Does the Central Limit Theorem apply to the VARIANCE of all n values?
Added by Stephanie R.
Step 1
By "the VARIANCE of all n values" I take it you mean the sample variance computed from n i.i.d. observations X1,...,Xn. The question asks whether a Central Limit Theorem–type result holds for that sample variance (i.e., whether it is asymptotically normal). Show more…
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The Central Limit Theorem states that "if a sample of size n is drawn from any random variable that has a mean (μ) and a standard deviation (σ), then the distribution of the sample mean approaches a normal distribution with mean (μ) and standard deviation (σ/√n) as the sample size, n, increases." Explain why the Central Limit Theorem is important in Statistics.
Adi S.
The central limit theorem says that, when a simple random sample of size n is drawn from any population with mean µ and standard deviation σ, then when n is sufficiently large the standard deviation of the sample mean is σ²/n. the distribution of the population is exactly Normal. the distribution of the sample mean is approximately Normal. the distribution of the sample mean is exactly Normal.
The variable x is normally distributed, and the sample size is 15. Does the Central Limit Theorem tell us that the sample mean (x) is normally distributed? Why or why not?
Sanchit J.
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