Develop and refine models
Identify omitted features and revise a model in a way that explains the available observations.
Original StudyForA scenarios and formative model answers; no Cambridge marks, predicted grades or certified mastery. Alternative reasoned answers may be valid. Full-length unseen papers, a wider mixed-problem bank and calibrated specialist teacher marking remain outstanding.
Before you begin
- Read tables and instructions in English
- Use arithmetic, ratios and whole-number constraints
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Before calculating, what should you identify?
What you will learn
- Choose relevant features of a real situation
- Compare refinements against data rather than adding arbitrary complexity
Look for a systematic residual
A model is a simplified representation with selected inputs and rules. Compare predicted and observed values. A consistent discrepancy suggests an omitted fixed term; a discrepancy growing with distance may suggest a wrong rate; a sudden jump may require a threshold rule.
The model cost=3 d predicts 6 and 12 credits at 2 and 4 km. Observed prices are 10 and 16, four higher in each case. Adding a 4-credit fixed charge gives cost=4+3 d and fits both observations.
Refine only what the evidence supports
A model with enough arbitrary parameters can fit almost anything. Prefer a relevant mechanism and test it with observations not used for fitting. Two matching points cannot establish that a refined model will work in every setting.
If new observations depart only after a certain hour or capacity, consider a piecewise rule. Explain the boundary and verify cases on both sides. Describe assumptions such as unchanged rates, identical items or no resource contention, and seek evidence before treating them as facts.
Sources for this lesson
Pause and explain
A constant observed-minus-predicted difference suggests which first refinement?
Worked example
Refine cost=3 d to fit the observed delivery prices.
Show the worked solution
- At 2 km the error is 10−6=4.
- At 4 km the error is 16−12=4.
- Introduce the fixed term 4 and check both.
Answer The refined model cost=4+3 d fits both observations. A 5 km prediction is 19 credits, conditional on the same tariff rule continuing.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Guided: Predict 5 km using the refined model and state its limitation.
Give me a hint
Keep the fixed and variable terms separate.
Compare my reasoning
- Variable cost is 3×5=15.
- Add the fixed 4.
- State that the tariff must continue.
The prediction is 19 credits. It relies on the refined rule still applying at 5 km and is not guaranteed by the two fitted observations alone.
Look for these in your work
- Uses both terms
- Calculates correctly
- States the model limit
Independent: A facility charges 2 credits per hour, but observations are 7 for 1 hour and 11 for 3 hours. Refine the model.
Give me a hint
Compare observed prices with 2 h.
Compare my reasoning
- The differences are 7−2=5 and 11−6=5.
- Add a 5-credit fixed charge.
- Check cost=5+2 h in both cases.
Cost=5+2 h fits the observations. It suggests a fixed 5-credit charge plus 2 per hour within the observed model, subject to further checks of how the facility actually charges.
Look for these in your work
- Finds the residual
- Builds the refined rule
- Checks both cases
Application: A model charges 4+3 d, but a rule adds 6 credits for journeys longer than 5 km. Write the piecewise rule and calculate 5 km and 6 km.
Give me a hint
Longer than excludes the boundary value 5.
Compare my reasoning
- For d≤5, cost=4+3 d.
- For d>5, cost=10+3 d.
- Evaluate each distance in its own branch.
At 5 km cost is 19; at 6 km it is 28. The rule is 4+3 d for d≤5 and 10+3 d for d>5, so the surcharge begins only beyond the threshold.
Look for these in your work
- States both branches
- Handles the boundary
- Calculates both values
Common mistakes
- Adding unexplained parameters
- Using fitted data as independent validation
- Ignoring systematic residuals
What can you explain now?
A constant observed-minus-predicted difference suggests which first refinement?
Compare with the explanation
An additional fixed term
A fixed term shifts all predictions by the same amount.
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Make it stick
A constant observed-minus-predicted difference suggests which first refinement?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.