Inference for Quantitative Data: Slopes

What this unit covers

The topics below follow the published Statistics course framework for Unit 9. This unit is worth 2–5% of the exam, so budget your time against that rather than against how long the unit takes to teach.

Lessons in this unit

• The Sampling Distribution of the Slope13 min · 3 objectivesDistinguish the sample slope b from the population slope β · Describe the sampling distribution of the slope · Interpret computer regression output for inference
• Confidence Intervals for the Slope14 min · 3 objectivesConstruct a t-interval for the population slope β · Read the slope and standard error from regression output · Interpret a slope interval in context
• Significance Tests for the Slope15 min · 3 objectivesState hypotheses for a test of the population slope · Compute the t statistic for the slope and find the p-value · Interpret regression output, including the reported p-value
• Conditions for Regression Inference13 min · 3 objectivesState the conditions (LINER) required for inference about a slope · Use residual plots to check the conditions · Explain the consequence of a violated condition

Formulas in Unit 9

Every term in Unit 9

All 12 terms we publish for Inference for Quantitative Data: Slopes, with definitions. Reading them through is the fastest way to find the ones you cannot define — then drill those in cram mode until you can produce them without the prompt.

Population regression model

y = α + βx + ε, with ε normally distributed with constant standard deviation. Inference is about β, the true slope.

Conditions for slope inference (LINER)

Linear relationship, Independent observations, Normal residuals, Equal variance about the line, and Random data collection.

Checking conditions with a residual plot

No curved pattern supports linearity; constant vertical spread supports equal variance. A histogram of residuals checks normality.

t-interval for the slope

b ± t*·SE(b) with df = n − 2, interpreted as capturing the true slope in context and units.

t-test for the slope

H₀: β = 0 means no linear relationship. t = b/SE(b) with n − 2 degrees of freedom.

Reading regression output

The coefficient column gives a and b, the SE column gives SE(b), and the printed t and p-value are for the two-sided test of zero slope.

Why df = n − 2

Two parameters, the slope and the intercept, are estimated from the data, so two degrees of freedom are used up.

Interpreting a significant slope

Convincing evidence of a linear relationship in the population — not evidence of causation unless the data came from a randomized experiment.

Standard error of the slope

Measures how much the estimated slope would vary across repeated samples. Read it from the SE Coef column of regression output.

Why the printed p-value may need halving

Software reports a two-sided p-value. A one-sided alternative uses half of it, provided the sample slope is in the hypothesized direction.

Confidence interval for the slope in context

State that you are confident the interval captures the true average change in the response per one-unit increase in the explanatory variable, with units.

Extrapolation warning in slope inference

Inference applies only over the range of x values observed; predictions beyond it have no support from the data.

What examiners penalize here

• A slope interval interpretation must name the *context and units*: "average change in [y-variable] per one-unit increase in [x-variable]." And always check whether **0 is inside** — that single fact tells you whether a linear relationship is plausible.
• On free response, name each condition and cite the *specific evidence* you used: "the residual plot shows no leftover curve (Linear) and roughly constant spread (Equal variance)." Vague statements like "the conditions are met" without evidence do not earn the point.

Practice Statistics

Our practice bank is drawn from across the whole course rather than filtered to one unit, which is closer to how the exam asks anyway — it will not tell you which unit a question is testing.

Questions about this unit

All 9 units of AP Statistics

1. Unit 1 · Exploring One-Variable Data
2. Unit 2 · Exploring Two-Variable Data
3. Unit 3 · Collecting Data
4. Unit 4 · Probability & Random Variables
5. Unit 5 · Sampling Distributions
6. Unit 6 · Inference for Proportions
7. Unit 7 · Inference for Means
8. Unit 8 · Inference for Categorical Data: Chi-Square
9. Unit 9 · Inference for Quantitative Data: Slopes