Unit 9: Inference for Quantitative Data: Slopes
AP Statistics: 42 practice questions with detailed explanations.
Unit Study Guide
Executive Summary
The last piece of inference: does x really predict y? Slope inference answers with t procedures on the regression slope.
Slope intervals and tests
A confidence interval estimates the true slope β: b ± t*·SEb, with df = n - 2. A hypothesis test usually checks H0: β = 0 — no linear relationship — with t = b / SEb. Small p-values give evidence of a linear relationship; intervals excluding 0 agree with rejecting H0.
Conditions
Linearity: scatterplot and residual plot show a linear pattern (no curves). Independence: random sampling, independent observations. Normality: residuals roughly normal. Equal variance: residual spread constant across x. Fanning or curving residual plots violate these.
Degrees of freedom and reading output
df = n - 2 because two parameters (slope and intercept) are estimated. Computer output gives b, SEb, t, and the p-value directly.
Transformations
When the relationship curves, transform to straighten it: log y for exponential growth, log x and log y for power models. Check the transformed scatterplot for linearity before proceeding.
Exam traps
df is n - 2, not n - 1. A slope interval containing 0 means no significant linear relationship at that level. Residuals, not the raw y-values, must be normal. Significance is not the same as a strong relationship — check r or r-squared for strength.