Central limit theorem states:

The central limit theorem states that, for sufficiently large sample sizes, the sampling distribution of the sample mean is approximately normal, regardless of the shape of the original population distribution, provided observations are independent and drawn appropriately. This is why option A is correct. The theorem does not require the population itself to be normal; if the population is normal, the sample mean is normally distributed for any sample size, but the central limit theorem is especially powerful because it applies broadly for large n. Option C is false because sample variance is an estimate of population variance, not automatically equal to it. Option D is false because standard error generally decreases as sample size increases but does not become exactly zero unless sample size is infinite or variability is absent. The central limit theorem supports confidence intervals and hypothesis tests for means. Study Guide references/topics: central limit theorem, sampling distribution, sample mean, normal approximation.

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