The problem presented involves a sampling distribution with a small population consisting of the values 2, 4, and 7. We are tasked with selecting all possible samples of size 2 with replacement, then calculating the mean of each sample.

Let’s analyze the answer choices one by one:

• Option A lists these samples:

  • $2, 4$, $2, 7$, $4, 2$, $4, 7$, $7, 2$, $7, 4$.
  • The means would be calculated as:
    • Mean of $(2,4) = \frac{2 + 4}{2} = 3$,
    • Mean of $(2,7) = \frac{2 + 7}{2} = 4.5$,
    • Mean of $(4,2) = \frac{4 + 2}{2} = 3$,
    • Mean of $(4,7) = \frac{4 + 7}{2} = 5.5$,
    • Mean of $(7,2) = \frac{7 + 2}{2} = 4.5$,
    • Mean of $(7,4) = \frac{7 + 4}{2} = 5.5$.
  • Therefore, the possible sample means are $3, 4.5, 5.5$.

• Option B lists these samples:

  • $2, 2$, $2, 4$, $2, 7$, $4, 2$, $4, 4$, $4, 7$, $7, 2$, $7, 4$, $7, 7$.
  • The means are:
    • Mean of $(2,2) = \frac{2 + 2}{2} = 2$,
    • Mean of $(2,4) = \frac{2 + 4}{2} = 3$,
    • Mean of $(2,7) = \frac{2 + 7}{2} = 4.5$,
    • Mean of $(4,2) = \frac{4 + 2}{2} = 3$,
    • Mean of $(4,4) = \frac{4 + 4}{2} = 4$,
    • Mean of $(4,7) = \frac{4 + 7}{2} = 5.5$,
    • Mean of $(7,2) = \frac{7 + 2}{2} = 4.5$,
    • Mean of $(7,4) = \frac{7 + 4}{2} = 5.5$,
    • Mean of $(7,7) = \frac{7 + 7}{2} = 7$.
  • The possible sample means are $2, 3, 4, 4.5, 5.5, 7$.

• Option C lists these samples:

  • $2, 4$, $2, 7$, $4, 7$.
  • The means are:
    • Mean of $(2,4) = \frac{2 + 4}{2} = 3$,
    • Mean of $(2,7) = \frac{2 + 7}{2} = 4.5$,
    • Mean of $(4,7) = \frac{4 + 7}{2} = 5.5$.
  • Therefore, the sample means are $3, 4.5, 5.5$.

Correct Answer:

The correct answer is B because it lists all the possible samples of size 2 with replacement and gives the correct sample means.

Would you like further clarification on sampling or another aspect of the problem?

Related Questions:

1. How do you calculate the mean of a sample?
2. What does sampling with replacement mean, and how does it affect possible samples?
3. How does the sample size impact the sampling distribution?
4. Why do we consider all possible combinations in a sampling distribution?
5. What is the difference between sampling with and without replacement?

Tip: In problems involving sampling, always ensure that you understand whether sampling is done with or without replacement, as this will significantly affect the number of possible samples.