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・拟合模型与残差进阶
The scatterplot shows the relationship between two variables, x and y.
Which of the following graphs shows a line of best fit that is most appropriate for the data shown in the scatterplot?
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. A line of best fit for the data is also shown.
For how many of the 9 data points is the actual y-value greater than the y-value predicted by the line of best fit?
原库参考解析
散点图・拟合模型与残差挑战
The scatterplot shows data set A, which consists of the weights y, in pounds, of a Labrador retriever puppy at various ages, x, in months. The equation of a line of best fit for the relationship in data set A can be written as \( y=-5.1+8.7x \) , where \( 2\le x\le6 \).
The puppy was weighed again at 9 months old and weighed 52 pounds. Data set B consists of all the data points in data set A as well as the data point (9, 52). The equation of a line of best fit for data set B can be written as \( y=r+sx \), where r and s are constants and \( 2\le x\le9 \). Assuming the equations of the lines of best fit are calculated in the same way, which of the following is the best estimate for the value of s?
原库参考解析
散点图・拟合模型与残差挑战
The scatterplot shows data set A, which consists of the weights y, in pounds, of a Labrador retriever puppy at various ages, x, in months. The equation of a line of best fit for the relationship in data set A can be written as \( y=-5.6+8.8x \), where \( 2\le x\le6 \) .
The puppy was weighed again at 9 months old and weighed 52 pounds. Data set B consists of all the data points in data set A as well as the data point (9, 52). The equation of a line of best fit for data set B can be written as \( y=r+sx \) , where r and s are constants and \( 2\le x\le9 \) . Assuming the equations of the lines of best fit are calculated in the same way, which of the following is the best estimate for the value of s?
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, t and d.
Which of the following equations is the most appropriate linear model for the data 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 \( x \) and \( y \) for the 10 data points in data set G.
Data set H is created by multiplying the y-value of each data point from data set G by 20. Which of the following equations is the most appropriate model for data set H?
原库参考解析
散点图・拟合模型与残差进阶
As part of a study on sedimentary habitats, the water temperature was recorded at different depths below the surface of the water in the northern Alboran Sea.
The data shown in the scatterplot give the recorded temperature, in degrees Celsius, for 7 depths, in meters, below the surface of the water.
Which of the following is closest to the slope of a line of best fit for the data shown?
原库参考解析
散点图・拟合模型与残差进阶
For the first 10 hours of an experiment, the scatterplot shows the temperature y, in degrees Fahrenheit (°F), of an object at various times x, in hours, since the start of the experiment. A line of best fit is also shown.
Which of the following is the best interpretation of the slope of the line of best fit?
原库参考解析
散点图・拟合模型与残差进阶
For the first hours of an experiment, the scatterplot shows the temperature y, in degrees Fahrenheit (°F) of an object at various times x, in hours, since the start of the experiment. A line of best fit is also shown.
Which of the following is the best interpretation of the slope of the line of best fit?
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 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
A group of 10 gardeners recorded data on the germination rates of their tomato crop for one growing season. The scatterplot shows the relationship between the number of tomato seeds planted, x, and the number of tomato seeds that germinated, y, for each of the gardeners. A line of best fit is also shown.
Which of the following is the best interpretation of the slope of the line of best fit in this context?
原库参考解析
散点图・拟合模型与残差进阶
The scatterplot shows the relationship between the square footage of 12 clothing stores and their annual sales, in millions of dollars. According to the trend shown by the data, which of the following is the best prediction for the annual sales, in millions of dollars, of a clothing store that is 4 thousand square feet?
原库参考解析
