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 at a value inside the range the data covers
    • drawing the line of best fit through as many points as it can possibly touch
    • extending the line beyond the last point that was plotted on the graph
    • working out how strong the correlation between the two variables is

  2. The first-aid tent logs sunburn cases, and the park logs paddling pools filled. Both rise through the summer. What is the most likely explanation?

    • Filling a paddling pool is what causes people to get sunburnt while they wait
    • Hot weather sends people to fill pools and out into the sun that burns them
    • People with sunburn fill paddling pools to cool themselves down afterwards
    • One of the two logs has almost certainly been filled in wrongly by somebody

  3. Using the same line y = 2x + 4, what is the prediction at x = 5?

    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. Data span x = 0 to 20. Which prediction is safest?

    • x = 10
    • x = 40
    • x = -10
    • x = 100

  6. Heights were measured for ages 10 to 15. The line predicts height at age 30. What is wrong?

    • The scatter points stop mattering
    • Predictions improve further from the data
    • Age 30 is outside the data range
    • Ages over 15 follow on for sure

  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 fitted to babies aged 0 to 12 months predicts height very well. Used on a 30-year-old it predicts 4 metres. What has gone wrong?

    • the line was drawn through the wrong points when the babies were first measured
    • growth is roughly steady over one year and then slows and stops, and the line never met that
    • the height measurements taken from the babies must have been recorded inaccurately
    • nothing has gone wrong here at all, since the line does fit the data it was originally drawn from

  10. Sam wants to predict a race time for someone who has done 20 training runs, when the log covers 2 to 14. What is the best thing to do?

    • extend the line to 20 and report the answer with the working shown alongside it
    • refuse to say anything at all about anybody who trained more than fourteen times
    • say the line does not cover 20, then log some runners who train that much
    • use the line at 14 instead, and report that figure as the answer for 20 runs
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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 at a value inside the range the data covers · Interpolation is reading the line between the smallest and largest measured values. Going beyond them is extrapolation, and neither has anything to do with how strong the correlation is.
  2. Hot weather sends people to fill pools and out into the sun that burns them · A third thing, the weather, pushes both counts up at the same time. Two quantities can move together with no arrow running between them at all.
  3. 14 · 2 times 5 plus 4 = 14.
  4. 66 · Substitute: 90 - 3 × 8 = 90 - 24 = 66. Since 8 lies between 4 and 15, this is an interpolation and the line was drawn through points on both sides of it.
  5. x = 10 · x = 10 sits inside the data range; the rest extrapolate beyond it.
  6. Age 30 is outside the data range · Age 30 is far past the 10 to 15 data, so the trend may not continue. This is risky extrapolation.
  7. 30 · The data stops at 50, so the prediction sits 80 - 50 = 30 beyond the last measured value.
  8. False · They might be, or a third thing might be lifting both, or it might be coincidence. Moving together is evidence worth following up, and it settles nothing on its own.
  9. growth is roughly steady over one year and then slows and stops, and the line never met that · Inside the first year a straight line is a fair description of growth. Beyond it the real relationship flattens off completely, and the line, which only ever saw the steep part, keeps climbing.
  10. say the line does not cover 20, then log some runners who train that much · The honest answer is that the question is outside the evidence, and the fix is more evidence rather than a longer line. Reporting the value at 14 as though it were the value at 20 hides the gap.
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