Interpreting p-Values
Introduced| English | Meaning |
|---|---|
| p-value/piː ˈvæljuː/ | The probability, assuming the null model, of a test statistic at least as extreme as the observed one in the direction or directions tested. |
How unusual are eighty-five on-time orders?
- For the lower-sided on-time test, $z\approx-1.667$. Its p-value P值 is approximately $P(Z\le-1.667)=0.0478$.
- This is an approximate probability under the null model: a sample proportion as low as observed or lower, if the true on-time proportion were 0.90.
Identify what is conditioned on
- The p-value measures tail probability assuming $H_0$ and the model assumptions. It is not the probability that $H_0$ is true.
- A small p-value suggests that the observed result is unusual under that model. It does not prove fraud, a specific cause or the size of a practical problem.
The p-value is the probability of a result at least as extreme as observed, assuming...
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p-value is computed under the assumption that H0 is true.
The alternative determines the tail
- For $H_a:p<p_0$, use the left tail; for $H_a:p>p_0$, use the right tail. With $z=-1.667$, those areas are about 0.0478 and 0.9522.
- The right-sided probability is large here. Do not use the smaller tail regardless of the planned direction.
A smaller p-value means stronger evidence against H0.
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Small p → data would be rare under H0 → strong evidence against it.
The p-value is the probability that H0 is true.
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It's P(data this extreme | H0), not P(H0 true).
Use both extremes for a two-sided test
- For a symmetric standard-normal two-sided test, use $P(Z\le-|z|)+P(Z\ge|z|)$, equivalently twice the smaller tail.
- For this statistic, the two-sided p-value is about $2(0.0478)=0.0956$. Doubling the larger directional tail would produce an invalid probability above one.
The lower-tail probability is 0.0478, but the upper-tail probability is 0.9522. The prechosen question decides which one is relevant.
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.
For a symmetric standard-normal z-test with a two-sided alternative, the p-value is...
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Use 2P(Z ≥ |z|). Doubling the larger tail gives the wrong answer and may exceed one.
Match the alternative to the approximate p-value for the rounded statistic z = -1.67.
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These areas use rounded z = -1.67. Keeping the worked statistic -1.6667 gives approximately 0.0478, 0.9522 and 0.0956.
For the worked statistic z = -1.6667, the left-tail area is approximately 0.0478. What is the right-tail area?
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The complementary area is 1 - 0.0478 = 0.9522. A right-sided alternative uses this larger tail.
Keep the probability in context
- Write: assuming a 90% on-time population rate and the checked model, results this low or lower occur about 4.8% of the time.
- For the same null comparison, a smaller p-value gives stronger incompatibility evidence. A large value does not prove the null or demonstrate that the study could detect every important effect.
A p-value conditions on H₀; it does not give the probability that H₀ is true.
A large p-value indicates that the data are...
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Large p → data unsurprising under H0 → weak evidence against it.
Separate evidence from a decision
- The p-value alone does not state the chosen decision threshold. Compare it with a significance level set for the study.
- A lower-sided p-value of 0.0478 and a two-sided value of 0.0956 can lead to different decisions at 0.05 because they answer different planned questions.
The p-value alone does not state the chosen decision threshold. Compare it with a significance level set for the study.
Write from what a p-value is not
- A test for an on-time rate gives a small tail probability.
- A reader says that tail probability is the chance the null hypothesis is true. The lesson tail probability 0.9522 is another test.
Which statements about a p-value of 0.0478 are justified?
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A p-value is conditional on the null model and tail rule. It does not supply the probability of the hypothesis or the size of an effect.
What is wrong with that reading?
The null is assumed when the p-value is computed.
What is a p-value?
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The calculation assumes the null.