Regression, financial models and residuals
導入済み| English | 意味 |
|---|---|
| residual/rɪˈsɪdʒuːəl/ | 残差 |
Is the prediction supported by the data?
- A line predicts bus travel time from distance. A small calculation error matters less than using a model outside its evidence.
- This lesson studies residual 残差: The observed value minus the value predicted by a fitted model.
Choose the mathematical structure
- A linear model y=a+bx has intercept a and slope b with contextual units. Inspect residuals and the data range. For repeated percentage change use a geometric model; for a loan distinguish principal, rate, repayment and period.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
The model t=8+2d fits observed journey times. An observed 21 minutes against a predicted 18 means what?
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
Residual = observed - predicted = 21-18 = 3. Residuals reveal systematic model weakness; a good fit on average can still hide poor regions.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
The fitted model t=8+2d predicts 18 minutes at d=5 km. If the observed time is 21, the residual is 3 minutes. A ¥1000 deposit at 5% compound annual interest becomes 1000×1.05³=1157.625 after 3 years.
Regression, financial models and residuals
A linear model y=a+bx has intercept a and slope b with contextual units
Use the positive residual to explain how the fitted model misses this observation.
For t=8+2d, find t at d=5.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
Substitute distance 5: 8+2×5=18 minutes.
Test a tempting shortcut
- A good fit does not prove a causal mechanism. A correlation coefficient measures linear association, not the gradient. Calculator output must be translated into a model, checked and interpreted.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
A high correlation permits reliable extrapolation to any distance. This claim is false. Explain which definition or assumption it violates.
Find the residual if observed t=21 and predicted t=18.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
Residual=observation-prediction=21-18=3 minutes.
A high correlation permits reliable extrapolation to any distance.
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
A good fit does not prove a causal mechanism. A correlation coefficient measures linear association, not the gradient. Calculator output must be translated into a model, checked and interpreted.
Interpret a new situation
- In IB AI, record the data source, domain, assumptions and calculator method. Compare an alternative model and judge predictions against residuals. A financial answer must state payment timing and whether interest is compounded.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the value of 1000 after 3 years at 5% compound annual interest.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
Compound annually: 1000×1.05³=1157.625.
Match each modelling quantity to its meaning.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
Model parameters carry contextual units, residuals measure per-point error, and repeated percentage change compounds multiplicatively.
Use this in your course
- edexcel IAL further mathematics; official unit S3. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
The observed value minus the value predicted by a fitted model. Choose the relationship, show the method, check its assumptions and interpret the result.