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Grade 8 · unit review

Review: Bivariate Trends, Predictions, and Residuals

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.

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Bring the unit skills together

Use these 4 new cases to check your reasoning. Choices are temporary and are not saved to an account, sent to staff or recorded as school grades. Download your current review choices before leaving or reloading.

Unit review questions
1. A supplied scatter record has pairs (1,7), (2,9), (3,11), (4,13). Which description fits?
Need a hint?
  • A scatter representation plots each input with its paired output.
  • Association does not establish causation.
2. A supplied trend model is y=3x+7. What y does it predict at x=8?
Need a hint?
  • A trend equation computes a predicted output from a supplied input.
  • Do not label a predicted value as a measurement.
3. At x=7, model y=2x+1 predicts 15, but the observed y is 8. Find observed−predicted.
Need a hint?
  • A residual is observed output minus predicted output.
  • Subtract predicted from observed in that order.
4. A model was fitted to measurements at x=2 through x=15. Which proposed prediction is extrapolation?
Need a hint?
  • Interpolation predicts within measured input bounds.
  • An equation can produce an extrapolated value without supporting its reliability.

The download contains your current choices and any checked feedback. Keep the separate transfer-task work in your own document; downloading its instructions does not mark it complete.

Use the skills in a new task

Create a four-part mathematical solution sheet for these supplied cases. 1. A supplied scatter record has pairs (1,9), (2,11), (3,13), (4,15). Which description fits? Candidate answers: (A) Negative linear association (B) No change in y as x changes (C) Positive linear association in this record (D) The record proves x causes y. 2. A supplied trend model is y=3x+9. What y does it predict at x=10? Candidate answers: (A) 38 (B) 39 (C) 40 (D) 78. 3. 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. 4. A model was fitted to measurements at x=2 through x=17. Which proposed prediction is extrapolation? Candidate answers: (A) Predicting at x=15 (B) Predicting at x=2 (C) Predicting at x=21 (D) Predicting at x=3.

Work on paper or in your own document. Explain the choices you made.

  • State each requested quantity or relationship and its input restrictions.
  • Show intermediate calculations or the applicable theorem with matching correspondences.
  • Give all four results, including exact values or requested rounding and units.
  • Check a result by substitution, reversal, dimensions, a representation or a counterexample as appropriate.
Compare with a model response

1. Positive linear association in this record. Each increase of 1 in x accompanies an increase of 2 in y; the four points lie on a rising line. Association in the record does not identify a cause. 2. 39. Substitute x=10: y=3(10)+9=39. This is the line model’s prediction, not a promised observation. 3. -9. Residual=observed−predicted=10−19=−9. The observed point is 9 output units below the line. 4. Predicting at x=21. 21 lies outside the measured interval 2 through 17. Extending a trend outside its data range is extrapolation and requires more caution than interpolation.

A different response can also meet the criteria. Compare the evidence and reasoning, rather than copying the wording.