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The ages of the 5 officers for a school club are [tex]$18, 18, 17, 16, $[/tex] and [tex]$15$[/tex]. The proportion of officers who are younger than [tex]$18$[/tex] is [tex]$0.6$[/tex]. The table displays all possible samples of size 2 and the corresponding proportion 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 \\
Proportion
\end{tabular} & 0 & 0.5 & 0.5 & 0.5 & 0.5 & 0.5 & 0.5 & 1 & 1 & 1 \\
\hline
\end{tabular}

Using the proportions in the table, is the sample proportion an unbiased estimator?

A. Yes, the sample proportions are calculated using samples from the population.
B. Yes, the mean of the sample proportions is [tex]$0.6$[/tex], which is the same as the population proportion.
C. No, [tex]$0.6$[/tex] is not one of the possible sample proportions.
D. No, [tex]$70 \%$[/tex] of the sample proportions are less than or equal to [tex]$0.5$[/tex].

Sagot :

GinnyAnswer
Let's go through the problem step-by-step to determine if the sample proportion is an unbiased estimator.

### Step 1: Understand the Population Proportion
The population includes the ages of the 5 officers: 18, 18, 17, 16, 15. We are given the proportion of officers who are younger than 18, which is \(0.6\).

### Step 2: Identify the Possible Samples of Size 2
All possible samples of size 2 from this population are listed in the table, including the repeated samples (e.g., 18,17 appears twice). We are interested in these samples to calculate the sample proportions.

### Step 3: List the Provided Sample Proportions
The table provides the sample proportions for each possible sample of size 2:
- 18,18: Proportion = 0
- 18,17: Proportion = 0.5
- 18,17: Proportion = 0.5
- 18,16: Proportion = 0.5
- 18,16: Proportion = 0.5
- 18,15: Proportion = 0.5
- 18,15: Proportion = 0.5
- 17,16: Proportion = 1
- 17,15: Proportion = 1

### Step 4: Calculate the Mean of the Sample Proportions
First, sum all the sample proportions:
[tex]\[ 0 + 0.5 + 0.5 + 0.5 + 0.5 + 0.5 + 0.5 + 1 + 1 \][/tex]

Next, count the number of sample proportions given, which is 9.

Now, find the mean by dividing the sum by the number of proportions:
[tex]\[ \text{Mean of sample proportions} = \frac{0 + 0.5 + 0.5 + 0.5 + 0.5 + 0.5 + 0.5 + 1 + 1}{9} = \frac{5}{9} \approx 0.5556 \][/tex]

### Step 5: Compare the Mean of Sample Proportions with Population Proportion
The population proportion is given as \(0.6\). The mean of the sample proportions we calculated is approximately \(0.5556\).

### Conclusion
The mean of the sample proportions (\(0.5556\)) is not exactly equal to the population proportion (\(0.6\)). Therefore, the sample proportion is not an unbiased estimator.

So, among the provided options, the correct answer is:
No, [tex]\(70\%\)[/tex] of the sample proportions are less than or equal to [tex]\(0.5\)[/tex].
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