P-Value Examples in Statistics

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 Statistics.

Concept Recap

The p-value is the probability of observing results at least as extreme as the actual data, calculated under the assumption that the null hypothesis is true. A small p-value (typically below 0.05) suggests the observed data is unlikely under the null, providing evidence against it.

P-value answers: 'If nothing special is really happening, how surprising is my data?' A tiny p-value (like 0.01) means your results would be very rare if the null were true - so maybe the null is wrong. A large p-value means your results aren't surprising under the null.

Read the full concept explanation →

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 how surprising the observed data would be if the null hypothesis were true. A very small p-value suggests the null is implausible given the evidence.

Common stuck point: The p-value is NOT the probability that the null hypothesis is true. It is the probability of seeing data this extreme IF the null hypothesis were already true.

Sense of Study hint: When interpreting a p-value, first state the null hypothesis clearly. Then compare the p-value to your significance level \alpha (usually 0.05). Finally, if p < \alpha, reject the null and conclude the result is statistically significant; if p \geq \alpha, fail to reject the null.

Worked Examples

Example 1

hard
A two-tailed z-test gives z = -2.65. The p-value is approximately 0.008. If \alpha = 0.05, should we reject H_0?

Solution

  1. 1
    Step 1: Compare the p-value to the significance level: 0.008 < 0.05.
  2. 2
    Step 2: Since p < \alpha, we reject H_0.
  3. 3
    Step 3: There is statistically significant evidence that the population mean differs from the claimed value.

Answer

Reject H_0. The result is statistically significant at the 0.05 level.
The p-value is the probability of obtaining results at least as extreme as observed, assuming H_0 is true. A small p-value (below \alpha) provides evidence against H_0.

Example 2

hard
A test gives a p-value of 0.12. Interpret this and state the decision at \alpha = 0.05.

Practice Problems

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

Example 1

hard
A researcher obtains p = 0.03. Would the result be significant at \alpha = 0.05? At \alpha = 0.01?

Example 2

hard
A hypothesis test gives a p-value of 0.20. What decision would you make at \alpha = 0.10 and at \alpha = 0.05?
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Background Knowledge

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

hypothesis testingprobability basicsampling distribution