P-Value
In short: P-value is the probability of seeing a sample result at least as far from the null claim as the one observed, calculated on the assumption that the null hypothesis is true.
AP StatisticsStatisticsAP Statistics Unit 3
What is P-Value?
A p-value puts a number on surprise. Imagine the null hypothesis is correct and the study is repeated endlessly; the p-value tells you what fraction of those repeats would land as far from \(H_0\) as your actual sample did.
When that fraction is tiny, your sample would be a rare event under \(H_0\), and the null becomes hard to believe. When it is large, the sample looks ordinary. Keep in mind what it is not: it is neither the chance that \(H_0\) is true nor the chance that results came from luck alone. It is a probability about the data, given \(H_0\).
How to spot it
- Computer output or a table prints a number labeled P beside a test statistic.
- The question asks you to interpret that number in context, in one sentence.
- A decision is required by weighing it against a significance level.
- A normal or t curve is shaded in one or both tails beyond the observed statistic.
The method
- Compute the test statistic (a \(z\) or \(t\) value) from the sample.
- Find the area in the tail beyond that value, in the direction of \(H_a\); for a two-sided test, double it.
- Interpret it in a sentence: “If \(H_0\) were true, the probability of a result this extreme is …”.
- Compare with \(\alpha\) to decide whether to reject \(H_0\).
P-Value: solved example
Problem
A game developer says a loot box yields a rare item half the time. A player opens 120 boxes and gets the rare item 72 times (60%). Run a two-sided test of \(H_0: p = 0.50\) and interpret the result.
Solution
Under \(H_0\) the standard error is \(\sqrt{\frac{0.5 \times 0.5}{120}} \approx 0.0456\), so \(z = \frac{0.60 – 0.50}{0.0456} \approx 2.19\).
One tail beyond 2.19 has area \(1 – 0.9857 = 0.0143\). Doubling for both directions gives 0.0285.
Meaning: if the true rate really were 50%, only about 2.85% of 120-box samples would deviate this much, high or low.
Try it
Question 1
A test reports 0.03. Is there a 3% chance that \(H_0\) is true?
Show the answer
No. That figure is the chance of data this extreme if \(H_0\) were true, not a probability for \(H_0\) itself.
Question 2
A test reports 0.30. What does it suggest about the data?
Show the answer
A result like this would show up about 30% of the time by chance under \(H_0\), so the data offer little evidence against it.
P-Value vs. Significance Level
The significance level \(\alpha\) is a cutoff chosen before the study. The p-value is computed from the data after the sample is collected. The decision rule compares them: reject \(H_0\) when the p-value is smaller than \(\alpha\).
Keep going
Lessons
Worksheets and quizzes
Related words
- Hypothesis Testing
- Significance Level (Alpha)
- Null and Alternative Hypotheses
- Z-Score
- Type I and Type II Errors
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