P-Value Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of P-Value.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The probability of observing a test statistic at least as extreme as the one computed from the sample data, assuming the null hypothesis H_0 is true.

The p-value answers: 'If nothing special is going on (H_0 is true), how surprising is my data?' A tiny p-value means the data would be very rare under H_0, so maybe H_0 is wrong. Think of it like this: you flip a coin 100 times and get 92 heads. If the coin is fair, the chance of that happening is astronomically small (tiny p-value)โ€”so you'd conclude the coin is probably not fair.

Read the full concept explanation โ†’

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: The p-value measures the strength of evidence against H_0: smaller p-value = stronger evidence against the null hypothesis. It is NOT the probability that H_0 is true.

Common stuck point: A p-value of 0.03 does NOT mean 'there's a 3\% chance H_0 is true.' It means 'if H_0 were true, there's a 3\% chance of seeing data this extreme.'

Worked Examples

Example 1

medium
A hypothesis test produces z=2.3 for a two-tailed test. Calculate the p-value and interpret it at both \alpha=0.05 and \alpha=0.01.

Solution

  1. 1
    Two-tailed p-value: p = 2 \times P(Z > 2.3) = 2 \times (1 - 0.9893) = 2 \times 0.0107 = 0.0214
  2. 2
    At \alpha=0.05: p=0.0214 < 0.05 โ†’ Reject H_0 (result is statistically significant)
  3. 3
    At \alpha=0.01: p=0.0214 > 0.01 โ†’ Fail to reject H_0 (result is not significant at 1% level)
  4. 4
    Interpretation: there's a 2.14% probability of getting a test statistic at least as extreme as 2.3 if H_0 is true

Answer

p=0.0214. Significant at \alpha=0.05 but not at \alpha=0.01.
The p-value is the probability of obtaining evidence as extreme or more extreme than observed, assuming Hโ‚€ is true. Small p-values (< ฮฑ) indicate data is unlikely under Hโ‚€. The same p-value can lead to different conclusions depending on the chosen significance level.

Example 2

hard
Correct the following misconceptions about p-values: (a) 'p=0.03 means there's a 3% chance Hโ‚€ is true.' (b) 'p=0.03 means the effect is large.'

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
A one-tailed test has z=1.7. Find the p-value and determine if we reject H_0 at \alpha=0.05.

Example 2

hard
Study A: n=50, p=0.04, effect size d=0.15 (tiny). Study B: n=10, p=0.06, effect size d=0.8 (large). Discuss which study's result is more practically meaningful and why we shouldn't rely solely on p-values.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

hypothesis testingprobability