Final answer:
The task involves calculating the sample proportion of even numbers in given samples and comparing it to the population proportion to see if it serves as a good estimator. The sample proportions mean should match the population's to ensure accurate estimation. The information provided should be sufficient to conclude whether sample proportions are good estimators or not.
Step-by-step explanation:
Understanding the Sampling Distribution of a Proportion
When dealing with a proportion problem, we are looking at categorical data that falls into two categories: success or failure. In this case, we're focused on the proportion of even numbers in a sample. Considering the households with numbers 3, 5, and 10, we acknowledge that only '10' is even, setting the population proportion of even numbers (p) to 1/3. For samples of size 2, we assess how many samples contain the even number '10'. We then calculate the sample proportion (p') for each sample, which equals the number of even numbers in the sample (x) divided by the sample size (n).
We create a probability distribution table illustrating each sample's event of success (having a '10') or failure (not having a '10'). To find whether the sample proportion makes a good estimator of the population proportion, we look at the mean of the sample proportions compared to the population proportion of even numbers. The mean of the sample proportions should equal the population proportion if the sample proportion targets the value of the population proportion correctly. After computing and averaging these sample proportions, if our mean matches 1/3, the sample proportions do indeed estimate the population proportion effectively (option A and C). If not, they may not be a good estimator (option B).
It is essential, however, to be cautious as the given data might not always be sufficient to reach a firm conclusion without performing proper statistical tests (option D).