Question 54·Hard·Two-Variable Data: Models and Scatterplots
The scatterplot titled "Used Car Price vs. Mileage" shows a line of best fit for the relationship between mileage (in miles) and price (in dollars).
Two points on the line of best fit are labeled and . A point labeled "Car" is at .
For a data point, the residual is defined as
Which choice is the residual for the point labeled "Car"?
For residual questions from a line of best fit on a scatterplot, first determine the line equation (often by using two labeled points), then plug in the given -value to get the predicted , and finally compute residual = actual − predicted. Check the sign: points above the line have positive residuals, and points below the line have negative residuals.
Hints
Use the two labeled points on the line
Compute the slope using and .
Find the predicted price for 55,000 miles
After you have the line equation, substitute to get the predicted .
Be careful with the subtraction order
Residual is defined as actual minus predicted. A positive residual means the actual point is above the line.
Desmos Guide
Enter the two points on the line
In Desmos, enter the labeled line points:
A=(20000,20500)
B=(60000,12500)
Create the line and predict at 55,000 miles
Find the slope and define the prediction function:
k=(y(B)-y(A))/(x(B)-x(A))
b=y(A)-k*x(A)
p(m)=k*m+b
Then evaluate:
pred=p(55000)
Compute the residual
Enter the actual price and subtract:
actual=15000
residual=actual-pred
The result is the residual for the car, in dollars.
Step-by-step Explanation
Find the slope of the line of best fit
Using the two labeled points and ,
Write an equation for the line
Use and substitute :
So the line of best fit is
Predict the price at 55,000 miles
Substitute :
Compute the residual (actual minus predicted)
The actual price of the car is , and the predicted price is , so
Therefore, the residual is $1,500.