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小悟空 · 2020年06月01日

问一道题：NO.PZ2015120604000118

  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

老师这道题查表时，怎么知道是查双尾还是单尾的啊？ 谢谢！

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已采纳答案

星星_品职助教 · 2020年06月01日

同学你好，

这道题目要求的是置信区间，正常而言，置信区间的查表都是双尾的。可以回忆一下置信区间的图像，是在分布中的一个“区间”，也就意味着这个区间左侧，右侧，都各有一个这个区间没有覆盖到的尾巴。

以本题为例，99%的置信区间指的是分布中间99%的面积，所以左右空出来的那两个尾巴的总面积就是1%，所以查t表要对应双尾。

但有可能t表只给了一个单尾的表，以原版书附表为例，给的t表就是one-tailed probability即单尾t表。这个时候就要根据对称性做一步转化，双尾1%对应的关键值就等同于单尾0.5%对应的关键值。所以查表时根据自由度去对应出单尾概率为0.5% 的关键值即可。

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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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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

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