SAT · Problem-Solving and Data Analysis · TOPIC LESSON
Read paired data, select a defensible model, and distinguish predictions from observations.
BY THE END, YOU CAN
BEFORE YOU START
Original educational lesson. Work through the examples before revealing the practice answers. This lesson supports SAT Math and Reading and Writing skill study; it does not reproduce official questions or guarantee a score.
LEARN THROUGH ORIGINAL PRACTICE
The teaching passages, worked examples and exercises in this lesson are original. They are not official College Board questions or past-exam reproductions. Blueprint and observed-practice analytics describe separate factual metadata; those source questions are not displayed here.
01 / UNDERSTAND & APPLY
A point (x,y) contains two measurements from the same individual, object or occasion. Keeping those measurements paired is essential: rearranging one column can destroy the relationship. A scatterplot shows the pairings; it does not connect points in an arbitrary row order.
Identify the explanatory quantity on the horizontal axis and the response on the vertical axis when the study gives those roles. A plotted association does not alone establish which variable causes the other.
02 / UNDERSTAND & APPLY
An upward pattern has positive association; a downward pattern has negative association. The form may be roughly straight, curved or without a clear pattern. Strength describes how tightly observations follow that form. An outlier or isolated high-leverage point may strongly affect a fitted model.
A curved pattern can have a strong relationship even if a straight-line summary is poor. Check axis scales and units before comparing steepness or concluding that a relationship is weak.
03 / UNDERSTAND & APPLY
A fitted line ŷ = mx + b estimates the response at input x. Its slope predicts m response-units of change for a one-unit input increase. The intercept predicts the response at x = 0, but zero may be outside the data range or meaningless in the context.
The line does not normally pass through every point. A residual is observed value minus predicted value, y − ŷ. Positive residuals are above the line and negative ones below. Residuals with systematic curvature suggest the model form may be missing a pattern.
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ORIGINAL WORKED EXAMPLE
A model predicts a plant height H = 2.5d + 8 centimeters after d days. At day 6, the observed height is 25 cm. Find the prediction and residual.
Answer Prediction 23 cm; residual +2 cm
04 / UNDERSTAND & APPLY
If two representative points are used to specify a linear model, compute their slope and then its intercept. A line through two selected points is not automatically the least-squares regression line for a larger dataset. Use an already-supplied model as given unless asked to fit one.
ORIGINAL WORKED EXAMPLE
A linear model passes through (2,11) and (8,29). Find its prediction at x = 5.
Answer 20
05 / UNDERSTAND & APPLY
A quadratic model has the form ŷ = ax² + bx + c. It can model data that rise then fall, fall then rise, or bend in one direction over the observed interval. Equal second differences in equally spaced exact data support a quadratic relationship; noisy data will only approximately follow it.
A quadratic fit should not be chosen merely because it can pass through more points. Context, residual pattern and the observed range matter, and a fitted curve can behave unrealistically far outside that range.
ORIGINAL WORKED EXAMPLE
A greenhouse uses H(t) = −t² + 8t + 12 to model average growth response over 0 ≤ t ≤ 8, where t is a treatment level. Which level maximizes the modeled response?
Answer t = 4; modeled maximum 28
06 / UNDERSTAND & APPLY
An exponential model ŷ = a·b^x predicts a constant multiplicative factor per equal input step. A linear model predicts a constant additive difference. An exponential decrease can slow in absolute amount while retaining the same percentage decrease.
Use the time interval attached to the factor: in 500(0.8)^(t/2), 0.8 is the factor every two time units, not every one.
ORIGINAL WORKED EXAMPLE
A model gives outputs 120,180,270 at inputs 0,1,2. Assuming exponential form, predict the output at input 4.
Answer 607.5
07 / UNDERSTAND & APPLY
At early inputs, a linear model may exceed an exponential one. A positive exponential with factor greater than 1 eventually outgrows any fixed positive linear rate as input grows, but “eventually” does not identify a particular practical crossover. Evaluate within the relevant range.
For models L(t) = 40 + 15t and E(t) = 40(1.2)^t, both start at 40. At t = 2, L = 70 while E = 57.6. The exponential’s factor describes a 20% increase per step, while the linear model adds 15 per step.
08 / UNDERSTAND & APPLY
Interpolation predicts within the observed input range. Extrapolation predicts outside it and may depend on a relationship that no longer holds. Neither is guaranteed accurate. A model may summarize a sample without accurately describing every individual.
A graph of a modeled relationship can convey a plateau, peak or changing rate even without a formula. Read the relevant coordinates and graph intervals; do not impose a straight line on a visibly curved model.
ORIGINAL WORKED EXAMPLE
A model was fitted to daily temperatures from 10°C to 25°C and predicts energy use at 40°C. Is that interpolation or extrapolation?
Answer Extrapolation
09 / UNDERSTAND & APPLY
Interpret parameters with their units.
ORIGINAL PRACTICE · 1
Negative association
Higher x tends to pair with lower y, without by itself proving a cause.
ORIGINAL PRACTICE · 2
37
Substitute: 4(7) + 9 = 37.
ORIGINAL PRACTICE · 3
−3
Residual is observed minus predicted: 16 − 19.
ORIGINAL PRACTICE · 4
Yes
The requested input lies within the observed range.
ORIGINAL PRACTICE · 5
1.5
12/8 and 18/12 both equal 1.5, supporting an exponential model if that family is assumed.
ORIGINAL PRACTICE · 6
20 at x = 3
The negative square term is at most zero.
ORIGINAL PRACTICE · 7
Predicted cost rises $6 per additional kilogram
Slope carries output units divided by input units.
ORIGINAL PRACTICE · 8
No
Other variables and study design matter; an association alone does not establish causation.
ORIGINAL PRACTICE · 9
The linear model leaves systematic structure unexplained
Residuals following a curve indicate a predictable deviation rather than unstructured scatter around zero.
KEEP THE BIG IDEAS
READ WITH CONTEXT
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