Measure Signed Prediction Error

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: At x=2, model y=2x+1 predicts 5, but the observed y is 3. Find observed−predicted. Candidate answers: (A) -2 (B) 0 (C) 2 (D) 8.
• Independent check inputs: At x=3, model y=2x+1 predicts 7, but the observed y is 4. Find observed−predicted. Candidate answers: (A) -3 (B) 0 (C) 11 (D) 3.

Learn the idea

Paired measurements describe how two variables vary together. A trend line summarizes association, predicted outputs can be compared with observations using residuals, and the observed input range limits how confidently that trend can be extended. A residual is observed output minus predicted output. Its sign identifies whether an observed point sits above or below the model, and its magnitude measures that vertical discrepancy.

Here is the method on the supplied teaching case. Residual=observed−predicted=3−5=−2. The observed point is 2 output units below the line.

Subtract predicted from observed in that order. A negative residual means overprediction, not a negative observation; compare residuals in the output’s units rather than treating them as input-coordinate differences.

1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
3. Solve the independent check with its changed inputs before revealing feedback.
4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.

Check your reasoning

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Take it further

Changed-input task: At x=9, model y=2x+1 predicts 19, but the observed y is 10. Find observed−predicted. Candidate answers: (A) -9 (B) 0 (C) 29 (D) 9. Show the mathematical reason, not only the selected answer. Expected result: -9. Residual=observed−predicted=10−19=−9. The observed point is 9 output units below the line.

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