If repeated random samples of size 40 are taken from an infinite population, the distribution of sample means

Question

BlackTom AI · Accepted Answer

Consider the statement: If repeated random samples of size 40 are taken from an infinite population, the distribution of sample means will reflect certain probabilistic principles.
Option 1: 'will always be normal because we do not know the distribution of the population.' This is not correct. The distribution of sample means is not guaranteed to be normal in every population, especially if the population is highly skewed or has heavy tails; the guarantee comes from the Central Limit Theorem under appropriate conditions, not from unknown population shape alone.
Option 2: 'will always be normal because the sample mean is always normal.' This is misleading. The sample mean itself is a random variable whose distribution can be approximately normal for large enough samples, but it is not inherently always normal for any sample size or population; the approximation depends on sample size and population characteristics.
Option 3: 'will be approximately normal because of the Central Limit Theorem.' This aligns with the Central Limit Theorem, which states that the distribution of the sample mean tends toward normality as the sample size increases, here n = 40, which is typically sufficient for many populations to yield an approximately normal distribution of the sample mean.
Option 4: 'will not be normal because we do not know the distribution of the population.' This overstates the limitation. While the population distribution matters, the CLT ensures approximate normality of the sample mean for reasonably large samples regardless of the population’s shape (provided certain conditions are met), so we are not guaranteed non-normality.
In summary, the most accurate assertion among the options is that the distribution of the sample means will be approximately normal due to the Central Limit Theorem, given a sample size of 40 from an infinite population and repeated sampling.

If repeated random samples of size 40 are taken from an infinite population, the distribution of sample means单项选择题

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