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Answer :
Final answer:
The central limit theorem states that the distribution of sample means will approach a normal distribution as the sample size increases, which is True, even if the population distribution is skewed.
Explanation:
The student's question pertains to the central limit theorem, which is a fundamental concept in statistics and probability theory. The main answer to the student's question is True. The central limit theorem describes that, regardless of the population's distribution, the distribution of sample means will approximate a normal distribution, given that the sample size is large enough. This principle holds true even if the original population distribution is skewed.
The explanation behind this is that as the sample size increases, the sampling distribution of the sample means becomes less influenced by the shape of the original population's distribution. By the time you have a large enough sample, typically considered as 30 or more, the distribution of the sample means will be nearly normal. This is due to the averaging out of individual sample variabilities, leading to a more stable and predictable pattern that resembles the normal distribution.
Standard error of the mean, which is the standard deviation of the sample means distribution, is also an important concept associated with the central limit theorem. It shows how much the sample mean is expected to vary from the actual population mean and is inversely related to the square root of the sample size.
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