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卓娅 · 2021年11月28日

置信区间

  1. CFA I
  2. Quantitative

NO.PZ2017092702000113

问题如下：

For a sample size of 17, with a mean of 116.23 and a variance of 245.55, the width of a 90% confidence interval using the appropriate t-distribution is closest to:

选项：

A.

13.23.

B.

13.27.

C.

13.68.

解释：

B is correct.

The confidence interval is calculated using the following equation:X‾±tα/2sn\overline X\pm t_{\alpha/2}\frac s{\sqrt n}

Sample standard deviation (s) = 245.55\sqrt{245.55} = 15.670.

For a sample size of 17, degrees of freedom equal 16, so t0.05 = 1.746.

The confidence interval is calculated as：


Therefore, the interval spans 109.5943 to 122.8656, meaning its width is equal to approximately 13.271. (This interval can be alternatively calculated as 6.6357 × 2).

样本标准差的计算如下：

245.55\sqrt{245.55} = 15.670.

当样本=17，自由度=16，那么 t0.05 = 1.746.

自信区间计算如下：


因此，自信区间为：109.5943 - 122.8656，这意味着其宽度大约等于 13.271。 （这个区间也可以计算为 6.6357 × 2）。

老师，最后的公式用的是置信区间，但是置信区间不是+/-1.746S么，怎么成了标准差/n根号

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星星_品职助教 · 2021年11月29日

同学你好，

通过题干描述可知这是在求一个sample mean的置信区间（For a sample size of 17, with a mean of.....）。所以对应的标准差也是sample mean的标准差，即标准误。

根据中心极限定理，sample mean的标准误为总体标准差σ除以根号n。

当总体标准差σ未知时，可用sample standard deviation s进行替代。

所以最终对应的标准差就是 s除以根号n。



卓娅 · 2021年12月01日

谢谢老师

星星_品职助教 · 2022年09月01日

@琳 已经在单独的提问中回复，同样的问题提问一次即可。

琳 · 2022年09月01日

请问老师 置信区间，置信水平，置信度分别代表什么，能用例子或者图示解答吗？谢谢

星星_品职助教 · 2021年12月01日

@卓娅 好的，学习加油~

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