Lines of best fit: prediction and its limits · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

How far a line may be trusted

Mathematics · Data & Statistics · ages 15-16
Name ______________________   Date ____________

  1. What does interpolation mean?

    • predicting within the measured x range
    • joining pairs of recorded points
    • predicting beyond the measured x range
    • measuring how closely points follow a line

  2. A park records more filled paddling pools and more sunburn cases in summer. Which is a possible shared explanation?

    • Pool filling directly increases sunburn.
    • Hot, sunny weather could affect both counts.
    • Sunburn directly increases the number of filled pools.
    • Different totals mean the logs cannot be compared.

  3. A fitted line predicts 17.6 visitors at a target inside its measured range. Report the estimate to the nearest whole visitor.

    Answer: ______________

  4. A line of best fit is y = 90 - 3x, and the data covers x from 4 to 15. What does the line predict when x = 8?

    Answer: ______________

  5. Recorded x values range from 5 to 12. The fitted line is drawn across an axis from 0 to 20. Which interval is the measured data range?

    • 5 to 12, from the recorded x values
    • 0 to 20, from the axis ends
    • 0 to 12, from zero to the last recorded x value
    • 5 to 20, from the line's end

  6. A line fitted to adult runners covers 2 to 14 training runs. Someone uses it for a child doing 7 runs. What check is still needed?

    • Whether 7 lies within 2 to 14, since that alone establishes suitability for any runner.
    • Whether substitution is correct, since the model's original population does not affect its use.
    • Whether the adult model is suitable for children.
    • Whether drawing the adult line farther supplies the missing observations about children.

  7. A study's data covers x from 10 to 50. A prediction is made at x = 80. How far beyond the data is it?

    Answer: ______________

  8. If two quantities rise together, one of them must be causing the other.

    Circle one:   True   False

  9. A line fits height data for babies aged 0 to 12 months closely. At age 30 years it predicts 4 metres. Which limitation does this illustrate?

    • The infant records must contain an error because a fitted line cannot otherwise give an unrealistic adult height.
    • A childhood trend need not continue unchanged into adulthood.
    • Checking the arithmetic justifies the adult prediction when the fit during infancy was close.
    • The measured age range extends to every age where the equation can be evaluated, whether or not it was observed.

  10. Sam needs a better-supported prediction for 20 training runs, but the log covers 2 to 14. Which action supplies relevant new evidence?

    • Treat the model's prediction at 20 as a new observation.
    • Use the prediction at 14 instead.
    • Record suitable runners near 20 runs, then reassess the model.
    • Repeat the existing calculations more precisely, without recording runners near the new target.
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Answer key

For grown-ups. Fold this page away before handing over the rest.

How far a line may be trusted W1-mt_CnYCx1a0j5-s1

  1. predicting within the measured x range · Interpolation estimates within the measured x range. Extrapolation estimates beyond that range.
  2. Hot, sunny weather could affect both counts. · Hot, sunny weather could affect pool use and sun exposure. The correlation alone does not establish which explanation is correct.
  3. 18 · 17.6 rounds to 18. Report about 18 visitors; rounding does not make the estimate a guaranteed count.
  4. 66 · 90 - 3 × 8 = 90 - 24 = 66. This is interpolation because 8 is within 4 to 15. It does not guarantee an observed value of 66.
  5. 5 to 12, from the recorded x values · The measured range is 5 to 12. Extending the axis or line does not add observations.
  6. Whether the adult model is suitable for children. · 7 is within the adult data's x range, but the child belongs to a different population. Range alone does not establish applicability.
  7. 30 · The data stops at 50, so the prediction sits 80 - 50 = 30 beyond the last measured value.
  8. False · A shared factor or chance could produce the association. Correlation alone does not establish a causal direction.
  9. A childhood trend need not continue unchanged into adulthood. · Thirty years is far beyond the first year covered by the data. A close fit during infancy does not justify extending that trend unchanged into adulthood.
  10. Record suitable runners near 20 runs, then reassess the model. · Suitable observations near 20 add evidence that the current log lacks. Check the model against those observations before using it.
Worksheet · LightMySky