Step by Step
1
What a p-value actually represents
The p-value is the probability of observing a result as extreme as (or more extreme than) what was actually found, ASSUMING the null hypothesis (H₀) is true.
2
The decision rule
If p < α (the significance level, commonly 0.05), reject H₀. If p ≥ α, fail to reject H₀.
3
Never say "accept H₀"
Failing to reject H₀ does not mean H₀ has been proven true — it only means there wasn't enough evidence to reject it. The correct phrasing is always "fail to reject H₀," never "accept H₀."
4
What a low p-value means, intuitively
A low p-value means the observed data would be quite unlikely if H₀ were actually true — this unlikeliness is what provides evidence against H₀ and in favor of the alternative hypothesis.
Applied Walkthrough
1
A hypothesis test yields a p-value of 0.02, with a significance level α = 0.05.
2
Since 0.02 < 0.05 (p < α), the decision is to reject H₀.
3
This means the observed data would be quite unlikely (only a 2% chance) if H₀ were actually true — providing convincing evidence in favor of the alternative hypothesis.
4
If instead the p-value had been 0.15 (greater than α = 0.05), the correct decision would be to fail to reject H₀ — NOT to say H₀ has been proven or "accepted," just that there wasn't enough evidence to reject it.
Exam Application
Exams test whether you correctly interpret what a p-value represents (probability of the observed result, assuming H₀ is true), whether you apply the p < α decision rule correctly, and whether you avoid the common phrase "accept H₀."
⚠ Common Trap
The most common and heavily tested trap is saying "accept H₀" instead of "fail to reject H₀" — these are not interchangeable phrases, since failing to reject H₀ never proves it true, only that there wasn't sufficient evidence against it.
✓ Quick Self-Check
1. What does a p-value represent?
The probability of observing a result as extreme as (or more extreme than) what was found, assuming H₀ is true.
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2. What is the decision rule for a hypothesis test using p-values?
If p < α, reject H₀; if p ≥ α, fail to reject H₀.
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3. Is it correct to say "accept H₀" when p ≥ α?
No — the correct phrasing is "fail to reject H₀," since failing to reject doesn't prove H₀ true.
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4. What does a low p-value suggest about the observed data, assuming H₀ is true?
That the observed data would be quite unlikely if H₀ were actually true, providing evidence against H₀.
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5. If p = 0.02 and α = 0.05, what is the decision?
Reject H₀, since p < α.
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