๐Ÿ  Homeโ€บ AP Precalculusโ€บ Unit 1โ€บ 1.14 Notes
๐Ÿ“ Topic 1.14 ยท Unit 1

Applications
of Modeling

Using regression models in context ยท predicting values ยท inverse problems ยท correlation coefficient ยท model selection

๐Ÿ“‹ Notes ๐Ÿƒ Flashcards ๐Ÿง  Quiz
๐ŸŽฏ Three Core Tasks with a Model
๐ŸŽฏ

Once you have the regression equation, you can:

1. Forward predict: Plug in an x-value โ†’ get predicted y
2. Inverse predict: Set equation = given y โ†’ solve for x
3. Find & interpret residuals: Actual โˆ’ Predicted โ†’ over or underestimate

๐Ÿ“Œ Example 1 โ€” Forward Prediction

Model: W(t) = 3.545 + 0.185t   (baby weight in kg, t = age in weeks)

Q
What is the predicted weight of a baby who is 10 weeks old?
1
Substitute t = 10 into the model
2
W(10) = 3.545 + 0.185(10) = 3.545 + 1.850 = 5.395 kg
Predicted weight at 10 weeks โ‰ˆ 5.395 kg
๐Ÿ“Œ Example 2 โ€” Inverse Prediction

Same model: W(t) = 3.545 + 0.185t

Q
A baby weighs 5.3 kg. What age does the model predict for this baby?
1
Set model equal to the given y-value: 3.545 + 0.185t = 5.3
2
Subtract 3.545 from both sides: 0.185t = 1.755
3
Divide: t = 1.755 รท 0.185 โ‰ˆ 9.49 weeks
Predicted age for a 5.3 kg baby โ‰ˆ 9.49 weeks
๐Ÿ“Š Correlation Coefficient r
๐Ÿ“Š
How Well Does the Model Fit?
r measures the strength and direction of a linear relationship (โˆ’1 to +1)
๐Ÿ“Š

The Correlation Coefficient r

r ranges from โˆ’1 to +1. Only applies to linear regression.
|r| close to 1: Strong fit โ€” data points are close to the line
|r| close to 0: Weak fit โ€” data is scattered
r > 0: Positive association (x up โ†’ y up)
r < 0: Negative association (x up โ†’ y down)

โˆ’1Perfect negative
โˆ’0.9Strong neg.
~0Weak / none
+0.9Strong pos.
+1Perfect positive
๐Ÿ“Œ Example 3 โ€” Interpreting r Values
a
r = 0.994 โ†’ Very strong positive linear relationship. Data points are very close to the regression line; x and y increase together.
b
r = โˆ’0.87 โ†’ Strong negative linear relationship. As x increases, y decreases; points are fairly close to the line.
c
r = 0.32 โ†’ Weak positive relationship. Some upward trend but data is very scattered; linear model may not be a good fit.
โญ AP Exam Tip โ€” rยฒ (Coefficient of Determination)

You may see rยฒ on some problems. rยฒ = the proportion of variation in y explained by the model. Example: rยฒ = 0.95 means 95% of the variation in y is explained by the linear relationship. To find r, take the square root (and check the sign from context).

๐Ÿ” Model Selection Framework
Data Pattern Context Clue Model
Straight line, constant changePrice per item, constant rateLinear
One turning pointArea (A = ฯ€rยฒ)Quadratic
Two turning points, S-shapeVolume (V = sยณ)Cubic
Approaches a limit (asymptote)Inverse rate (y = k/x)Rational
๐Ÿ“Œ Example 4 โ€” Complete Multi-Part Application

Using the baby weight model W(t) = 3.545 + 0.185t from Topic 1.13:

a
Is a linear model appropriate here? Yes โ€” the data shows a roughly constant increase in weight per week, and the context (weight growing at a roughly steady rate in early weeks) supports a linear model.
b
Predict W(10): W(10) = 3.545 + 0.185(10) โ‰ˆ 5.395 kg
c
Residual at t = 5: Actual = 4.4, Predicted = 4.47 โ†’ Residual = 4.4 โˆ’ 4.47 = โˆ’0.07 kg (overestimated)
d
Inverse โ€” find t when W = 5.0: 3.545 + 0.185t = 5.0 โ†’ 0.185t = 1.455 โ†’ t โ‰ˆ 7.86 weeks
โญ AP Exam Strategy
โญ Always include UNITS

Write "5.395 kg" not just "5.395". The AP exam awards interpretation points for proper context and units.

โญ Residual interpretation = full sentence

Don't write just "โˆ’0.07". Write: "The model overestimated the weight of the 5-week-old baby by 0.07 kg." The sign โ†’ direction, the magnitude โ†’ how much.

โญ Justify model choice with BOTH context AND data

Full justification: "A quadratic model is appropriate because [context: area involves rยฒ] AND [data: y-values increase to a maximum then decrease โ€” one turning point]." One reason alone is usually not enough for full credit.

โญ Extrapolation warning

A model is only reliable within (or near) the range of the data used to build it. Predicting far outside that range (extrapolation) may give unreliable or nonsensical results โ€” always note this limitation if relevant.