Unit 9: Inference for Quantitative Data: Slopes

Significance Test for Slope

Testing whether a linear relationship exists in the population.

Advanced25 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Confidence Interval for Slope

Testing Whether a Linear Relationship Really Exists

The confidence interval for the hours-studied/exam-score slope excluded 0 entirely. Before reading on: what would it mean, formally, to test the specific claim that the true population slope actually is 0 — no linear relationship at all?

Definition — Significance Test for a Population Slope

H₀: β₁ = 0 (no linear relationship in the population). The test statistic is t = (b₁ − 0)/SE(b₁), with df = n−2, compared to a t-distribution to find a p-value.

t = \frac{b_1 - 0}{SE(b_1)}

Worked Example — Conducting a Significance Test for the Slope

H₀: β₁ = 0. Ha: β₁ > 0 (one-sided, since more study time is expected to raise scores, not lower them). Using b₁ ≈ 5.886 and SE(b₁) ≈ 0.410 from the previous lesson: t = 5.886/0.410 ≈ 14.4, with df = 4. Even the most extreme value a standard t-table lists for df=4 (t = 4.604, upper-tail area 0.005) is far smaller than 14.4 — so the p-value is well below 0.0005.

Since the p-value is far smaller than any reasonable significance level, reject H₀: there is very strong evidence of a genuine positive linear relationship between hours studied and exam score in the population. This significance test only establishes that the slope is convincingly non-zero — it says nothing by itself about how large or practically important that relationship is; r² from Unit 2 (about 98.1%) is what actually speaks to strength. A tiny, practically unimportant slope can still be statistically significant with a large enough sample, and a large, meaningful-looking slope can fail to reach significance with too small a sample — statistical significance and practical importance are separate questions.

Tip

When a test statistic is far more extreme than anything a standard table lists, report the p-value as bounded (e.g., 'p < 0.0005') rather than guessing a specific decimal value beyond what the table can actually support.

Common Mistakes

Treating a small, statistically significant p-value as proof the relationship is practically important.
Statistical significance answers 'is the slope convincingly non-zero,' not 'is this relationship large or useful in practice' — r² (or the size of the slope itself, in context) addresses practical importance separately.
Using a two-sided alternative when the direction of the relationship was already anticipated from context.
As with the proportion test earlier in this course, the alternative hypothesis should match the actual question — anticipating more study time raises scores calls for the one-sided Ha: β₁ > 0, not a two-sided test.

Key Takeaways

A significance test for slope uses t = b₁/SE(b₁) with df = n−2 to test H₀: β₁ = 0.
A p-value more extreme than any standard table entry is reported as bounded, not as a fabricated precise value.
Statistical significance (a convincingly non-zero slope) and practical importance (how large or useful the relationship is) are separate questions, answered by different tools.

Summary

This closes AP Statistics: exploring data, collecting it well, reasoning about probability and random variables, describing sampling distributions, and using all of it to estimate and test claims about an unknown population — each unit building the statistical reasoning the next one depends on. The consistent thread throughout is distinguishing what the data actually shows from what it doesn't: association from causation, statistical significance from practical importance, and a probability about a procedure from a probability about a fixed, unknown truth.

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