Suppose that Quackers now runs 200 people in each experiment. Which of the panels A-E (as shown in Q29 above) most accurately captures what you would expect the sampling distribution of the mean to look like?

Question

BlackTom AI · Accepted Answer

A large sample size (n = 200) makes the sampling distribution of the sample mean approximately normal and much tighter (less spread) around the population mean—so Panel E.

1) Relevant concepts:
- Central Limit Theorem (CLT): For sufficiently large n, the sampling distribution of the mean is approximately normal regardless of the population distribution.
- Standard error (SE) of the mean: SE = σ / sqrt(n), where σ is the population standard deviation. As n increases, SE decreases in proportion to 1/sqrt(n).
- Law of Large Numbers: Larger samples yield sample means that concentrate closer to the true population mean.

2) Step-by-step reasoning:
- With n = 200, sqrt(n) = sqrt(200) ≈ 14.142.
- The standard error becomes SE = σ / 14.142. This is a small fraction of the original population spread (for example, compared to σ itself).
- Because n is large, CLT implies the distribution of sample means will be approximately normal (bell-shaped), centered at the population mean, and much narrower than the original population distribution.

3) Example calculations (intermediate steps):
- If σ = 10, SE = 10 / 14.142 ≈ 0.707.
- If σ = 5, SE = 5 / 14.142 ≈ 0.354.
- Compare to a smaller sample, say n = 25: SE_25 = σ / 5. Ratio SE_200 / SE_25 = (σ/14.142) / (σ/5) = 5 / 14.142 ≈ 0.354. So the spread with n=200 is about 35% of the spread with n=25.

4) Why Panel E is correct:
Panel E matches an approximately normal distribution that is relatively narrow (low variability) around the mean—exactly what the CLT and reduced SE predict for n = 200.

5) Why the other options are incorrect (briefly):
- Panel A: likely skewed or non-normal; CLT predicts near-normal for large n.
- Panel B: likely too wide; contradicts the reduced SE for n = 200.
- Panel C: may be uniform or flat; sampling distributions of means are not uniform for large n.
- Panel D: could be bimodal or otherwise irregular; large n gives a single-peaked normal shape.

Conclusion: Panel E correctly represents the approximately normal, tightly concentrated sampling distribution of the mean when experiments use n = 200, confirming the provided answer: Panel E.

Suppose that Quackers now runs 200 people in each experiment. Which of the panels A-E (as shown in Q29 above) most accurately captures what you would expect the sampling distribution of the mean to look like?单项选择题

类似问题

Suppose that Quackers only runs 1 person in each experiment. Which of the panels A-E (as shown in Q29 above) most accurately captures what you would expect the sampling distribution of the mean to look like?

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If you increase the sample size from 25 to 100 while keeping the population standard deviation constant, what happens to the standard deviation of the sampling distribution of the mean?

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