Choose a representation
Calculate an observed-minus-predicted residual at the same input. When projecting from subgroups, use the correct population weight for each representative sample. Reconstruct missing table cells before conditioning.
Worked example
A model predicts y=2 x+6. An observation at x=4 has y=11. What is its residual, observed minus predicted? Evaluate the prediction at the observed input. Subtract the predicted output from the observed output. Substitute the given numbers: 11-(2 × 4+6) = -3. The requested value is -3.
Contrast a wrong operation
In that example, 3 follows this error: Reverse observed and predicted in the subtraction. The correct requested value is -3. Explain which quantity the mistaken method is actually calculating.
Connect a second form
A school has 400 juniors and 800 seniors. Representative subgroup samples suggest 30% of juniors and 60% of seniors support a plan. What is the estimated total number supporting it? Project each sample proportion onto its own subgroup population. Add the two subgroup estimates; do not give the groups equal population weights. Substitute the given numbers: 400 × 0.3+800 × 0.6 = 600. The requested value is 600.
Transfer and check
A model predicts y=2 x+7. An observation at x=5 has y=14. What is its residual, observed minus predicted? Pause before checking. Answer: -3. Evaluate the prediction at the observed input. Subtract the predicted output from the observed output. Substitute the given numbers: 14-(2 × 5+7) = -3.
Boundaries of the method
A fitted line is not a causal claim. A sample estimate depends on representativeness, and a conditional probability uses the conditioned group rather than the whole table. This lesson samples a PSAT/NMSQT objective; its examples are original and do not claim official question status or full coverage.
A model predicts y=2 x+6. An observation at x=4 has y=11. What is its residual, observed minus predicted?
- Evaluate the prediction at the observed input.
- Subtract the predicted output from the observed output.
- Substitute the given numbers: 11-(2 × 4+6) = -3.
- The requested value is -3.
Compare the other answer choices
Reverse observed and predicted in the subtraction.
Report the predicted output rather than the residual.
Multiply the residual by the model slope a second time.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
Math specifications