For which of the following situations would the Central Limit Theorem for Sample Means NOT apply?

Question

BlackTom AI · Accepted Answer

The Central Limit Theorem (CLT) states that the distribution of the sample mean approaches a normal distribution as sample size increases, provided observations are independent and the population has finite variance. A common practical rule-of-thumb: for moderately skewed populations you typically need n ≥ 30 for the sample mean to be approximately normal; if the original population is already bell-shaped (normal), the sample mean is normal for any sample size.

1) Relevant knowledge points and concepts
- CLT: sample means → approximately normal as n grows.
- Requirements: independence, finite variance; if population is normal (bell-shaped), no large-n requirement.
- Practical cutoff: n ≈ 30 often used for skewed distributions.

2) Step-by-step reasoning
- Option a: n = 20 from a bell-shaped population. Because the population is bell-shaped (normal), the sampling distribution of the mean is exactly normal even for n = 20. CLT (or rather properties of the normal distribution) holds.
- Option b: n = 50 from a skewed population. n = 50 is larger than the practical cutoff (30), so by CLT the sample mean will be approximately normal.
- Option c: n = 50 from a bell-shaped population. Population is bell-shaped and sample size is large; sample mean is normal (even exact if population is exactly normal).
- Option d: n = 20 from a skewed population. Here the population is skewed and sample size is small (n < 30), so the sampling distribution of the mean may not be approximately normal; CLT’s large-sample justification does not apply.

3) Intermediate “calculation” step
- Compare sample sizes to the rule-of-thumb: 20 < 30 (insufficient for skewed), 50 ≥ 30 (sufficient).

4) Why the correct answer is correct
- Only option d violates the CLT practical conditions: small sample from a skewed population, so the sample mean’s distribution may remain skewed rather than approximately normal.

5) Why other options are incorrect
- a and c: bell-shaped population ⇒ sample mean normal regardless of n (a) or certainly for n = 50 (c).
- b: n = 50 is large enough for the CLT to apply despite skewness.

6) Conclusion
Therefore the CLT for sample means would NOT apply in d. This matches the provided correct answer: d.

For which of the following situations would the Central Limit Theorem for Sample Means NOT apply?单项选择题

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