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?

Question

BlackTom AI · Accepted Answer

The sample mean for samples of size n has expectation equal to the population mean and variance equal to the population variance divided by n. Formally, E(X̄) = μ and Var(X̄) = σ^2 / n. These two facts are the key concepts: unbiasedness of the sample mean and the dependence of its variability on sample size. The Central Limit Theorem explains how the shape of the sampling distribution approaches normality as n increases, but that does not apply when n = 1.

Step-by-step reasoning:
1. Let individual observations Xi be drawn from a population with mean μ and variance σ^2. The sample mean for sample size n is X̄ = (1/n) Σ Xi.
2. Compute expectation: E(X̄) = (1/n) Σ E(Xi) = (1/n) · nμ = μ. So the sampling distribution is centered at the population mean.
3. Compute variance: Var(X̄) = (1/n^2) Σ Var(Xi) = (1/n^2) · nσ^2 = σ^2 / n.
4. Substitute n = 1: Var(X̄) = σ^2 / 1 = σ^2 and SD(X̄) = σ. Thus the sampling distribution of the mean when n = 1 has the same mean, same variance, and the same shape as the original population distribution. There is no “shrinkage” of spread or smoothing toward normality for n = 1.

Why Panel C is correct:
- Panel C must depict a distribution of sample means that matches the original population distribution in shape and spread. That matches the derived results (mean μ, variance σ^2) for n = 1.

Why the other options are incorrect:
- Any panel showing a narrower distribution (reduced spread) would imply Var(X̄) < σ^2, which would be true only for n > 1, so those panels are wrong for n = 1.
- Any panel showing a more normal (smoothed) shape is reflecting the CLT effect for larger n; this does not occur at n = 1.
- Any panel showing a different center (shifted mean) or greater spread would contradict unbiasedness or the variance formula.

Therefore, the correct choice is Panel C, which matches the sampling distribution characteristics when only one observation is taken per experiment. This confirms the provided answer: Panel C.

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?单项选择题

类似问题

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?

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