Question 18·Hard·Two-Variable Data: Models and Scatterplots
Each dot in the scatterplot above represents an apartment listing in a city, where is the distance from downtown (in kilometers) and is the monthly rent (in hundreds of dollars). A line of best fit for the data is shown, and one data point is labeled .
The residual for a point is defined as
Suppose the monthly rent for the apartment represented by point decreases by $200, and the line of best fit is shifted upward by (hundreds of dollars) at every -value.
Which choice gives the new residual for point , in dollars?
For harder residual questions, compute the original residual structure first (actual minus predicted at the same ), then apply any stated changes to the actual value and/or the prediction separately, staying in the graph’s units (here, hundreds of dollars). Only after the final subtraction should you convert to dollars by multiplying by .
Hints
Start with actual minus predicted
Read point ’s actual -value and the line’s predicted -value at the same before making any changes.
Convert the $200 change into the graph’s units
Because is in hundreds of dollars, convert $200 into “hundreds of dollars” and adjust the actual rent accordingly.
Account for the line shift, then convert back to dollars
An upward shift of adds to the prediction. After finding the new residual in hundreds of dollars, multiply by to get dollars.
Desmos Guide
Enter the line of best fit
Using the two points on the line, compute the slope and intercept , then enter
.
Find the predicted value at
Evaluate the line at to get the original predicted value (hundreds of dollars).
Apply the changes and compute the residual
Decrease the actual value by (because $200 is hundreds of dollars) and increase the predicted value by (due to the vertical shift). Compute (actual − predicted), then multiply by to convert the residual to dollars.
Step-by-step Explanation
Find the original actual and predicted rents at point
From the graph, point is at , so the original actual rent is (hundreds of dollars).
Using the line of best fit (from the graph, it passes through and ), the predicted value at is (hundreds of dollars).
Adjust the actual rent (in hundreds of dollars)
A decrease of $200 is a decrease of (hundreds of dollars).
So the new actual rent is
(hundreds of dollars).
Adjust the predicted rent for the shifted line
Shifting the line upward by (hundreds of dollars) increases the predicted value by at every .
So the new predicted rent at is
(hundreds of dollars).
Compute the new residual and convert to dollars
Residual (in hundreds of dollars):
Convert to dollars:
Therefore, the new residual for point is dollars.