Which of the following best describes the Central Limit Theorem in the context of a sampling distribution of the sample mean?

Question

BlackTom AI · Accepted Answer

1) Relevant knowledge points and concepts
- Central Limit Theorem (CLT): For iid random samples, the sampling distribution of the sample mean has mean equal to the population mean μ and standard error σ/√n (where σ is the population standard deviation). If the population distribution is normal, the sample mean is exactly normally distributed for any sample size n. If the population is not normal, the distribution of the sample mean becomes approximately normal as n grows (the approximation improves with larger n). Assumptions: independent, identically distributed samples (random sampling).

2) Step-by-step reasoning
- Start from the population with mean μ and standard deviation σ. Take samples of size n and compute each sample’s mean.
- The sampling distribution of those sample means has:
  - Mean: E(X̄) = μ.
  - Standard deviation (standard error): SD(X̄) = σ/√n.
- If the original population is normal, X̄ ~ Normal(μ, σ/√n) exactly for any n.
- If the population is not normal, the CLT says X̄ tends toward Normal(μ, σ/√n) as n increases.

3) Calculations (intermediate steps)
Example: σ = 10.
- If n = 25: SE = σ/√n = 10/√25 = 10/5 = 2.
- If n = 100: SE = 10/√100 = 10/10 = 1.
This shows the sampling distribution gets narrower as n increases (standard error decreases).

4) Why the correct answer is correct
Option b states: "It is exactly normal if the population is normal; otherwise, it becomes approximately normal as the sample size increases." This matches the CLT’s precise conditions and conclusions (exact normality when population is normal; approximate normality for large n otherwise).

5) Why the other options are incorrect
- a: Incorrect — the sampling distribution of the mean is not identical to the population distribution in general; it centers at μ and has smaller spread σ/√n.
- c: Incorrect — “greater than 30” is a rule-of-thumb, not a strict threshold; also if the population is normal, exact normality holds for any n.
- d: Incorrect — the standard deviation of the sampling distribution decreases (σ/√n) as n increases, not increases.
- e: Incorrect — the mean of the sampling distribution equals the population mean μ, not always greater.

6) Conclusion
The correct choice is b, which matches the Central Limit Theorem as described above.

Which of the following best describes the Central Limit Theorem in the context of a sampling distribution of the sample mean?单项选择题

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