Unit 2: Exploring Two-Variable Data
AP Statistics: 52 practice questions with detailed explanations.
Unit Study Guide
Executive Summary
Scatterplots reveal relationships between two quantitative variables; regression summarizes them; residuals diagnose the fit.
Scatterplots and correlation
Explanatory variable (x) predicts the response (y). Correlation r measures strength AND direction of LINEAR association; r runs -1 to 1, has no units, and is unaffected by swapping x and y. r near 0 means no linear pattern — a perfect curve can still have r near 0.
Least-squares regression
The regression line y-hat = a + bx minimizes squared residuals. It passes through (x-bar, y-bar). Slope b = r · sy / sₓ: predicted change in y per one-unit change in x. Intercept a predicts y when x = 0.
Residuals
Residual = observed - predicted. Positive: above the line. Residual plots should show random scatter; curvature means nonlinear, fanning means unequal variance. Extrapolation beyond the data range is risky.
r-squared and influence
r-squared is the percent of y's variation explained by the linear model. An influential point (often extreme in x) substantially changes the line; an outlier has a large residual. Fit with and without to detect influence.
Two-way tables
Joint, marginal, and conditional distributions describe relationships between two categorical variables. Compare conditional distributions to detect association; segmented bar charts visualize them. Simpson's paradox: associations can reverse when groups are combined.
Exam traps
Correlation is not causation. r = 0.8 is NOT twice r = 0.4. r-squared is r SQUARED. Switching axes changes the line but not r.