看不懂这道题的提干意思以及求解逻辑，可否请老师解答的详细一些，谢谢

NO.PZ2015120604000178

问题如下：

BIM is a well-developed public company and its market capitalization growth rates over past few decades are normally distributed.The population has a mean of 10% and a standard deviation of 5.4%. An investor wants to figure out the probability that BIM's average capitalization growth rate will over 12.32% in the next three years (all the data are monthly statistics).Which of the following option is most accurate?

选项：

0.5%.

1%.

5%.

解释：

A is correct.

It is appropriate to use the central limit theroy to calculate. Standard error=5.4%/ $\sqrt {\rm{36}}$ P(X＞12.32%)=P(Z＞ (12.32%-10%)/0.009)=P(Z＞2.58). So the corresponding probability is 0.5%.

看不懂这道题的提干意思以及求解的顺序和逻辑，请老师解答的详细一些，谢谢

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品职助教_七七 · 2024年01月03日

嗨，从没放弃的小努力你好：

这道题求的是the probability that BIM's average capitalization growth rate will over 12.32% in the next three years (all the data are monthly statistics)，关键词是average。

根据这个描述可以看出，要求的是X的“均值”，即X bar大于12.32%的概率（X为BIM 's capitalization growth rate）。

X bar大于12.32%的概率无法直接得出，但根据中心极限定理可知，X bar服从正态分布，所可以通过正态分布标准化的公式，将P(X＞12.32%)转化成可以查表的标准正态分布，即z分布的形式，进而查表求出概率。

P(X bar＞12.32%)=P(Z＞(12.32%-X bar的均值)/X bar的标准差)

同样根据中心极限定理，X bar的均值=总体均值=10%； X bar的标准差（也就是standard error）=总体标准差/ √n=5.4%/ √36=0.9%，

所以P(X bar＞12.32%)=P(Z＞(12.32%-10%)/0.9%)= P(Z＞2.58)

可以通过标准正态分布表查出2.58对应的左侧概率是0.9951，所以P(Z＞2.58)=0.0049≈0.5%。

也可以通过上课要求背诵的正态分布置信区间关键值，直接得出2.58对应99%的置信区间，所以右侧尾部面积为1%/2=0.5%。

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看不懂这道题的提干意思以及求解逻辑，可否请老师解答的详细一些，谢谢

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