Concluding a Test
| English | Chinese | Pinyin |
|---|---|---|
| significance level | 显著性水平 | xiǎn zhù xìng shuǐ píng |
| reject | 拒绝 | jù jué |
| fail to reject | 不拒绝 | bù jù jué |
Compare p to alpha
- Set a significance level 显著性水平 $\alpha$ before the test (commonly $\alpha = 0.05$).
- It's the threshold for "small enough to act on."
- Decision rule: if p-value $\le \alpha$, the result is statistically significant.
- If p-value $> \alpha$, it isn't.
Reject or fail to reject
- If p-value $\le \alpha$: reject 拒绝 $H_0$ (the evidence against it is strong enough).
- If p-value $> \alpha$: fail to reject 不拒绝 $H_0$ (not enough evidence).
- We never say "accept $H_0$" — only "fail to reject."
- The decision is a yes/no verdict on the null.
Conclude in context
- Translate the decision into a sentence about the alternative, in context.
- "There is convincing evidence that $p < 0.90$" (if rejected), or
- "There is not convincing evidence that $p < 0.90$" (if not rejected).
- Always mention the actual proportion, population, and direction.
Failing to reject ≠ proof
- Failing to reject $H_0$ does not prove $H_0$ is true.
- It only means the data didn't give strong enough evidence against it.
- Absence of evidence is not evidence of absence — maybe $n$ was just too small.
- So report "no convincing evidence," never "$H_0$ is true."
"Fail to reject" is not "accept." A large p-value means we lack evidence against $H_0$ — perhaps because $H_0$ is true, or perhaps because our sample was too small to detect a real effect. Concluding "$H_0$ is true" overstates what a test can do. State the conclusion in terms of the alternative, in context, either way.
Test of $H_a: p < 0.90$ gives p-value $= 0.047$, with $\alpha = 0.05$.
- Compare: $0.047 \le 0.05$ → reject $H_0$.
- Conclude: there is convincing evidence that the on-time rate is below $90\%$.
- Had the p-value been $0.12$, we'd fail to reject — no convincing evidence.
Compare the p-value to the significance level $\alpha$: if p $\le \alpha$, reject $H_0$; if p $> \alpha$, fail to reject. State the conclusion about the alternative in context. Remember failing to reject does not prove $H_0$ — it only means insufficient evidence against it.
p-value vs. the significance level
If the tail area (p) is at most α, reject H0.
The decision rule for a significance test is: reject H0 when...
Reject when the p-value is at most the significance level.
Failing to reject H0 proves that H0 is true.
It only means insufficient evidence against H0.
With p-value = 0.047 and α = 0.05, the decision is...
0.047 ≤ 0.05, so reject H0.
In statistics we say 'accept H0' when the p-value is large.
We say 'fail to reject' — never 'accept'.
The threshold p-value set before a test is the significance ___ (one word).
The significance level α is the pre-set threshold.