Students continue their exploration of residuals. In this lesson, students build on their knowledge of calculating residuals and expand their practice by creating residual plots. Additionally, students reason abstractly by thinking about how a particular pattern in a scatter plot will be represented in the residual plot.
The predicted y-value is calculated using the equation of the least-squares line.
The residual is calculated using:
residual = actual y value - predicted y value
The sum of the residuals provides an idea of the degree of accuracy when using the least-squares line to make predictions.
To make a residual plot, plot the x-values on the horizontal axis and the residuals on the vertical axis.
Example 1: Calculating Residuals
The curb weight of a car is the weight of the car without luggage or passengers. The table below shows the curb weights (in hundreds of pounds) and fuel efficiencies (in miles per gallon) of five compact cars.
Using a calculator, the least-squares line for this data set was found to have the equation:
y = 78.62 - 1.5290x
where is the curb weight (in hundreds of pounds) and is the predicted fuel efficiency (in miles per gallon).
1. Will the residual for the car whose curb weight is 25.33 be positive or negative? Roughly what is the value of the residual for this point?
2. Will the residual for the car whose curb weight is 27.79 be positive or negative? Roughly what is the value of the residual for this point?
3. Calculate the residuals.
It is often useful to make a graph of the residuals, called a residual plot. You will make the residual plot for the compact car data set.
1. Suppose you are given a scatter plot (with least-squares line) that looks like this:
What would the residual plot to look like? (Make a quick sketch on the axes given below. There’s no need to plot the points exactly.)2. Suppose the scatter plot looked like this:
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