Explain trends and fit model parameters
Fit a specified model to observations while distinguishing numerical agreement from a causal explanation.
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
- Find parameters of a stated model
- Suggest and assess explanations for a trend or change point
A model family constrains the calculation
For output=a+bx, two distinct input observations determine a and b. Check additional observations rather than assuming the fit continues. If the family is not specified, many models can fit a few points, and extending beyond the observed range requires caution.
A fictional delivery model is cost=a+bd. At 2 km the price is 9 credits and at 6 km it is 17. Thus b=2 credits/km and a=5. A 4 km delivery is predicted to cost 13 under this rule.
Describe before explaining
A trend says what changes; an explanation suggests why. A sudden increase may reflect a tariff threshold, an event, a changed measurement or a different population. These are hypotheses unless the problem supplies confirming evidence.
A fitted line does not prove causation or guarantee extrapolation. Explain which model feature accounts for the trend, identify the observations it fits, and state what new observation would discriminate between rival explanations.
Sources for this lesson
Pause and explain
Does an increasing graph alone establish the cause of the increase?
Worked example
Fit the stated linear delivery model and predict 4 km.
Show the worked solution
- The price difference 8 over distance difference 4 gives b=2.
- At 2 km, a=9−4=5.
- At 4 km, cost=5+8=13.
Answer The fitted model is cost=5+2 d and predicts 13 credits at 4 km. This is a prediction within the specified model, not proof of a real operator’s tariff.
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: Check that cost=5+2 d matches both observations.
Give me a hint
Substitute both distances.
Compare my reasoning
- At 2 km, 5+4=9.
- At 6 km, 5+12=17.
- Both are exact matches.
The model gives 9 and 17 credits at 2 and 6 km respectively, matching both observations. This checks the numerical fit without establishing why the tariff has that form.
Look for these in your work
- Checks both points
- Uses the units
- Limits causal claims
Independent: Within y=a+bx, observations are (1,7) and(4,16). Find a,b and the prediction atx=5.
Give me a hint
Find change per input unit.
Compare my reasoning
- b=(16−7)/(4−1)=3.
- a=7−3=4.
- At 5, y=4+15=19.
a=4 andb=3, so the prediction is 19. The model fits both supplied points; use of it at 5 is conditional on the same linear relation continuing.
Look for these in your work
- Finds both parameters
- Calculates the prediction
- States the continuation assumption
Application: Visitor counts jump on the day a new entrance counter is installed. Suggest two explanations and a useful check.
Give me a hint
A change in measurement can resemble a change in attendance.
Compare my reasoning
- Actual attendance may have risen.
- The new counter may count repeat entries differently.
- Compare manual counts or overlapping old/new measurements.
The jump could reflect more visitors or a changed counting method, such as counting re-entries. Independent manual counts or an overlap comparison would help distinguish the explanations. The timing alone does not prove a real attendance increase.
Look for these in your work
- Suggests alternatives
- Links each to the observation
- Proposes discriminating evidence
Common mistakes
- Confusing a trend with its cause
- Extrapolating without qualification
- Assuming a model family that was not supplied
What can you explain now?
Does an increasing graph alone establish the cause of the increase?
Compare with the explanation
No, several mechanisms may fit it
A pattern needs additional evidence to distinguish explanations.
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Make it stick
Does an increasing graph alone establish the cause of the increase?
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.