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书茜 · 2022年02月04日

不理解

  1. CFA I
  2. Quantitative

NO.PZ2015120604000118

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 for the 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


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。

本题只涉及一次抽样，一次抽样中抽了25个学生的上学时间，这25个学生的平均时间和方差都已知，为什么不能用40+-2.58*8来计算？

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1 个答案

星星_品职助教 · 2022年02月04日

同学你好，

已知的只是这个样本中，学生的平均时间和标准差，而抽样的目的是要推断总体。所以要建立置信区间。

由于置信区间是基于样本均值建立出来的，所以对应的标准差也要是样本均值的标准差（标准误），即8 / 根号25

对应公式如下：

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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

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2022-09-22 16:13 2 · 回答

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