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Answer: For simple random samples of size n from a population with mean μ and finite variance σ², the sampling distribution of the sample mean approaches the normal distribution with mean μ and variance σ²/n, as the sample size becomes large
The Central Limit Theorem states that the sampling distribution of the sample means approaches a normal distribution as the sample size gets larger — no matter what the shape of the population distribution. This fact holds especially true for sample sizes over 30. All this is saying is that as you take more samples, especially large ones of size 30 or more, your graph of the sample means will look more like a normal distribution. Please note that it's the sample mean that's normally distributed, not the sample itself. This is why option C is incorrect. This handy result considerably simplifies the computation of probabilities and the construction of a statistical hypothesis. So long as we have sample statistics, we can draw relevant conclusions about the actual population regardless of the population's distribution, provided n is sufficiently large (n is usually taken to be ≥ 30).
Author: Nikitesh Somanthe
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Which of the following best describes the central limit theorem?
A
When the sample size is large, the sum of independent and identically distributed (i.i.d.) random variables are normally distributed
B
The sum of n independent and identically distributed random variables approaches the normal distribution as n becomes large
C
For simple random samples of size n from a population with mean μ and finite variance σ², the sampling distribution approaches the normal distribution with mean μ and variance σ²/n, as the sample size becomes large
D
For simple random samples of size n from a population with mean μ and finite variance σ², the sampling distribution of the sample mean approaches the normal distribution with mean μ and variance σ²/n, as the sample size becomes large