AP Statistics · Unit 3 · Topic 3.6

p-Values Study Lab

A p-value answers one question: if the null hypothesis were true, how surprising is our sample? Drag the observed p̂ and watch the shaded tail area, then check it against a simulation of samples from the null model.

1. See the p-value as an area

The p-value is the probability, assuming H₀ is true, of getting a sample proportion at least as extreme as the one observed, in the direction of Hₐ. Move p̂ and watch the shaded area change.

z = (p̂ − p₀) ÷ √( p₀(1 − p₀) ÷ n ) p-value = area beyond z in the direction of Hₐ
Preset studies

Alternative
Significance level
Check it by simulation

Null model

p-value areaObserved p̂Simulated p̂ under H₀
Test statistic z
p-value (normal model)
p-value (simulation)
Compared with α

Show your work

2. Key ideas

Definition

Probability, assuming H₀

The p-value is the probability of getting a statistic at least as extreme as the observed one, in the direction of Hₐ, assuming H₀ is true.

Evidence

Small p-value, strong evidence

A small p-value means the data would rarely happen if H₀ were true, which is evidence against H₀ and for Hₐ.

Direction

One tail or two

For Hₐ: p > p₀ use the upper tail; for p < p₀, the lower tail; for p ≠ p₀, double the tail beyond |z|.

Decision

Compare with α

If p-value < α, reject H₀ (statistically significant). If p-value ≥ α, fail to reject H₀. Never "accept" H₀.

What it is NOT

Not P(H₀ is true)

The p-value does not give the probability that a hypothesis is true or that results are due to chance. It assumes H₀ is true.

Simulation

p-values without formulas

Simulate many samples assuming H₀ and find the proportion that are at least as extreme as the observed result.

3. Quick reference

From z to p-value. Each row assumes the conditions are met.

Alternativezp-valueAt α = 0.05
Hₐ: p > p₀1.50P(Z ≥ 1.50) = 0.0668Fail to reject
Hₐ: p > p₀2.00P(Z ≥ 2.00) = 0.0228Reject
Hₐ: p < p₀−1.80P(Z ≤ −1.80) = 0.0359Reject
Hₐ: p ≠ p₀2.002 × 0.0228 = 0.0455Reject
Hₐ: p ≠ p₀−1.802 × 0.0359 = 0.0719Fail to reject

4. On the FRQ

How to earn the points

  • Interpret with three parts: the assumption ("assuming p = 0.10"), the probability, and "a sample proportion of 0.153 or greater by chance alone."
  • Link the decision to the comparison: "Because the p-value of 0.015 < α = 0.05, we reject H₀."
  • State the conclusion in context about the population: "There is convincing evidence that more than 10% of students at the school are left-handed."
  • For a simulation-based p-value, count the dots at least as extreme and divide by the number of simulated samples.
  • Show the area you found: "P(Z ≥ 2.16) = 0.0154."

Common mistakes

  • Saying the p-value is the probability that H₀ is true.
  • Forgetting to double for a two-sided alternative.
  • Writing "accept H₀" or "H₀ is true" when the p-value is large.
  • Using the wrong tail, e.g., P(Z ≤ z) for Hₐ: p > p₀.
  • Interpreting a p-value without the "assuming H₀ is true" condition.

5. Practice questions

AP-style multiple choice. Pick an answer to see the explanation.

0 / 0 correct

6. Key terms

The vocabulary for topic 3.6. Read the list, flip through flashcards, or quiz yourself. For every term in the unit, see the Unit 3 terminology guide.