AP Statistics Unit 9 Test: Inference for Quantitative Data — Slopes

What Unit 9 Covers in AP Statistics

Unit 9 combines the regression analysis from Unit 2 with the inferential reasoning developed in Units 6 through 8. The result is a t-test and confidence interval for the true slope of a population regression line — one of the most conceptually integrated topics in the entire AP Statistics course.

The Population Regression Model

In Unit 2, the LSRL is a descriptive tool for a specific dataset. Unit 9 extends this to the population level: there is a true linear relationship between x and y in the population, described by a regression model with a true slope (beta) and true intercept. The sample LSRL is an estimate of this population relationship, and the goal of inference is to draw conclusions about the true slope.

t-Test for the Regression Slope

The null hypothesis for this test is typically that the true slope equals zero — meaning there is no linear relationship between x and y in the population. The test statistic is the sample slope divided by its standard error, and it follows a t-distribution with n − 2 degrees of freedom. A small p-value provides evidence that the true slope is not zero, suggesting a linear relationship exists in the population.

Confidence Interval for the Regression Slope

A confidence interval for the true slope estimates the range of plausible values for beta. Like all t confidence intervals, it uses the sample statistic (the sample slope) plus or minus a t critical value times the standard error of the slope. Interpretation must be in context: you are estimating the true change in the mean value of y for each one-unit increase in x, in the population.

Reading Computer Regression Output

AP exam FRQs for Unit 9 frequently present computer output from a regression analysis. Students must be able to extract the sample slope, its standard error, the t-statistic, and the p-value from a labeled table, then use these values to complete an inference procedure or interpret results.

Linking Unit 9 to Earlier Units

Unit 9 is notable for how many earlier concepts it synthesizes:

• Slope and y-intercept interpretation from Unit 2
• Residual plots for checking the linearity condition
• The four-step inference structure from Units 6 and 7
• The t-distribution and degrees of freedom from Unit 7

Key AP FRQ Patterns for Unit 9

• Writing hypotheses about the population slope using correct notation (beta, not b)
• Identifying and verifying the four conditions for regression inference: Linear, Independent, Normal, Equal variance (LINE)
• Interpreting a slope confidence interval with the phrase 'on average' and in the context of the original variables
• Distinguishing between the sample slope (b) and the population slope (beta) in written explanations