Unit 9: Inference for Quantitative Data: Slopes

Confidence Interval for Slope

Constructing a confidence interval for the true slope of a linear relationship.

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

How Confident Can We Be in a Sample's Slope?

Unit 2's regression of exam score on hours studied, from a sample of just 6 students, gave a slope of about 5.886 points per hour. Before reading on: if a different sample of 6 students had been collected instead, would you expect its regression line to have exactly the same slope — or some other, nearby value?

Definition — Confidence Interval for a Population Slope

A confidence interval for the true population slope β₁ has the form b₁ ± t*·SE(b₁), with df = n−2 (two fewer than n, since both a slope and an intercept are estimated from the sample). Conditions (sometimes remembered as LINER): a Linear relationship, Independent observations, Normally distributed residuals, Equal variability of residuals across x, and a Random sample.

b_1 \pm t^* \cdot SE(b_1)

Worked Example — Constructing a Confidence Interval for the Slope

From the Unit 2 regression (n = 6), statistical software reports slope b₁ ≈ 5.886 and SE(b₁) ≈ 0.410 (computed from the residuals' spread around the line, divided by the spread of the x-values). df = 6−2 = 4; for 95% confidence, t* ≈ 2.776. Margin of error = 2.776 × 0.410 ≈ 1.14. The interval: 5.886 ± 1.14 = (4.75, 7.02).

Interpretation: 'We are 95% confident that the true slope of the relationship between hours studied and exam score, for the population this sample represents, is between about 4.75 and 7.02 points per hour.' Since this interval doesn't come close to including 0, it suggests a genuine, non-zero relationship exists in the population — not just an artifact of this particular sample of 6 students.

Tip

A confidence interval for a slope that excludes 0 already hints at what a significance test would formally confirm — the interval and the test are two views of the same underlying evidence.

Common Mistakes

Using df = n−1 for a slope's confidence interval, as with a one-sample mean.
Regression estimates two quantities from the data — the slope and the intercept — so two degrees of freedom are used up, leaving df = n−2, not n−1.
Skipping the LINER conditions because the regression itself already produced a high r².
A strong r² describes how well the line fits this sample — it doesn't by itself confirm the residuals are Normally distributed or equally variable across x, both of which the inference procedure still requires checking (typically via a residual plot).

Key Takeaways

A confidence interval for a slope uses b₁ ± t*·SE(b₁), with df = n−2.
An interval that excludes 0 suggests a genuine population relationship, not just sample-specific noise.
The LINER conditions (linearity, independence, Normal residuals, equal variability, random sample) must be checked before trusting the inference.

Summary

The final lesson tests this same slope directly against the specific claim of no relationship at all.

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