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四喜丸子 · 2023年04月12日

关于自由度

  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

请问为什么 No.PZ2017092702000113 (选择题)，这道题的置信区间计算需要使用“分母根号下n-1”，即需要用自由度求解，但是这道题不需要呢？


请问老师区别在哪里？谢谢老师

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星星_品职助教 · 2023年04月12日

同学你好，

两道题的计算方法是一致的。对于提问中涉及其中standard error的计算也是一样的。

1）对于你提到的No.PZ2017092702000113 这道题里，standard error的计算是s / 根号n，分母不是n-1。即standard error=√ (245.55/17)，

具体可见题目截图及对应答案解析。

2）对于本题而言，standard error=standard error=√ (529/65)。同样是standard erro=s / 根号n 的算法。

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此后根据Confidence interval = Point estimate ± Reliability factor × Standard error。代入point estimate（即题干中给出的mean）和reliability factor（本题因是正态分布，直接使用99%对应的2.58；上一道题是t分布，需要查表），即可计算出最终的置信区间。

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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

2023-06-06 23:49 1 · 回答

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 请问老师，我有两个问题，第一个是如果题干没有告诉我应该用到什么分布的话，我怎么判断是应该用z分布还是t分布呢？第二个问题是，我现在很混乱z分布和t分布的计算有什么区别呢》唯一的区别就是t分布的计算是自由度-1是吗，z分布就直接用sample zise的数量？

2022-12-13 06:49 1 · 回答

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 ​为什么样本标准差不是除以根号64呢？

2022-11-06 16:45 1 · 回答

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 ​为什么这个n用的65，但是上一题t分布里面n用的n37，查表用的 36呢？

2022-07-07 15:19 1 · 回答