Setting Up a Test for the Slope
The slope hypotheses
- A test about a slope asks whether there's a real linear relationship.
- Null $H_0: \beta = 0$ — the slope is zero.
- Alternative $H_a$: $\beta \ne 0$ (some relationship), or one-sided $\beta > 0$ / $\beta < 0$.
- The default $H_0$ is almost always "$\beta = 0$."
Why β = 0 means "no relationship"
- If the true slope $\beta = 0$, the regression line is flat.
- A flat line predicts the same $y$ no matter what $x$ is → $x$ tells you nothing.
- So $\beta = 0$ is exactly "no linear relationship."
- Rejecting $H_0$ means the data support a real linear link.
Verify the conditions
- Check the same LINER conditions as for the interval:
- Linear, Independent, Normal, Equal SD, Random.
- Use residual plots for the linear / normal / equal-SD parts.
- Only then is the $t$-model for the slope valid.
The right test
- The procedure is a $t$-test for the slope, with $df = n - 2$.
- It's the regression counterpart of the one-mean $t$-test.
- The test statistic (next lesson) compares $b$ to $0$ in units of $SE_b$.
- Everything is read from the same computer output.
The null is $\beta = 0$ (a flat line = no linear relationship), and the $df$ is $n - 2$. Don't test "$b = 0$" — $b$ is the sample estimate; the hypothesis is about the population slope $\beta$. And choose a one- or two-sided $H_a$ from the question before seeing the output.
Does studying more relate to higher scores?
- $H_0: \beta = 0$ (no linear relationship between hours and score).
- $H_a: \beta > 0$ (more hours → higher scores) — a one-sided claim.
- Conditions checked, we'll run a $t$-test for the slope with $df = n - 2$.
A slope test states $H_0: \beta = 0$ (no linear relationship) vs. a one- or two-sided $H_a$, using a $t$-test for the slope with $df = n - 2$ after checking the LINER conditions. A slope of $0$ means a flat line — $x$ gives no help predicting $y$.
The null: a flat line (β = 0)
H0 says the true slope is 0 — no linear relationship.
The usual null hypothesis for a slope test is...
H0 is about the population slope β being 0.
A true slope β = 0 means...
Flat line → x doesn't help predict y.
The slope t-test uses what degrees of freedom for n = 20?
df = n − 2 = 18.
The hypotheses are about the population slope β, not the sample slope b.
b is the estimate; β is the parameter tested.
Rejecting H0: β = 0 provides evidence that...
Rejecting the flat-line null supports a real linear link.