Significance Test for the Slope

Unit 9 · Inference for Quantitative Data: Slopes

What AP Stats asks here

A slope t-test asks whether $\beta = 0$ — equivalently, whether the linear association is statistically distinguishable from zero. The mechanics live in the regression printout: $b$ , $SE_b$ , $t = b/SE_b$ , df $= n - 2$ . The recurring AP traps: using df $= n - 1$ (the one-sample t df) and reporting 'no relationship' on a failure to reject (the test only rules out a linear effect of detectable size).

Slope t-test

Hypotheses

H_0: \beta = 0 \;\;\text{vs.}\;\; H_a: \beta \neq 0 \text{ (or one-sided)}

Test statistic

t = \dfrac{b - 0}{SE_b}

Degrees of freedom

\text{df} = n - 2

AP Tip: Always df = n − 2 for slope inference. The −2 comes from estimating both slope and intercept; one-sample t uses n − 1 because only the mean is estimated.

Caution: Failing to reject H₀: β = 0 rules out a linear association of detectable size, not all associations. Non-linear patterns can hide here — look at the scatterplot.

Type 1

Set up the test and read the output

Read $b$ and $SE_b$ from the slope row; compute $t = b/SE_b$ with df $= n - 2$ .

Example 1

An ice-cream-sales-on-temperature regression with

n = 22

shows

b = 7.834

and

SE_b = 0.412

. For a two-sided test of

H_0: \beta = 0

: (A)

t \approx 19.0

, df = 20, p-value ≈ 0 — reject; convincing positive linear association. (B)

t \approx 19.0

, df = 21, p-value ≈ 0 — reject. (C)

t \approx 0.05

, df = 20, p-value ≈ 0.96 — fail to reject. (D)

t \approx 5.0

, df = 20, p-value ≈ 0.05 — reject.

Practice more of this type— AI-generated · always-new problems

Generate Problems →

Type 2

Fail to reject ≠ no relationship

The test only rules out a linear association of detectable size. Hidden non-linear patterns and small linear effects remain possible.

Example 2

A regression of resting heart rate on height (

n = 30

) gives

b = -0.18

SE_b = 0.21

, two-sided p ≈ 0.40. A student writes 'We have shown that height has no effect on heart rate.' Which critique is correct? (A) The student is correct — the p-value is very large. (B) Failing to reject only means there is no evidence of a linear effect of detectable size. The slope CI includes 0 along with a range of plausible slopes; non-linear patterns or smaller linear effects could still exist. 'No effect' overstates the result and 'effect' is causal language. (C) The student is correct only at

\alpha = 0.10

. (D) The student is wrong because

r^2

was not reported.

Practice more of this type— AI-generated · always-new problems

Generate Problems →

Type 3

df and sample size affect the p-value

Same $t$ from different samples gives different p-values because df changes. Larger n shrinks the tail.

Example 3

Study A:

b = 1.5

SE_b = 0.30

n = 10

. Study B:

b = 1.5

SE_b = 0.30

n = 50

. Both two-sided. Which is correct? (A) Both p-values are identical because t is the same. (B) Study A has df = 8 and p ≈ 0.001 (heavier t tails); Study B has df = 48 and p ≈ 9 × 10⁻⁶ (t nearly normal). Both reject, but B more decisively. (C) Study B fails to reject because n is too large. (D) Study A is wrong because df should be 9.

Practice more of this type— AI-generated · always-new problems

Generate Problems →