Fit or select a linear model for data and interpret slope in context. | Two-variable data: Models and scatterplots | SAT Math

核心知识

The slope of a linear model tells you how much the predicted y-value changes for each one-unit increase in x — and you must state it in the context of the problem.

深入理解

On the SAT, interpreting slope means translating the number into a real-world statement. If the equation is $𝑦 = 3.2 𝑥 + 10$ and $𝑥$ is hours worked and $𝑦$ is dollars earned, then the slope 3.2 means: "For each additional hour worked, the predicted earnings increase by $3.20."

Three elements of a correct interpretation:

"For each additional one unit of $𝑥$ " — slope is always per unit.
"The predicted $𝑦$ " — it's an estimate, not a guarantee.
Direction — positive slope means increase; negative slope means decrease.

When the SAT asks you to select the best linear model, look for two things: the line should pass through the middle of the data, and roughly half the points should be above and half below.

Watch the units. If $𝑥$ is in months and $𝑦$ is in thousands of dollars, the slope isn't "dollars per month" — it's "thousands of dollars per month." Wrong units are a classic trap in the answer choices.

分步讲解

Identify what x and y represent in context, including units.
Read the slope from the equation (the coefficient of x).
State the interpretation: 'For each additional [one unit of x], the predicted [y] increases/decreases by [slope value] [y-units].'
If selecting among models, choose the line that splits the data points roughly evenly above and below.

常见误解

Interpreting slope as a total rather than a rate. Slope is per unit of x, not the overall change.
Forgetting to match units. If y is in thousands, the slope is in thousands per unit of x, not raw units.
Describing slope as what 'will happen' instead of what the model 'predicts.'

题目

示例解析

A nutritionist collected data on the daily sugar intake $𝑠$ (in grams) and body mass index $𝑏$ for 40 adults. The line of best fit is $𝑏 = 0.15 𝑠 + 20.8$ . Which of the following is the best interpretation of the slope of this line?

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