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文志(Wenzhi) · 2023年07月14日

怎么理解这个题目呢

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NO.PZ2023062701000012

问题如下：

A researcher is conducting a study to examine the relationship between the amount of time spent studying and the scores achieved on a final exam. The researcher collects data from a sample of 100 students. Which of the following assumptions of linear regression is most likely to be violated in this study?

选项：

A.

Linearity

B.

Homoskedasticity

C.

Independence

解释：

In this study, the assumption of homoskedasticity is most likely to be violated. Homoskedasticity in linear regression assumes that the variance of the residuals (the difference between observed and predicted values) is constant across all levels of the independent variable. However, in the context of studying, it is plausible to assume that the variance of scores achieved on the final exam may differ depending on the amount of time spent studying. For example, students who study for longer periods might have more consistent and less variable exam scores compared to those who study less. Therefore, the assumption of homoskedasticity may not hold in this study.

X应该是学习时间，Y是分数，为什么会是学习时间长方差就小，学习时间短方差大，这个结论是怎么来的。即使是这样，那怎么解决，难道是学习时间短的学生多抽样学习时间长的少抽样么？请帮忙解惑，谢谢

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1 个答案

星星_品职助教 · 2023年07月14日

同学你好，

1）同方差指的是残差项的方差是否为固定的值。本题中残差为：实际考试成绩 - 模型预测出的考试成绩，如果该残差比较稳定，则残差的方差也趋于固定值，满足同方差。但如果该残差随着学习时间的改变呈现一会大一会小，则残差的方差不稳定，不是同方差；

2）本题的思路为学习时间长，成绩会比较稳定。学习时间短，成绩波动比较大。这是一个比较符合现实的假定。学的时间短，成绩偶然性比较大；

3）由于学习时间短造成成绩不稳定，会导致残差波动比较大，此时方差要比学习时间长时（成绩稳定时）的方差大。构成异方差；

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本题异方差的解决方案不是考察的内容，实际情况往往会比较复杂，经常不能简单的用修正异方差的技术去强行解决，其中一种可能的解决方案是将学习时间短和学习时间长的两部分学生分开建立两个不同的模型。这需要根据研究的具体目的和可获取的数据而定，超出了本题的范围，了解即可。

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