Interpreting p-Values
| English | Chinese | Pinyin |
|---|---|---|
| p-value | P值 | P zhí |
What a p-value measures
- The p-value P值 is the probability of a result at least as extreme as observed, assuming $H_0$ is true.
- It answers: "if the null were true, how often would we see data like this (or more so)?"
- Small p-value → our data would be rare under $H_0$.
- It is computed from the test statistic and the alternative.
Computing it from z
- Find the area in the tail(s) of the standard normal beyond your $z$.
- One-sided $H_a$: the area in one tail (the direction of $H_a$).
- Two-sided $H_a$: double the one-tail area (both directions count as extreme).
- The alternative hypothesis decides which tail(s) to use.
Interpret in context
- State it plainly: "if $H_0$ were true, there's a p-value chance of a result this extreme."
- Tie it to the actual study — the proportion, the population, the direction.
- A p-value is a conditional probability (given $H_0$), not the probability that $H_0$ is true.
- Context keeps the number meaningful.
Small p, strong evidence
- The smaller the p-value, the stronger the evidence against $H_0$.
- A tiny p-value says the data would be very surprising if $H_0$ held.
- A large p-value says the data are quite consistent with $H_0$.
- Next lesson: how small is "small enough" to act on.
The p-value is NOT the probability that $H_0$ is true. It's the probability of data this extreme or more, assuming $H_0$. A small p-value is evidence against $H_0$; it never proves $H_0$ false, and a large one never proves $H_0$ true. And for a two-sided test, remember to double the one-tail area.
A one-sided test ($H_a: p < 0.90$) gives $z = -1.67$.
- p-value: the area to the left of $z = -1.67 \approx 0.047$.
- Interpret: if truly $p = 0.90$, we'd see $\hat{p}$ this low (or lower) about $4.7\%$ of the time.
- That's fairly rare → moderate evidence against $H_0$.
The p-value is the probability of a result at least as extreme as observed, assuming $H_0$ is true — computed as a tail area from the test statistic (doubled for a two-sided $H_a$). A smaller p-value means stronger evidence against $H_0$. It is not the probability that $H_0$ is true.
The p-value is a tail area
Shade beyond the test statistic; two-sided doubles it.
The p-value is the probability of a result at least as extreme as observed, assuming...
p-value is computed under the assumption that H0 is true.
A smaller p-value means stronger evidence against H0.
Small p → data would be rare under H0 → strong evidence against it.
The p-value is the probability that H0 is true.
It's P(data this extreme | H0), not P(H0 true).
For a two-sided test, the p-value is...
Both directions count as extreme, so double the tail area.
A large p-value indicates that the data are...
Large p → data unsurprising under H0 → weak evidence against it.