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广超 · 2020年10月29日

问一道题：NO.PZ2015120604000118 [ CFA I ]

  1. CFA I
  2. Quantitative

问题如下：

Deli Bur is a high school. A recent survey of 25 student of the high school indicates that the mean time they spend going to school is 40 minutes . This sample's standard deviation is 8 minutes. The distribution of the population is supposed to be normal. The 99% confidence interval for the mean time that all Deli Bur students  spend going to the school is:

选项：

A.

30.72 to 34.55

B.

38.52 to 54.48

C.

35.52 to 44.48

解释：

C is correct.

Because the simple size is less than 30, so the confidence interval fo rthe  population whose variance is unknow is :

x - ± t α/2 (s/ n ) .

To calculate critical value: t 0.005  and df = 24 is 2.797.

So, the confidence interval is 40 ± 2.797(8 / 5) = 40 ± 4.48 = 35.52 to 44.48

老师，答案中这个公式，老师写在讲义里了，可是基础班的课程里没讲这个公式是怎么来的。基础班Estimation的三个视频里没有关于这个公式的内容。Z a/2 是怎么来的？代表什么？
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已采纳答案

星星_品职助教 · 2020年10月30日

同学你好，

置信区间的公式在正态分布知识点里讲过原理。以99%的置信区间为例，通过查表，可知落在在均值μ周围±2.58个标准差范围内的概率为99%。即99%置信区间对应[μ-2.58σ，μ+2.58σ]。其余68%，90%，95%置信区间同理，其余概率基本上都不需要考虑。

这里面2.58对应的就是Z α/2。在置信区间公式里，这个Z α/2叫做reliability factor，也可以叫做critical value，后者更常用。

α是significance level，表示置信区间外的总面积（拒绝域）为1%。由于对称，那么单尾拒绝域的面积就是α/2=1%/2=0.5%. 在正态分布（用Z表示）下，查表即可得这个概率面积对应的分位点为2.58.

-------------

对应到这道题，可知均值X bar=40，对应的X bar的标准差（即标准误）=8/√25=1.6。 

而critical value需要查表求得，如果是正态分布，这个值就是2.58。但这道题对应的是t分布，需要查的是t表（总体方差未知用t）。

本题α=1%，则α/2=0.005。t分布需要考虑自由度，df=n-1=24。通过对应单尾概率0.005和自由度24查表可得t critical value=2.797.

代入公式40±2.797×1.6即可得到答案C选项。

 

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NO.PZ2015120604000118 问题如下 li Bur is a high school. A recent survey of 25 stunt of the high school incates ththe metime they spengoing to school is 40 minutes . This sample's stanrviation is 8 minutes. The stribution of the population is supposeto normal. The 99% confinintervfor the metime thall li Bur stunts spengoing to the school is: A.30.72 to 34.55 B.38.52 to 54.48 C.35.52 to 44.48 C is correct.Because the simple size is less th30, so the confinintervfor the population whose varianis unknow is : x - ± t α/2 (s/ n ) . To calculate criticvalue: t 0.005 an = 24 is 2.797.So, the confinintervis 40 ± 2.797(8 / 5) = 40 ± 4.48 = 35.52 to 44.48本题由于样本数量小于30，且总体方差未知。可知均值X bar=40，对应的X bar的标准差（即标准误）=8/√25=1.6。 而criticvalue需要查表求得，如果是正态分布，这个值就是2.58。但这道题对应的是t分布，需要查的是t表（总体方差未知用t）。本题α=1%，则α/2=0.005。t分布需要考虑自由度，=n-1=24。通过对应单尾概率0.005和自由度24查表可得t criticvalue=2.797.代入公式40±2.797×1.6即可得到答案 如何看出这道题总体方差未知呢？

2023-04-12 19:27 1 · 回答

NO.PZ2015120604000118问题如下 li Bur is a high school. A recent survey of 25 stunt of the high school incates ththe metime they spengoing to school is 40 minutes . This sample's stanrviation is 8 minutes. The stribution of the population is supposeto normal. The 99% confinintervfor the metime thall li Bur stunts spengoing to the school is:A.30.72 to 34.55B.38.52 to 54.48C.35.52 to 44.48 C is correct.Because the simple size is less th30, so the confinintervfor the population whose varianis unknow is : x - ± t α/2 (s/ n ) . To calculate criticvalue: t 0.005 an = 24 is 2.797.So, the confinintervis 40 ± 2.797(8 / 5) = 40 ± 4.48 = 35.52 to 44.48本题由于样本数量小于30，且总体方差未知。可知均值X bar=40，对应的X bar的标准差（即标准误）=8/√25=1.6。 而criticvalue需要查表求得，如果是正态分布，这个值就是2.58。但这道题对应的是t分布，需要查的是t表（总体方差未知用t）。本题α=1%，则α/2=0.005。t分布需要考虑自由度，=n-1=24。通过对应单尾概率0.005和自由度24查表可得t criticvalue=2.797.代入公式40±2.797×1.6即可得到答案百分之99对应的不是2.58吗，为啥答案是2.7几

2023-04-10 19:58 1 · 回答

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2022-12-05 20:52 2 · 回答

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2022-09-22 16:13 2 · 回答

NO.PZ2015120604000118问题如下li Bur is a high school. A recent survey of 25 stunt of the high school incates ththe metime they spengoing to school is 40 minutes . This sample's stanrviation is 8 minutes. The stribution of the population is supposeto normal. The 99% confinintervfor the metime thall li Bur stunts spengoing to the school is:A.30.72 to 34.55B.38.52 to 54.48C.35.52 to 44.48 C is correct.Because the simple size is less th30, so the confinintervfor the population whose varianis unknow is : x - ± t α/2 (s/ n ) . To calculate criticvalue: t 0.005 an = 24 is 2.797.So, the confinintervis 40 ± 2.797(8 / 5) = 40 ± 4.48 = 35.52 to 44.48本题由于样本数量小于30，且总体方差未知。可知均值X bar=40，对应的X bar的标准差（即标准误）=8/√25=1.6。 而criticvalue需要查表求得，如果是正态分布，这个值就是2.58。但这道题对应的是t分布，需要查的是t表（总体方差未知用t）。本题α=1%，则α/2=0.005。t分布需要考虑自由度，=n-1=24。通过对应单尾概率0.005和自由度24查表可得t criticvalue=2.797.代入公式40±2.797×1.6即可得到答案老师好，请问下，是单尾、为什么查表不是按1%查

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