这部分需要会什么？
- 从散点图识别正相关、负相关与非线性趋势
- 解释拟合直线的斜率和预测值
- 用观测值减预测值计算残差
核心概念与公式
散点图表示两变量的成对观测。拟合模型 \(\hat y=mx+b\) 给出预测值；残差定义为 \(y-\hat y\)。残差为正表示点在模型上方，为负表示点在模型下方。
拟合线描述整体趋势，不必通过每个点，也不保证预测准确。相关性不自动证明因果关系。远离观测范围的外推可能失效；明显弯曲的趋势通常不适合只用直线概括。
做题时怎样使用
- 先看变量与单位，再判断趋势方向和模型形式
- 把输入代入模型得到预测值
- 若求残差，使用实际值减预测值；若问系数含义，用单位解释
容易混淆的地方
残差的减法顺序不能颠倒。把拟合线看成所有观测都必须满足的精确等式，会误解数据。
用一个例题串起方法
知识点教学例题拟合模型为 \(\hat y=2.5x+8\)。当 \(x=6\) 时，实际观测值为 26，求残差。
展开解题思路 ＋
预测值 \(\hat y=2.5\times6+8=23\)。残差为 \(26-23=3\)，观测点位于拟合线上方。
本知识点题库
先独立作答，再查看答案与解析。
散点图・拟合模型与残差基础
The scatterplot shows the relationship between the weight, in pounds, of each of 9 female gray wolves on April 30 and the number of offspring each gray wolf produced.
How many offspring did the 50-pound gray wolf produce?
原库参考解析
散点图・拟合模型与残差挑战
An orchard owner recorded the weight, in pounds, of all nectarines that grew on a dwarf nectarine tree during each growing season after the tree's transplantation. The scatterplot shows this weight, in pounds, for each growing season after the tree's transplantation.
What was the weight, to the nearest pound, of all nectarines that grew on the tree during the 4th growing season after the tree's transplantation?
原库参考解析
散点图・拟合模型与残差进阶
The scatterplot shows the relationship between x and y. A line of best fit is also shown.
Which of the following is closest to the slope of the line of best fit shown?
原库参考解析
散点图・拟合模型与残差进阶
The scatterplot shows the relationship between two variables, x and y.
Which equation is the most appropriate linear model for this relationship?
原库参考解析
散点图・拟合模型与残差挑战
The scatterplot shows the relationship between two variables, x and y. A line of best fit is also shown. For how many of the 11 data points does the line of best fit predict a greater y-value than the actual y-value?
原库参考解析
散点图・拟合模型与残差进阶
Which of the following equations is the most appropriate linear model for the data shown in the scatterplot?
原库参考解析
散点图・拟合模型与残差挑战
The scatterplot shows the relationship between two variables, x and y. A line of best fit is also shown.
Which of the following equations best represents the line of best fit shown?
原库参考解析
散点图・拟合模型与残差挑战
The scatterplot shows the relationship between two variables, x and y. A line of best fit is also shown.
Which of the following equations best represents the line of best fit shown?
原库参考解析
散点图・拟合模型与残差进阶
The scatterplot shows the relationship between two variables, x and y.
Which of the following equations is the most appropriate linear model for the data shown?
原库参考解析
