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1.14 · Forward prediction
Model: W(t) = 3.545 + 0.185t. Predict the weight when t = 10 weeks.
Substitute t = 10
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1.14 · Inverse prediction
Model: W(t) = 3.545 + 0.185t. A baby weighs 5.0 kg. Find the predicted age.
Set = 5.0, solve for t
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1.14 · What does r measure?
What does the correlation coefficient r measure?
Range: −1 to +1
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1.14 · Interpret r = 0.994
r = 0.994. Interpret this value.
Close to +1 means...
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1.14 · Interpret r = −0.87
r = −0.87. Interpret this value.
Negative sign means...
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1.14 · Interpret r = 0.15
r = 0.15. What does this suggest about the linear model?
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1.14 · r vs r²
What is r²? How does it differ from r?
Coefficient of determination
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1.14 · Units on AP exam
Why must you always include units when interpreting a predicted value?
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✓ Inverse prediction
3.545 + 0.185t = 5.0 → 0.185t = 1.455 → t = 1.455/0.185 ≈ 7.86 weeks.
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✓ Forward prediction
W(10) = 3.545 + 0.185(10) = 3.545 + 1.850 = 5.395 kg
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✓ Interpret r = 0.994
Very strong POSITIVE linear relationship. Data points are very close to the regression line. As x increases, y increases.
r² = 0.988 → 98.8% of variation explained
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✓ What does r measure?
r measures the STRENGTH and DIRECTION of a linear relationship. Ranges from −1 to +1. |r| close to 1 = strong fit. Only applies to LINEAR regression.
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✓ Interpret r = 0.15
Very WEAK positive relationship. Data is scattered. A linear model is NOT a good fit — consider a different model type.
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✓ Interpret r = −0.87
Strong NEGATIVE linear relationship. As x increases, y decreases. Points are fairly close to the line but the trend is downward.
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✓ Units on AP exam
Units connect math to the real-world context. AP exam awards interpretation points that require units. '5.395' is incomplete; '5.395 kg' is the full answer.
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✓ r vs r²
r = strength + direction of linear relationship. r² = proportion of variation in y explained by the model. r² = (r)². E.g. r = 0.95 → r² = 0.9025 = 90.25% explained.
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1.14 · Full residual sentence
Predicted: 4.47 kg. Actual: 4.4 kg. Write a complete interpretation.
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1.14 · Projectile — model type
A ball is thrown upward. Height vs. time is recorded. Which regression model?
Physics: h = −16t² + v₀t + h₀
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1.14 · Extrapolation warning
A model built from x ∈ [0, 12] is used to predict at x = 50. What is the concern?
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1.14 · Two justifications
AP FRQ asks you to select and justify a model. What must your answer include?
Context AND data
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✓ Projectile — model type
QUADRATIC. Physics formula h = −16t² + v₀t + h₀ is quadratic in t. Also: data rises to a peak then falls — one turning point.
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✓ Full residual sentence
Residual = 4.4 − 4.47 = −0.07 kg. The model OVERESTIMATED the weight of the 5-week-old baby by 0.07 kg.
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✓ Two justifications
(1) Context reason (e.g. 'area involves r²') AND (2) Data reason (e.g. 'one turning point — y increases to max then decreases'). Both required for full credit.
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✓ Extrapolation warning
Extrapolating far beyond the data range (x ∈ [0,12]) is unreliable. The model was only validated within that range. Predictions at x = 50 may be wildly inaccurate.