Choose the test that could prove you wrong
Where should you measure next when two explanations fit the existing observations?
Entering midway? Revisit When two improvements cancel each other for the preceding worked example, or start here and use the explanation below.
Start with a concrete question
Original synthetic scenario. Two candidate mechanisms agree at inputs zero and one. One grows linearly; the other curves upward. A selectable synthetic truth produces a new observation.
Try the initial inputs, then write a prediction before changing one setting. All calculations run locally on your CPU.
What do you expect?
Will another measurement at x=1 distinguish the two explanations?
Change one thing. Inspect what happens.
- Set x to zero and one; verify both explanations predict the same values.
- Choose a new input where predictions separate and compare residuals.
- Vary measurement disturbance and find a test input where it can obscure the difference.
- Write a precommitted prediction for both hypotheses and a decision rule that allows an inconclusive result.
Your settingsMechanism
Calculation belowEvidence
Compare outcomes
Compare the current results
Every bar starts at zero and uses the same numeric scale in this chart. The upper bound automatically adjusts to the largest current value (at least 1), so compare the numbers when changing inputs. Units and quantities come from each label; different kinds of quantity are not interchangeable. Displayed values are rounded to at most six significant digits.Follow the calculation
- Both candidate explanations agree at x=0 (10 units) and x=1 (12 units). Those observations alone cannot choose between them.
- New test x=2: linear predicts 10 + 2 × 2 = 14.
- Curved predicts 10 + 2 × 2² = 18.
- Constructed observation = selected hidden mechanism + measurement disturbance 0 = 18.
- Residuals: |18 − 14| = 4.000; |18 − 18| = 0.000.
A useful new test makes competing explanations predict different outcomes. One close fit supports a comparison under these assumptions, not universal proof.
Captures keep your inputs, prediction and observation. Results are recalculated from those inputs. Resetting inputs keeps your captures and challenge history.
Explain the mechanism
What the experiment explains
Original synthetic scenario. Two candidate mechanisms agree at inputs zero and one. One grows linearly; the other curves upward. A selectable synthetic truth produces a new observation.
Fitting observations is not the same as uniquely explaining them. Different mechanisms can agree on a limited set of measurements.
At x=2, the linear mechanism predicts 14 and the curved mechanism predicts 18. A result near 18 favors the curved explanation in this restricted comparison; x=1 would yield 12 for both.
A residual is observed minus predicted value. We compare absolute residuals, but a smallest residual alone is not proof: the candidate set, measurement uncertainty, and validity of the test range matter.
Read the relationship
Linear prediction = 10 + 2x. Curved prediction = 10 + 2x². Absolute residual = |observed − predicted|, in output units.
Inspect the evidence
The numerical trace is recalculated from your current settings. Compare a controlled pair of trials before changing several inputs together. Inspect intermediate quantities as well as final metrics; state which assumptions make the comparison meaningful.
Boundaries of the model
- Only two candidate models; known functional forms and bounded additive disturbance.
- The truth selector is an instructional reveal, not information available during a real blinded study.
Return to the scientific reasoning and experimental design course and project to connect this mechanism to the broader investigation.
Try a fresh case
New candidates predict 5 + 3x and 5 + x². At x=4, how far apart are their predictions?
This is a public practice check. Retries support learning; matching a number does not certify independent proficiency.
Take the investigation further
Choose any extension. These are project prompts for you to carry out; they are not automatically assessed.
School investigation
Invent two rules that agree on two examples and choose a third example that separates them.
College investigation
Design sequential tests using uncertainty bounds and held-out observations.
Workplace investigation
Propose a product experiment that could reject the favored explanation rather than only collect supporting anecdotes.
Keep the evidence, choose your next step
Completion records your own decision that you have worked through this investigation. Practice checks and independent project evidence are separate.