To solve the problem, let’s go through each part step by step.

Part 1: Proportion of Even Numbers in the Population

The ages of the children are 2, 3, and 10.

• Even numbers: 2 and 10
• Total numbers: 2, 3, and 10

The even numbers in the population are 2 and 10, so there are 2 even numbers out of 3 total.

$\text{Proportion of even numbers} = \frac{\text{Number of even ages}}{\text{Total ages}} = \frac{2}{3} \approx 0.667$

Rounded to three decimal places:
The proportion is 0.667.

Part 2: Proportion of Even Numbers in Each Sample

Now, let's determine the proportion of even numbers for each of the nine samples:

1. (2, 2): 2 even → Proportion = 1.0
2. (2, 3): 1 even → Proportion = 0.5
3. (2, 10): 2 even → Proportion = 1.0
4. (3, 2): 1 even → Proportion = 0.5
5. (3, 3): 0 even → Proportion = 0.0
6. (3, 10): 1 even → Proportion = 0.5
7. (10, 2): 2 even → Proportion = 1.0
8. (10, 3): 1 even → Proportion = 0.5
9. (10, 10): 2 even → Proportion = 1.0

Summary of the sampling distribution:

• Proportions and their frequencies:
  • 0.0: 1 occurrence (from sample 5)
  • 0.5: 4 occurrences (from samples 2, 4, 6, 8)
  • 1.0: 4 occurrences (from samples 1, 3, 7, 9)

Probability Distribution:

Sample Proportion	Probability
0.0	$\frac{1}{9}$ ≈ 0.111
0.5	$\frac{4}{9}$ ≈ 0.444
1.0	$\frac{4}{9}$ ≈ 0.444

Part 3: Mean of the Sampling Distribution

To find the mean of the sampling distribution, calculate the weighted average:

$\text{Mean} = (0.0 \cdot \frac{1}{9}) + (0.5 \cdot \frac{4}{9}) + (1.0 \cdot \frac{4}{9})$ $= 0 + \frac{2}{9} + \frac{4}{9} = \frac{6}{9} = \frac{2}{3} \approx 0.667$

Rounded to three decimal places:
The mean is 0.667.

Part 4: Unbiased Estimator

To determine if the sample proportion is an unbiased estimator of the population proportion, we compare the mean of the sampling distribution (0.667) with the population proportion (also 0.667).

Thus, the answer is: A. The sample proportions target the proportion of even numbers in the population, so sample proportions make good estimators of the population proportion.

Summary

1. The proportion of even numbers in the population is 0.667.
2. The probability distribution of sample proportions is as follows:
   • 0.0: ≈ 0.111
   • 0.5: ≈ 0.444
   • 1.0: ≈ 0.444
3. The mean of the sampling distribution is 0.667.
4. The sample proportions are unbiased estimators of the population proportion.

If you have any questions or need further details, feel free to ask! Here are some related questions to consider:

1. How do you calculate the variance of the sampling distribution?
2. What would happen to the proportions if we included another age?
3. How does sampling without replacement affect the results?
4. What is the significance of using a sample size of 2?
5. How could this method be applied in real-world scenarios?

Tip: Understanding sampling distributions is crucial for inferential statistics, as it helps in estimating population parameters based on sample statistics.