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JY123 · 2023年11月12日

可以

NO.PZ2023061301000003

问题如下:

A population has a non-normal distribution with mean μ and variance σ2. The sampling distribution of the sample mean computed from samples of large size from that population will have:

选项:

A.

the same distribution as the population distribution.

B.

its mean approximately equal to the population mean.

C.

its variance approximately equal to the population variance.

解释:

B is correct. Given a population described by any probability distribution (normal

or non-normal) with finite variance, the central limit theorem states that the

sampling distribution of the sample mean will be approximately normal, with the

mean approximately equal to the population mean, when the sample size is large.

请问, 什么时候 sample mean 等于population mean, 什么时候   standard deviation of the sample 等于 standing deviation of the population ?  此题可以考虑  standard deviation of the sample 等于 standard deviation of the population 吗? 谢谢啦

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星星_品职助教 · 2023年11月12日

同学你好,

本题考查中心极限定理的应用,针对的是样本均值(sample mean)的分布。

当N>30时,样本均值作为一个随机变量服从正态分布,其均值等于总体均值μ,其标准差(也称标准误)等于总体标准差σ/根号n。

此外,如果总体标准差σ未知,可以用样本标准差s来替代。在这种情况下,样本均值的标准差(标准误)=s/根号n。


提问中的几个问题都可以根据上述原则来判断。sample mean作为一个随机变量的mean等于population mean(本题B选项);standard deviation of the sample 可以在standard deviation of the population未知时做替代。但sample mean的standard deviation≠ standard deviation of the sample(C选项错误),而是等于standard deviation of the sample / 根号n。

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