Key takeaway: A p-value is the chance, computed while assuming the null hypothesis is true, of seeing sample results at least as far from the claim as the results actually obtained.
Grade 12 – AP StatisticsAP Statistics Unit 3Statistics and Data
Definition of P-Value
A p-value measures surprise. Suppose the null hypothesis were exactly right. How often would random sampling alone give a result as far from the claim as ours, or farther? That fraction is the number reported.
Small values mean the data would be rare under the null, which counts against it. Large values mean the data look quite ordinary. It is a probability about data given a hypothesis, and not a probability that the hypothesis is true.
Formula and notation
For a left-tailed z-test with observed statistic \(z_0\):
\[ P(Z\le z_0)\quad\text{computed with }H_0\text{ assumed true}. \]
A right-tailed test uses the upper area \(P(Z\ge z_0)\), and a two-sided test doubles the smaller tail.
Key facts
- The smaller it is, the stronger the evidence against the null hypothesis.
- Reject the null when the value is at most the significance level alpha.
- It is never the probability that the null hypothesis is true.
- A tiny p-value can accompany a trivially small real effect when the sample is huge.
P-Value example
A machine is supposed to fill bottles with a mean of 500 mL, with known standard deviation 6 mL. A random sample of 36 bottles averages 498.2 mL. Test whether the machine underfills, and find the p-value.
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Hypotheses: \(H_0:\mu=500\) and \(H_a:\mu\lt 500\).
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Standard error: \(\frac{6}{\sqrt{36}}=1\), so \(z=\frac{498.2-500}{1}=-1.80\).
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Left-tail area from the normal table: \(P(Z\le -1.80)=0.0359\).
The p-value is about 0.0359: if the machine were correct, a sample this low would occur about 3.6% of the time.
Common mistake
Watch out
A student says, “The result is 0.0359, so there is a 3.59% chance that the machine is working correctly.”
Why it is wrong
That number is computed by assuming the machine is correct, so it cannot also be the probability of that assumption. It is the chance of data at least this extreme if the null holds. The right reading: results like these are unusual under the null.
Related terms and lessons
- Hypothesis Testing
- Significance Level (Alpha)
- Null and Alternative Hypotheses
- Z-Score
- Normal Distribution
- AP Statistics · Inference for ProportionsLessonWorksheetQuiz
- AP Statistics · Inference for MeansLessonWorksheetQuiz
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