Worked solution
Compare two models using squared residuals
A random sample of five employees has ages years and monthly earnings thousand dollars.
| Employee | ||
|---|---|---|
| A | 19 | 3.16 |
| B | 21 | 3.21 |
| C | 26 | 3.34 |
| D | 28 | 4.03 |
| E | 32 | 3.99 |
(i) Alex uses . Calculate the sum of squared residuals for Alex’s line.
(ii) Bertrand uses another straight line whose sum of squared residuals is 2. State, with a reason, which model gives a better fit.
CJC · Lecture · Example 6
(i) Square each observed-minus-predicted difference
Answer
(in squared thousands of dollars).
The source prints all five predicted values: . Pair them with the observed earnings and calculate:
| Age | Observed | Predicted | Residual |
|---|---|---|---|
| 19 | 3.16 | 3.70 | −0.54 |
| 21 | 3.21 | 3.90 | −0.69 |
| 26 | 3.34 | 4.40 | −1.06 |
| 28 | 4.03 | 4.60 | −0.57 |
| 32 | 3.99 | 5.00 | −1.01 |
The residuals, squares and final sum complete the blank Student working space. Square each residual before adding; do not square their sum or take a square root.
(ii) Compare the same error measure on the same data
Answer
Bertrand’s model gives the better fit because .
The source states that Bertrand’s smaller sum of squared residuals makes it a better fit. Both values compare vertical earnings errors for these same five observations. This comparison does not establish that Bertrand’s line is the least-squares optimum among every possible line, or that either model predicts other populations accurately.
Solution notes
Printed predictions and model-comparison conclusion retained. Residual arithmetic and units authored for the blank sum. No unsupported equation for Bertrand’s line invented; no printed marks.