Concluding a Test
Introduced| English | Meaning |
|---|---|
| significance level/sɪɡˈnɪfɪkəns ˈlevl/ | The chosen probability of rejecting a true null hypothesis in a statistical test. |
| reject/rɪˈdʒekt/ | To decide the evidence is strong enough against the null hypothesis. |
| fail to reject/feɪl tə rɪˈdʒekt/ | To decide the evidence is not strong enough against the null hypothesis. |
The study must decide how much evidence is enough
- The on-time study chose a significance level 显著性水平 $\alpha=0.05$ before testing. Its planned lower-sided p-value is about 0.0478.
- The threshold is a decision rule, not the probability that this particular conclusion is wrong.
Apply the preselected rule
- Reject 拒绝 $H_0$ when the p-value is at most $\alpha$. Here $0.0478\le0.05$, so reject the 90% null claim for the planned lower-sided test.
- If the preselected threshold had been 0.01, these data would not reject. Do not raise the threshold after seeing the result.
The decision rule for a significance test is: reject H0 when...
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Reject when the p-value is at most the significance level.
The threshold p-value set before a test is the significance ___ (one word).
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The significance level α is the pre-set threshold.
State the conclusion about the population
- The checked test gives evidence that the on-time proportion for orders in the specified population is below 0.90.
- Distinguish the population claim from the observed sample: 85% was observed, but the test does not prove that the true population rate is exactly 85%.
Failing to reject H0 proves that H0 is true.
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It only means insufficient evidence against H0.
In statistics we say 'accept H0' when the p-value is large.
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We say 'fail to reject' — never 'accept'.
A non-rejection leaves uncertainty
- With p-value 0.12 and $\alpha=0.05$, fail to reject 不拒绝 the null. The study has insufficient evidence for its stated alternative at this threshold.
- Fail to reject does not mean the null is proved or that the rate meets the promise. Limited precision or low power can hide a real difference.
The same p-value 0.0478 rejects at 0.05 and fails to reject at 0.01. Both decisions use thresholds chosen in advance.
Lower-tail evidence for the worked test
The worked lower-sided test has z ≈ -1.67. The shaded left tail is approximately 0.0475 using this rounded statistic.
With p-value = 0.047 and α = 0.05, the decision is...
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0.047 ≤ 0.05, so reject H0.
For p-value 0.0478, match the prechosen threshold to the decision.
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Compare the same p-value with the threshold chosen before seeing the result.
How far below α = 0.05 is a p-value of 0.0478?
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0.05 - 0.0478 = 0.0022. This small difference does not make the decision a statement of certainty.
Evidence need not imply a useful effect
- A very large sample can detect a small departure. Whether the departure matters depends on its size, uncertainty and consequences.
- A test decision does not establish causation in an observational survey. Report the study design and avoid making a stronger causal claim.
Failing to reject means insufficient evidence at the chosen threshold. It does not establish equality.
Report the whole reasoning chain
- State the hypotheses, check conditions, calculate the statistic and p-value, compare with the preselected level, then conclude in context.
- For a non-rejection, report insufficient evidence rather than accepting the null as fact. Keep the alternative and study population explicit.
State the hypotheses, check conditions, calculate the statistic and p-value, compare with the preselected level, then conclude in context.
Write from rejecting at the 0.05 level
- The p-value is 0.01.
- The significance level was set at 0.05 before the data. The lesson p-value 0.0022 is another test.
A random-sample test rejects H0: p = 0.90 in favour of p < 0.90. Select justified conclusions.
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Use evidence language, name the population and threshold, and avoid certainty or causal claims from a sample alone.
What is the decision?
A p-value below the preselected level leads to rejection.
What is the decision?
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0.01 is below 0.05.