A filled point on a scatterplot shows what happened, while a line of best fit predicts what the model expects at the same horizontal position. Use two labeled points on the line to find its slope, then compare the prediction with the filled point. If the vertical axis is in hundreds, convert the difference to individual customers before multiplying by revenue per customer.
Hints
- Hint 1
The observed value comes from the filled dot , but the prediction comes from the solid line at the same ad spending. Which two labeled open circles give you points on that line?
- Hint 2
A line’s slope is its change in output divided by its change in input. Find it from the two marked points on the line, then use it to predict the height of the line at ’s horizontal position.
- Hint 3
The residual compares the actual output with the predicted output at the same input. Once you find that difference, remember that each vertical-axis unit represents one hundred customers, not one customer.
Step-by-step
Compare the dot with the fitted line
Step 1Separate the observation from the prediction
Read as : at that ad spending, the company actually gained hundreds of customers. The open circles and lie on the solid line. They describe the model, while the filled dot is the observation. Using as a point on the line would erase the gap you need to find.
- Step 2
Find the line’s slope
The slope, , is the change in predicted customer hundreds divided by the change in ad-spending thousands. Type in Desmos. It shows , so the line rises by 2 hundreds of customers for each additional thousand dollars of spending.
- Step 3
Predict customers at Q’s spending
Start at the line’s point . To reach ’s spending of thousand dollars, add times the horizontal change to . Type ; Desmos shows . So the model predicts hundreds of customers at the same spending level where shows hundreds.
- Step 4
Find the extra customers
The residual is actual minus predicted. Subtract the line’s value from the dot’s value:
That’s 5 hundreds of extra customers, not 5 individual customers.
- Step 5
Convert the difference to revenue
Each of those 5 hundreds contains customers, and each customer generated $30. Type ; Desmos shows . The actual first-month revenue was $15,000 greater than the model predicted. Choice C.