AP Statistics Linear Regression Models — AP Statistics
1. Introduction to Linear Regression Models ★★☆☆☆ ⏱ 3 min
Linear regression models are statistical models that describe the linear relationship between an explanatory (independent) variable $x$ and a response (dependent) variable $y$. We distinguish between the population model, which describes the true underlying relationship with unknown parameters $\beta_0, \beta_1$ and a random error term $\varepsilon$, and the estimated sample model, which uses calculated statistics $b_0, b_1$ from sample data.
The most common method for fitting a linear model to sample data is least squares regression, which produces the line that minimizes the sum of squared vertical distances between observed data points and the line. This topic makes up approximately 5-7% of the total AP Statistics exam weight, and appears in both multiple-choice and free-response sections.
2. The Least Squares Regression Line (LSRL) ★★☆☆☆ ⏱ 4 min
A residual is the vertical difference between the observed response value and the predicted response value for any $x$: $e_i = y_i - \hat{y}_i$, where $\hat{y}_i$ is the predicted $y$ for observation $i$. The goal of least squares is to minimize $\sum e_i^2 = \sum (y_i - \hat{y}_i)^2$.
b_1 = r \cdot \frac{s_y}{s_x}
b_0 = \bar{y} - b_1\bar{x}
where $r$ is the correlation coefficient between $x$ and $y$, $s_y$ is the sample standard deviation of $y$, $s_x$ is the sample standard deviation of $x$, $\bar{y}$ is the sample mean of $y$, and $\bar{x}$ is the sample mean of $x$. A key property of the LSRL is that it always passes through the point $(\bar{x}, \bar{y})$, which can be used to check calculations. The LSRL is defined for predicting $y$ from $x$, so swapping $x$ and $y$ produces an entirely different line.
Exam tip: Always round your slope and intercept to 2-3 significant digits matching the input data; over-rounding early leads to calculation errors, and too many digits wastes time on the exam.
3. Interpreting Slope and Intercept in Context ★★★☆☆ ⏱ 3 min
One of the most frequently tested skills on the AP Statistics exam is correctly interpreting the slope and intercept of a linear regression model in context. Unlike pure math problems, AP requires interpretation tied directly to the scenario, not just a generic description.
The slope $b_1$ is the predicted *average* change in the response variable $y$ for a 1-unit increase in the explanatory variable $x$. It always has units of (units of y) per (unit of x). The intercept $b_0$ is the predicted average value of $y$ when $x = 0$. The intercept only has a practical, meaningful interpretation if $x = 0$ is a plausible possible value in the context of the problem. If $x = 0$ is impossible or far outside the range of observed data, the intercept is only a mathematical anchor for the line and has no practical meaning.
Exam tip: If you are asked to compare slopes of two models, a steeper slope (larger absolute value) always means a larger predicted change in $y$ per 1-unit change in $x$, regardless of the sign.
4. Residual Analysis and Coefficient of Determination ★★★☆☆ ⏱ 4 min
After fitting a linear regression model, we need to check if a linear model is actually appropriate for the data, and measure how much variation in $y$ the model explains. This is done with residual plots and the coefficient of determination ($R^2$).
A residual plot graphs residuals on the $y$-axis against the explanatory variable $x$ on the $x$-axis. For a linear model to be appropriate, residuals should be randomly scattered around the horizontal line at 0 with no clear pattern. A curved pattern means the true relationship between $x$ and $y$ is non-linear, so a linear model is a poor fit. A fan-shaped pattern (residuals getting wider or narrower as $x$ increases) means non-constant error variance, which violates regression assumptions.
The coefficient of determination $R^2$ (equal to $r^2$ for simple linear regression) measures the proportion of variation in the response variable $y$ that is explained by the linear relationship with $x$. It ranges from 0 (no linear explanation) to 1 (all variation explained), or 0% to 100% when expressed as a percentage. Higher $R^2$ means a stronger linear relationship.
Exam tip: Residual plots only check if a linear model is appropriate, not how strong the relationship is. A weak linear relationship can still have a random residual pattern, meaning a linear model is appropriate but not very predictive.
Common Pitfalls
Why: Correlation is symmetric, but regression is not, and students often mix up which variable is which
Why: Students forget regression predicts the average response, not an exact outcome for every person
Why: Students confuse association (what regression measures) with causation, which requires a randomized experiment
Why: Students think every intercept needs an interpretation by default
Why: Students assume the linear relationship holds everywhere, which is almost never true
Why: Students confuse model adequacy (linearity) with strength of relationship