Slope t-Test & p-Value Mastery

Part 1 — Understanding the Concept

What is this question actually asking?

In regression, a computer output gives you a slope b and its standard error SE_b. The question "can we really say this slope isn't zero?" is exactly what the t-test for the slope answers.

WHAT

b (the slope)

The estimated average change in y for a 1-unit increase in x. Because it comes from a sample, it carries some uncertainty.

WHAT

SE_b (standard error)

How much the estimate b would bounce around from sample to sample. Smaller SE_b means b is more trustworthy.

Why do we use a t-test here?

We never know the true population slope β, so we set up the null hypothesis H₀: β = 0 (meaning "x and y have no linear relationship") and ask how many standard errors away from 0 our sample's b actually is. That distance, measured in SE_b units, is the t-statistic.

Step 1 — Compute t: t = b / SE_b tells you how many standard errors b is from 0.
Step 2 — Compute degrees of freedom: df = n − 2 (we subtract 2 because regression estimates two parameters: the intercept and the slope.)
Step 3 — Locate t in the t-table: find the row for that df, then see which two critical values your t falls between.
Step 4 — Read off the p-value range: the larger |t| is, the smaller the p-value. Since this is a two-sided test, double the one-tail probability from the table.

Part 2 — Worked Example (Original Problem)

Worked Example

Given: b = 2.5, SE_b = 0.8, n = 22

Step-by-step

t = b / SE_b	2.5 / 0.8 = 3.125
df = n − 2	22 − 2 = 20
t-table (df=20) row	... 2.845 → 3.125 → 3.552 ...
one-tail p	between 0.001 and 0.005
two-tail p (× 2)	between 0.002 and 0.01

In the df=20 row, t=3.125 falls between the critical values 2.845 (one-tail p=.005) and 3.552 (one-tail p=.001). Since this is a two-sided test, doubling both bounds gives an exact t-table range of 0.002 < p < 0.01.

Answer: 0.002 < p < 0.01. (The exact calculator value is p ≈ 0.0053, which falls right inside this range — solving it by hand with the t-table gets you to the same conclusion.)

Reference t-table excerpt (df=20)

df	0.10	0.05	0.025	0.01	0.005	0.001
one-tail p →	two-tail = ×2 (e.g. 0.05 one-tail = 0.10 two-tail)
20	1.325	1.725	2.086	2.528	2.845	3.552

The 20 questions below each use a different df value (6–33), so apply the same method each time: find the matching df row and work from there. Every answer was pre-verified with exact computation (scipy's t-distribution).

Part 3 — Practice Set

20-Question Drill

Get it right and confetti pops 🎉 Get it wrong and the explanation shows up immediately, plus a similar problem to try again.

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