Whenever the population has a normal distribution, the sampling distribution of sample mean is a normal or near normal distribution

Question

BlackTom AI · Accepted Answer

Question restatement: When the population is normally distributed, the sampling distribution of the sample mean is normal (or near normal). 
Option analysis: 
Option: 'for any sample size' — This statement is correct because if the population distribution is normal, then the sampling distribution of the mean is exactly normal for any sample size n. The mean of a sample from a normal population is itself normally distributed with mean equal to the population mean and variance equal to the population variance divided by n. There is no need to rely on the central limit theorem here; the exact distribution holds for all n when the parent distribution is normal. 
Why the other typical alternatives (if present) would be incorrect: If there were an option suggesting that only large sample sizes yield normal sampling distribution, that would be false in this context because the normal population guarantees normality of the sample mean for all n, not just large n. If there were an option claiming the sampling distribution is never normal or only approximately normal, that would contradict the property of normal populations. Since the provided options include only one choice, the reasoning concludes that the single statement aligns with the known result.

Whenever the population has a normal distribution, the sampling distribution of sample mean is a normal or near normal distributionSingle choice

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