UnofficialAP StatisticsUnit 9 of 9

AP Statistics Unit 9: Inference for Quantitative Data: Slopes

Intervals and tests for the slope of a regression line.

Unit overview

The slope $b$ of a sample regression line estimates the true slope $\beta$ of the population relationship. Because $b$ varies from sample to sample, you can build a confidence interval $b \pm t^* SE_b$ or test $H_0: \beta = 0$ with $t = b / SE_b$ and $n - 2$ degrees of freedom.

Conditions are summarised as LINER: the true relationship is linear, observations are independent, residuals are approximately normal, the spread of residuals is roughly equal for all $x$ , and the data come from a random sample or randomised experiment. Almost every question gives computer output, so reading the slope and its standard error from a table is part of the skill.

Key topics

Sampling distribution of $b$: Centred at $\beta$ with standard error $SE_b$ from computer output.
Confidence interval for slope: $b \pm t^* SE_b$ with $df = n - 2$ .
Test for slope: $t = (b - \beta_0)/SE_b$ , usually with $\beta_0 = 0$ .
Checking conditions: Residual plot for linearity and equal spread, a histogram or dotplot of residuals for normality.

Worked example

Computer output gives slope $b = 1.8$ with $SE_b = 0.6$ . Find the test statistic for $H_0: \beta = 0$ .

$t = \dfrac{b - 0}{SE_b}$ .
$t = \dfrac{1.8}{0.6} = 3$ .

Answer: $t = 3$

Common mistakes

Using $n - 1$ degrees of freedom instead of $n - 2$ .
Reading the standard error of the intercept instead of the slope from output.
Interpreting the interval as describing individual points rather than the true slope.

Exam tips

Interpret the slope interval as a range of plausible values for the change in the mean response per unit increase in $x$ .
If the interval contains 0, there is no convincing evidence of a linear relationship at that confidence level.

Descriptive Statistics Study Guide (study guide)
Statistics Formulas cheat sheet (one-page formulas)
Descriptive Statistics Quiz (quick quiz)
Statistics practice problems (with worked solutions)

See the AP Statistics exam format and every unit on the course page.