Regression, financial models and residuals · 回归、财务模型与残差
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| residual/rɪˈsɪdʒuːəl/ | 残差 | cán chà |
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.
Which description correctly defines residual? · 哪一项描述正确定义了残差？
The observed value minus the value predicted by a fitted model. · 观测值减去拟合模型预测的值。
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.
Regression, financial models and residuals · 回归、财务模型与残差
A linear model y=a+bx has intercept a and slope b with contextual units · 线性模型 y=a+bx 具有截距 a 和斜率 b，并带有相应的实际单位
Use the positive residual to explain how the fitted model misses this observation. · 利用正残差解释拟合模型如何遗漏该观测点。
For t=8+2d, find t at d=5. · 对于 t=8+2d，求当 d=5 时的 t 值。
Substitute distance 5: 8+2×5=18 minutes. · 代入距离 5：8+2×5=18 分钟。
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. · 若观测到的 t=21 且预测的 t=18，求残差。
Residual=observation-prediction=21-18=3 minutes. · 残差=观测值-预测值=21-18=3 分钟。
A high correlation permits reliable extrapolation to any distance. · 高相关系数允许对任意距离进行可靠的外推。
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. · 拟合良好并不证明存在因果机制。相关系数衡量的是线性关联而非斜率。计算器输出必须转化为模型，并经过检验和解释。
Interpret a new situation
- Use the AQA large data set and suitable statistical displays. Interpret residuals, correlation and model limitations; association alone does not establish causation.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
For t=8+2d, find the predicted time when d=7. · 对于 t=8+2d，求当 d=7 时的预测时间。
Substitute d=7: t=8+2×7=22 minutes. · 代入 d=7：t=8+2×7=22 分钟。
Match each part of a complete solution to its purpose. · 将完整解答的每个部分与其目的相匹配。
An assumption justifies the model; a check tests the result; interpretation connects it to the question. · 假设用于论证模型的合理性；检查用于验证结果；解释用于将其与问题建立联系。
Use this in your course
- 7357 · A-level · L. Match the target tier and specification before assigning extensions.
- 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.