Topic 9.3 Justifying a Claim About the Slope of a Regression Model Based on a Confidence Interval: use a slope interval to judge whether a linear relationship exists and to evaluate claims about the size and direction of the slope.

What this topic is asking

The College Board (Topic 9.3) wants you to use a slope confidence interval to justify a claim: judge whether a linear relationship exists (does the interval contain $0$?), and assess claims about the size and direction of the slope, accounting for confidence level and sample size.

The zero-check for a slope

This is the plausible-value reasoning applied to the slope, with the relevant null value being $0$ (no association). It directly answers "do those points align?": if $0$ is implausible, the points show a real linear trend; if $0$ is plausible, the observed slope could be chance. The sign of an interval that excludes $0$ gives the direction of the relationship. A two-sided test of $H_0: \beta = 0$ at $\alpha = 1 - C$ reaches the same verdict, the duality of Topic 9.5.

Judging size-of-slope claims

A magnitude claim, "each extra unit of $x$ raises mean $y$ by at least $2$," must be checked against the entire interval, not the point estimate $b$. If part of the interval falls short of the claimed size, then a smaller slope is still plausible and the claim is not established, even when $b$ meets it. Only if the whole interval satisfies the claim is it supported. The interval, the set of plausible slopes, carries the justification. As elsewhere, separate "is there a relationship?" (the zero-check) from "is the slope at least this big?" (the whole-interval check); both are commonly asked and scored differently.

Confidence level, sample size, and decisiveness

A higher confidence level widens the slope interval (more often correct, less precise), which can pull a once-decisive interval back across $0$ and weaken the relationship conclusion. A larger sample, or a wider spread of $x$-values, shrinks $SE_b$ and narrows the interval, sharpening conclusions about both the existence and the size of the slope. These trade-offs mirror the mean case. Note that interpreting a slope always stays in context and units (per unit of $x$), and any conclusion about causation requires a randomised experiment, not just a slope that excludes $0$.

Try this

Q1. A $95\%$ slope interval is $(-2.1, -0.3)$. What does it say? [1 point]

• Cue. Entirely below $0$: convincing evidence of a negative linear relationship between $x$ and $y$.

Q2. Why does a slope interval containing $0$ fail to support a relationship? [1 point]

• Cue. A slope of $0$ means no linear relationship; if $0$ is plausible, the observed slope could be due to chance.