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chuziyang · 2022年04月29日

2.58/2

  1. CFA I
  2. Quantitative

NO.PZ2017092702000114

问题如下：

For a sample size of 65 with a mean of 31 taken from a normally distributed population with a variance of 529, a 99% confidence interval for the population mean will have a lower limit closest to:

选项：

A.

23.64.

B.

25.41.

C.

30.09.

解释：

A is correct.

To solve, use the structure of Confidence interval = Point estimate ± Reliability factor × Standard error, which, for a normally distributed population with known variance, is represented by the following formula:X‾±zα/2σn\overline X\pm z_{\alpha/2}\frac\sigma{\sqrt n}

For a 99% confidence interval, use z0.005 = 2.58. Also, σ = 529\sqrt{529} = 23.

Therefore, the lower limit =31−2.58×2365=23.639831-2.58\times\frac{23}{\sqrt{65}}=23.6398

我们需要用到【置信区间结构】的计算公式解决本题：

the structure of Confidence interval = Point estimate ± Reliability factor × Standard error

即：X‾±zα/2σn\overline X\pm z_{\alpha/2}\frac\sigma{\sqrt n}

当置信区间=99%的时候，Z0.005=2.58，且 σ = 529\sqrt{529} = 23.

所以，下限为： =31−2.58∗2365=23.6398=31-2.58*\frac{23}{\sqrt{65}}=23.6398

請問2.58爲何沒有除以2，公式中不是Zα/2的嗎？謝謝


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星星_品职助教 · 2022年04月29日

同学你好，

α是significance level，也就是本题中的1-99%=1%。而2.58（以及1.65,1.96等）是关键值（critical value），这两者不是同一个概念，α/2是significance level/2，不是critical value/2。

对于数量中的置信区间，左右侧各有一个尾部，这两个尾部的面积共同构成了这1%的significance level，也就是单尾面积为1%/2=0.5%。

所以查表时，要按照单尾0.5%去查对应的关键值（critical value），这就是“α/2”的含义正态分布下，这个关键值查出来就是2.58.

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NO.PZ2017092702000114 问题如下 For a sample size of 65 with a meof 31 taken from a normally stributepopulation with a varianof 529, a 99% confinintervfor the population mewill have a lower limit closest to: A.23.64. B.25.41. C.30.09. A is correct.To solve, use the structure of Confininterv= Point estimate ± Reliability factor × Stanrerror, which, for a normally stributepopulation with known variance, is representethe following formula:X‾±zα/2σn\overline X\pm z_{\alpha/2}\frac\sigma{\sqrt n}X±zα/2​n​σ​ For a 99% confininterval, use z0.005 = 2.58. Also, σ = 529\sqrt{529}529​ = 23. Therefore, the lower limit =31−2.58×2365=23.639831-2.58\times\frac{23}{\sqrt{65}}=23.639831−2.58×65​23​=23.6398 我们需要用到【置信区间结构】的计算公式解决本题the structure of Confininterv= Point estimate ± Reliability factor × Stanrerror即X‾±zα/2σn\overline X\pm z_{\alpha/2}\frac\sigma{\sqrt n}X±zα/2​n​σ​当置信区间=99%的时候，Z0.005=2.58，且 σ = 529\sqrt{529}529​ = 23. 所以，下限为 =31−2.58∗2365=23.6398=31-2.58*\frac{23}{\sqrt{65}}=23.6398=31−2.58∗65​23​=23.6398 老师，这种题型的计算思考步骤是什么呢，总是容易搞混，看题找不到方向？谢谢老师

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