Suppose there exists some variable whose true population distribution is shown in the diagram below. Because of a wealthy benefactor, Quackers is able to run 5000 experiments; in each of these, he samples independently from the true population distribution and measures the mean of the sample.

Suppose Quackers runs 200 people in each experiment. Which of the panels A-E above most accurately captures what you would expect the sampling distribution of the mean to look like?

Question

Suppose there exists some variable whose true population distribution is shown in the diagram below. Because of a wealthy benefactor, Quackers is able to run 5000 experiments; in each of these, he samples independently from the true population distribution and measures the mean of the sample.

Suppose Quackers runs 200 people in each experiment. Which of the panels A-E above most accurately captures what you would expect the sampling distribution of the mean to look like?

BlackTom AI · Accepted Answer

By the Central Limit Theorem (CLT), the distribution of sample means tends toward a normal (bell‑shaped) distribution and its spread shrinks with sample size.

1. Relevant knowledge points and concepts
- Central Limit Theorem: for large n, the sampling distribution of the mean is approximately normal regardless of the population’s shape.
- Standard error (SE) of the mean: SE = σ / sqrt(n), where σ is the population standard deviation and n is sample size. Larger n makes the sampling distribution much narrower.
- The sampling distribution is centered at the population mean.

2. Step‑by‑step reasoning
- The true population is clearly bimodal (two peaks) and has mean roughly at the center (0).
- Quackers takes samples of n = 200 in each experiment, which is a large sample size.
- By CLT, the distribution of the sample means will be approximately normal (one peak), not bimodal.

3. Calculations / intermediate steps
- SE = σ / sqrt(n). Here n = 200 so sqrt(200) ≈ 14.14.
- Thus SE ≈ σ / 14.14 — the spread of the sample means is about 1/14 of the population SD. Example: if σ were 0.5, SE ≈ 0.5/14.14 ≈ 0.035; if σ were 1, SE ≈ 0.071. In any case the sampling distribution is much narrower than the original.

4. Why the correct answer is correct
- Panel E shows a single, narrow, symmetric peak centered at the population mean — exactly what CLT plus the small SE predict for n = 200.

5. Why the other options are incorrect
- A and C retain bimodality (two peaks) — wrong because averaging many observations destroys the original bimodal shape.
- B shows a bimodal but flattened distribution — still bimodal and too wide.
- D shows a wide unimodal distribution — unimodal but not narrow enough given the large sample size.

6. Conclusion
- Therefore Panel E is the best depiction of the sampling distribution of the mean for samples of 200. This matches the provided answer: Panel E.

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