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苏·Xu · 2023年02月27日

卡方

  1. CFA I
  2. Quantitative

NO.PZ2015120604000144

问题如下：

Quintina conducts a hypothesis to test whether the variance of the annually return is 0.0144. Then she selects a random 25 year's data as a sample, Quintina figures out the variance of the sample is 0.013924. Assume the level of significance is 5%, which of the following conclusion is least accurate?

选项：

A.

The population variance is not significant different from 0.0144.

B.

The population variance is significantly different from 0.0144.

C.

This hypothesis is appropriate to use the chi-square test.

解释：

B is correct.

The null hypothesis is H0: σ2=0.0144. It is appropriate to use the two-tailed chi-square test.

The test statistic is:=(25-1)*0.013924/0.0144=23.21

The critical chi-square critical values are 12.4 and 39.36.

Because the test statistic falls between these two values,so Quintina fails to reject the null hypothesis.

So we conclude that the population variance is not significantly different from 0.0144.

题干是问which is the least accurate?所以选B。

critical value 根据公式不应该是 24*23.21 吗？。

critical value 地两个值如何算出来的？

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1 个答案

星星_品职助教 · 2023年02月28日

同学你好，

critical value的值都是查表所得，不需要计算。答案解析中的公式为卡方统计量（Chi-squared statistic）的计算。

-------

附查表过程如下：

根据significant level为5%，可判定左右尾的拒绝域面积各为2.5%。根据sample size=25，可以得到该卡方统计量的自由度为df=n-1=24.

原版书给出的卡方分布表针对的是右尾概率，所以首先查出右尾概率2.5%的关键值为39.364

左尾概率2.5%相当于右尾概率为97.5%，所以查出左侧关键值为12.401


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