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海伦岛主 · 2020年07月22日

问一道题:NO.PZ2018101001000014

问题如下:

Harry is a supermarket’s employer and he is interested in using math model to solve some problems. Recently, he wants to know the relationship among commodity demand volume(V), commodity prices(P) and consumer income level(I). The dependent variable is commodity demand volume and the other two is the independent variables. After doing the multiple regression analysis, he gets the following results:

What is the upper confidence interval of consumer income level’s regression coefficient at 0.1 level?

选项:

A.

3.467

B.

8.919

C.

9.798

解释:

B is correct.

考点: Interpret Regression Coefficient, Assumptions and F-test.

解析: Regression Coefficient Confidence Interval的公式是b1^±tcSb1^,这里的b1^等于6.193,Sb1^ 等于1.652。已知n为150,所以自由度(df)=n-k-1=147。当样本容量足够大(大于120)时,在α =0.1的情况下,tc近似于1.65,所以对于本题来说,置信区间上限等于6.193+1.65*1.652=8.919,选择B选项。

查表(T分布)可得,单尾90%、自由度120的T:1.289,双侧90%、自由度120的T:1.658,此处为何用1.658?

1 个答案

星星_品职助教 · 2020年07月22日

同学你好,

这道题用1.658不是很精确,因为原版书提供的t表并没有147这个自由度对应的值。只能近似得到t critical value在1.653和1.658之间。

但解决这道题并不用这么精细,随意代入一个值然后找最接近的就可以。或者直接用正态分布的1.645替代然后选最接近的选项也行。

从考试的角度而言,一般critical value会给出几组值,让考生选用最合适的那组,根据自由度和significance level正确选择数字代入即可。

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

如果自行查表,过程附录如下:

查t值需要两个要素,第一自由度,第二对应的significance level

对于本题,“regression coefficient”对应的自由度为n-k-1,通过表格可以看出来n=150,同时有两个自变量,即k=2,所以自由度为150-2-1=147,

对于significance level而言,要求的是“the upper confidence interval ”,所以知道这是一个置信区间的查表问题,对于置信区间而言,significance level对应的都是双尾的面积,所以如果查原版书的单尾t表,单尾面积就为0.1/2=0.05.

对应以上两者查表即可得到t critical value在1.658和1.653之间。但这个其实也不用查,因为n很大的时候,t分布近似于正态分布,根据我们背诵的正态分的一组值,可以直接对应出来1.645,直接用这个求近似也行。

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