散布図、モデルの残差と予測
導入済み| English | 意味 |
|---|---|
| residual/rɪˈsɪdʒuːəl/ | 残差 |
| extrapolation/ekˈstræpəleɪʃn/ | 外挿 |
A decision before an answer
- A line predicting 18 when the observed value is 21 has a residual of +3. It does not pass through every observation.
- Your goal: Interpret slope and intercept of a fitted model in context.
Read the relationship
- A scatterplot pairs two quantitative variables, one point per paired observation. A fitted line summarises a trend rather than connecting each point. In a model y=a+bx, b predicts the change in y for one-unit increase in x; its units are y-units per x-unit. The intercept a predicts y when x=0, which may lie outside the data range.
- Calculate a residual and distinguish interpolation from extrapolation.
Model predicts 30; observation is 26. Residual:
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26-30=-4.
Use the defining rule
- A residual is observed y minus predicted y. Positive means the observation lies above the model; negative means below. Substitute the observed x into the model before subtracting. A large residual marks a poor prediction for that point, not automatically an error in measurement or proof that the trend is absent.
- Choose linear or exponential patterns from equal-step data.
Observed x ranges from 2 to 8. Predicting at x=5 is:
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5 is inside the observed range.
Check the conditions
- Interpolation predicts inside the observed x-range; extrapolation goes beyond it and needs stronger caution because the relationship may change. Use a model’s contextual domain as well as its algebraic form. A negative predicted travel time, for example, can reveal an unjustified extrapolation rather than a physically possible outcome.
- Choose linear or exponential patterns from equal-step data.
For study time x hours and predicted practice score y=12+3x, the slope is 3 score points/hour. At x=2 the prediction is 18. If the observed score is 21, residual=21-18=3. With x-values observed from 1 to 5, predicting at x=3 interpolates and x=12 extrapolates. Outputs 5,10,20,40 at equal input steps suggest factor-two exponential growth.
For y=4+2x, prediction at x=7 is ____.
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4+2·7=18.
Apply the task format
- For equal input steps, nearly constant output differences suggest a linear model; nearly constant output ratios suggest an exponential model. Scatter can make either pattern approximate. Association does not identify a cause: both variables may reflect a third factor. A fitted trend is evidence about prediction, not by itself about intervention.
- Choose linear or exponential patterns from equal-step data.
Keep observed minus predicted in that order. Do not turn a model slope into a causal treatment effect.
このケースに当てはまる答えはどれか？
文脈の中で適合モデルの傾き切片を解釈する
A strong scatterplot association alone proves x causes y.
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Confounding or other explanations can produce association.
Keep the distinctions
- residual 残差 — Observed response minus the model’s predicted response.
- extrapolation 外推 — Prediction outside the range of observed inputs.
- Interpret slope and intercept of a fitted model in context.
- Calculate a residual and distinguish interpolation from extrapolation.
- Choose linear or exponential patterns from equal-step data.
Match each term with its precise meaning in this lesson.
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Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
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The order follows the stated process; check each stage before the next.