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书茜 · 2022年02月04日

样本的标准差和样本均值的标准误是同一个公式吧？

  1. CFA I
  2. Quantitative

NO.PZ2015120604000117

问题如下：

A random sample is 100 CFA candidate's exam scoring. The mean of the scoring is 64. The standard deviation of the population scoring is 15. The distribution is supposed to be normal. The 95% confidence interval for the population mean should be:

选项：

A.

61.06 to 66.94.

B.

61.06 to 69.94.

C.

65.06 to 66.94.

解释：

A is correct.

Because the variance of the population is know and n ≥ 30,so The confidence interval is:

x¯ ± z α/2 (σ/ n )

z α/2 = z 0.025 =1.96

64 ± 1.96(15/10) = 64 ± 2.94 = 61.06 to 66.94.

因为样本方差已知且n>30，我们直接使用置信区间计算：

64 ± 1.96(15/10) = 64 ± 2.94 = 61.06 至 66.94.

如果只抽一次，一次数量大于等于30，那么，这一次数据的标准差等于总体标准差除以根号n么？如果抽100次，每一次数量都大于等于30，每一次都会有一个均值，这些均值又形成一个新的分布，这个新的分布的方差叫做标准误，也等于总体的标准差除以根号n吗？

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已采纳答案

星星_品职助教 · 2022年02月04日

同学你好，

1）只抽一次，只能得到这组数据自身的标准差s。但可以知道的是这组数据的均值服从正态分布，样本均值 的均值为总体均值μ，标准差是总体标准差σ除以根号n。这是中心极限定理的内容。

如果此时σ不知道，就用这组数据自身的标准差s来做近似替代。此时标准误=s/根号n。

2）中心极限定理是对“抽100次”的情况做的总结。有了这个定理后，此后就不需要反复抽样。即使只抽一次我们也知道样本均值的标准差（标准误）就是总体标准差σ除以根号n。



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