Interpreting p-Values
Introduced| English |
|---|
| p-value/piː ˈvæljuː/ |
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...
p-value is computed under the assumption that H0 is true.
The alternative determines the tail
- For $H_a:p
- 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.
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).
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...
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.
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?
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...
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.
Which statements about a p-value of 0.0478 are justified?
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.