SAT: Scatterplots & Modeling

The straight line $\hat{y} = a + bx$ that minimizes the sum of squared residuals for a scatterplot. It summarizes the overall trend in the data.

The slope $b$ in a regression equation $\hat{y} = a + bx$ represents the predicted change in the response variable $y$ for each 1-unit increase in the explanatory variable $x$. The SAT always asks you to interpret slope using the units of both variables.

The constant $a$ in $\hat{y} = a + bx$ is the predicted value of $y$ when $x = 0$. On the SAT, you must state its meaning in the scenario and note whether $x = 0$ makes practical sense.

The difference between an observed value and its predicted value: $e = y - \hat{y}$. A positive residual means the actual value is above the line; a negative residual means the actual value is below.

A graph of residuals vs. the explanatory variable $x$. If residuals scatter randomly around 0, a linear model is appropriate. A curved pattern suggests a nonlinear model would fit better.

A number from $-1$ to $1$ that measures the strength and direction of a linear relationship. Values near $\pm 1$ indicate strong linear association; values near 0 indicate weak or no linear association.

$R^2$ is the proportion (0 to 1) of the variance in $y$ that is explained by the linear model. It equals the square of $r$.

A linear model has constant additive change: $y = a + bx$. An exponential model has constant multiplicative (percent) change: $y = a \cdot b^x$ with $b > 0, b \neq 1$. For large $x$ with $b > 1$, exponential growth always overtakes linear growth.

Interpolation predicts $y$ for an $x$-value within the observed data range -- generally reliable. Extrapolation predicts $y$ for an $x$ outside the data range -- less reliable because the trend may not continue.

A strong correlation between two variables does not prove that one causes the other. There may be lurking (confounding) variables or the relationship may be coincidental. The SAT tests this distinction with answer choices that use causal language.