A p-Value Is a Probability About Samples, Computed in a World Where the Null Is True

You'll learnwhat a p-value is — a tail area of the null's own sampling distribution — how the alternative decides which tail counts, why the number moves when the sample does, and why every standard misreading is the definition with its assuming-clause dropped.

A company claims a 40% renewal rate and a manager suspects less. Of 250 customers, 88 renewed — a proportion of 0.352. Is that low enough to matter? The p-value answers one very specific version of that question: if the rate really were 0.40, how often would a random sample of 250 land at 0.352 or lower? The whole calculation happens inside the world that if builds — a curve centered at 0.40 with a spread computed from 0.40 — and the answer, about 6%, is the area of that curve's left tail. It is not the probability the claim is true, not the probability the result was chance, and not the probability of this exact sample; each of those is the same definition with one clause removed, and this page computes each one so you can watch it come out as a different number. Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · THE ASSUMING CLAUSE · TAIL AREAS · ONE- AND TWO-SIDED · v2

Assuming p = 0.40: the sample sits at z = −1.55 and 0.0607 of this curve lies below it.

Before you start

What you're looking at

The samples of 250 that a true rate of 0.40 would produce — the claim's own world, drawn before any data arrive.

The question

The survey gave 0.352. How often does this world go that low, and what exactly does that number mean?

Step 1 / 12