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*Baby 8 肖闲* · 2021年02月05日

问一道题：NO.PZ2015120604000137 [ CFA I ]

  1. CFA I
  2. Quantitative

问题如下：

Ida Lee and his team collect a lot of analysis of statistical data about the car market .According to the data, they come to the conclusion that the mean price of SUV is more than $46,435 in his country. Then they randomly select 64 SUVs whose mean price is $46,760. Assuming the standard deviation of the sample is 1370. Ida Lee conducts a hypothesis test to check their conclusion. At which significant level should Ida Lee reject the null hypothesis?

选项：

A.

0.5%.

B.

5%.

C.

1%.

解释：

B is correct.

If the significant level is 5%, the critical value=1.669<1.8978, Ida Lee should reject the null.

If the significant level is 0.5%, the critical value=2.656>1.8978, Ida Lee will fail to reject the null.

If the significant level is 1%, the critical value=2.387>1.8978, Ida Lee will fail to reject the null.


老师，这道题的原假设和备责假设分别都是什么呢？而且，我的理解是这个比较是单边的，如果是单边的话，那 level  是5%的话，对应的不是1。96 吗？还是我的理解上有什么错误？谢谢
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星星_品职助教 · 2021年02月05日

同学你好，

这道题首先要建立原假设和备择假设，题干中描述“they come to the conclusion that the mean price of SUV is more than $46,435”，可以看出大于$46,435是Ida Lee相信并想去证明的。

建立假设的原则是“想要拒绝的放在原假设，想要证明的放在备择假设”

所以原假设是H0：μ≤46435，备择假设为Ha：μ＞46435。通过备择假设可以得到这是一个拒绝域在右侧的单尾的假设检验。

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

上课要求背的正态分布的关键值对应的都是 置信区间。例如1.96对应95%的置信区间.

在数量科目里，所有的置信区间都位于分布中央位置，所以左右两侧各有一个对称的“尾巴”。对于95%的置信区间而言，这两个尾巴的总面积是5%，所以单侧尾巴面积是2.5%。

如果单尾检验的 significant level=5%，说明单尾面积就是5%，所以这个时候正态分布的关键值就不是1.96了，而是1.65.

可以验证一下，1.65对应的是正态分布下90%的置信区间，所以双尾的面积为10%，单侧尾部面积即为5%。

这道题严格来说应该用t分布查表求关键值，不过一般这个值和正态的差别不是很大，所以对于非精细计算的题目，n很大时，用正态的关键值也可以去做判断。

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