问一道题：NO.PZ2015120604000178 [ CFA I ]

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.

$Standatd error = \frac{5.4 %/}{\sqrt{6}} = 0.009$

P(X＞12.32%)=P(Z＞ (12.32%-10%)/0.009)=P(Z＞2.58). So the corresponding probability is 0.5%.

这题看完题目没有思路，不知道从哪里着手。请老师讲解一下？

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星星_品职助教 · 2020年04月20日

同学你好，

这道题要求是BIM的 average capitalization growth rate超过12.32%的概率，这里的关键词是“average”，说明要求的是是 X bar也就是X的均值＞12.32%的概率。所以就要看X bar是什么分布（如果是正态分布，就可以做标准化）。然后通过题干可以看出，这道题满足中心极限定理的条件 1. 总体方差已知 2. N＞30 （因为是月度数据取值，所以sample size N=3*12=36）。所以可以得到X bar服从正态分布，且X bar的均值就为总体均值10%，X bar的标准差 (标准误) 即为已知的总体标准差除以根号N，也就是答案解析中的0.009。接下来就是正常的正态分布标准化计算了，可以得到X bar＞12.32%标准化后相当于Z＞2.58，查表或者用背的2.58对应99%置信区间的规律都可以得出概率为0.5%，

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H. · 2020年04月21日

谢谢老师，突破点突然就清楚了。

星星_品职助教 · 2020年04月21日

加油～

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问一道题：NO.PZ2015120604000178 [ CFA I ]

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