A least-squares line fitted to a sample is an estimate. Another sample would give a different slope. Interpreting a slope is only half the job; we also need to know how much it could vary, and whether a "relationship" might be nothing but chance. This lesson uses the t procedures you already know, and it is the doorway to econometrics.
Key idea
Model: $y = \alpha + \beta x + \varepsilon$, where $\beta$ is the true slope. The sample slope $b$ has standard error $SE_b$ (from software). To test $H_0: \beta = 0$ (no linear relationship): $t = \frac{b}{SE_b}$ with $df = n - 2$. A confidence interval for $\beta$: $b \pm t^* SE_b$. Conditions: linear form, independent observations, roughly normal residuals with equal spread, random sample or experiment.
Why it works
Imagine drawing many samples from the same population and fitting a line to each. The slopes $b$ pile up around the true slope $\beta$, with a spread that shrinks with more data and with x-values that are more spread out (a wide base makes a line steadier). $SE_b$ estimates that spread. Standardizing, $(b - \beta)/SE_b$ follows a t distribution, with $n - 2$ degrees of freedom because two numbers, the intercept and the slope, were estimated from the data before the residual spread could be measured.
| Predictor | Coefficient | SE | t | p |
|---|---|---|---|---|
| Constant | 12.0 | 3.1 | 3.87 | 0.001 |
| hours of practice | 2.4 | 0.8 | 3.00 | 0.008 |
Worked example: Reading the output
Use the output for $n = 20$ players (score against hours of practice). Test $H_0: \beta = 0$.
- $t = \dfrac{b}{SE_b} = \dfrac{2.4}{0.8} = 3.0$ with $df = 20 - 2 = 18$.
- The two-sided p-value is about $0.008$ (the output's "p" column).
- $0.008 < 0.05$: reject $H_0$. There is convincing evidence of a positive linear relationship between practice and score in the population.
Answer: t = 3.0, p ≈ 0.008.
Worked example: An interval for the slope
Build a 95% confidence interval for $\beta$. ($t^* = 2.101$ for 18 df.)
- $2.4 \pm 2.101 \times 0.8 = 2.4 \pm 1.68$.
- From about $0.72$ to $4.08$ points per hour.
- Each extra hour of practice is associated with between about 0.7 and 4.1 more points, on average. The interval excludes 0, matching the test.
Answer: $(0.72, 4.08)$
Worked example: No convincing evidence
Another regression with $n = 20$ has $b = 0.5$ and $SE_b = 0.4$. What do you conclude?
- $t = 0.5/0.4 = 1.25$, df $= 18$; two-sided p-value about $0.23$.
- Fail to reject: the sample slope could easily come from a population with no linear relationship.
Answer: Not convincing evidence.
Common mistake
Two overreaches. A significant slope from observational data does not prove that x causes y (lurking variables still apply), and failing to reject does not prove the slope is 0, only that this sample cannot tell. Check what was randomized, and phrase conclusions about the population slope.
The econometrics course picks up exactly here: Inference in Regression shows thousands of sample slopes piling up around the true one, then multiple regression adds more x-variables.