Sampling Distribution of the Sample Mean Concerning estimations/intervals: True or

loraliyeruxi

loraliyeruxi

Answered question

2022-03-31

Sampling Distribution of the Sample Mean
Concerning estimations/intervals:
True or False: if the three conditions Random, Normal, and Independent for using a confidence interval for a population mean are not met, then the sampling distribution of x is unknown.
I'm unsure about the criteria - would you not know the distribution of the sample mean even if these criteria are unfilled? Are there specific conditions for doing so?

Answer & Explanation

Cailyn Hanson

Cailyn Hanson

Beginner2022-04-01Added 11 answers

It is difficult to make valid true-false questions about general statistical principles, especially ones as detailed as this one. In making such questions, the danger is that one might not have thought of a situation beyond the material in the current chapter. My guess is that the authors of this one intended the answer to be True, but I don't think so.
Let X1,X2,,X10 be a random sample from Exp(rate=λ), the exponential distribution with rate λ and mean μ=1λ. Then XΓ(shape=n,rate=nλ), which has mean E(X)=μ=1λ, variance V(X)=μ2n=1λ2n and SD(X)=μn=1λn. [The reason for showing both μ's and λ's is that some books define the exponential distribution using the mean and others using the rate.]
Then when using data X1,X2,,X10 to estimate the population mean μ, the standard error (standard deviation) of the mean is μn.
Thus you have (1) random sampling and (3) independence, but not (2) normality. However, the standard error is known.
For sufficiently large n, by the Central Limit Theorem, X is approximately normal. So for large n, one might use the X±1.96Xn as an approximate 95% confidence interval for μ.
A better 95% CI for μ, based on the gamma family of distributions, is valid for all n: Let L and U cut 2.5% from the lower and upper tails of Gamma(n,n) (found using software). Then the confidence interval for μ is (XU,XL).. This method does not use the standard error.

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