which is the same as the population proportion.