P-Values and Statistical Significance
A free Statistics and Data Analysis lesson from the “Inference and Conclusions” unit, with a worked example and practice problems including step-by-step solutions.
A p-value measures how unusual the sample result would be if the null hypothesis were true. Statistical significance happens when the p-value is small enough under the chosen significance level. This lesson builds the habit of reading the context first, choosing the right statistical tool, calculating carefully, and then writing what the result means. By the end, students should be able to do the computation and explain why that computation answers the question.
What you'll learn
- Interpret a p-value
- Use a significance level
- Avoid saying the p-value is the probability the null is true
Worked example
Problem. A test has p-value 0.04 and alpha = 0.05. Is the result statistically significant?
- Worked Example: First identify exactly what the question is asking: A test has p-value 0.04 and alpha = 0.05. Is the result statistically significant?
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- A result is significant when p-value < alpha.
- Here 0.04 < 0.05.
Answer: yes, because 0.04 is less than 0.05
Practice problems
1. Practice case A: What does a p-value describe?
Choices: the probability the alternative is true · the sample mean itself · the confidence level · how unusual the data would be if the null were true
Show solution
- Warm-up: First identify exactly what the question is asking: What does a p-value describe?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- The p-value is calculated under the null hypothesis.
- Small values mean the result is unusual under that assumption.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: how unusual the data would be if the null were true
2. Practice case B: At alpha = 0.01, p-value = 0.006 is:
Choices: statistically significant · not statistically significant · proof of the null · a margin of error
Show solution
- Warm-up: First identify exactly what the question is asking: At alpha = 0.01, p-value = 0.006 is:
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- A result is significant when p-value is below alpha.
- Each listed p-value is below alpha.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: statistically significant
3. Practice case C: At alpha = 0.05, p-value = 0.07 is:
Choices: a sample proportion · not statistically significant · statistically significant · proof of the alternative
Show solution
- Warm-up: First identify exactly what the question is asking: At alpha = 0.05, p-value = 0.07 is:
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- A result is not significant when p-value is above alpha.
- The evidence is not strong enough at that level.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: not statistically significant
4. Practice case D: Which p-value is more surprising under the null?
Choices: 0.65 · 0.90 · 0.01 · 0.20
Show solution
- Warm-up: First identify exactly what the question is asking: Which p-value is more surprising under the null?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Smaller p-values are more surprising under the null.
- That means stronger evidence against it.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: 0.01
5. Practice case E: A statistically significant result may still be:
Choices: unlikely under the null · evidence against the null · based on a significance level · important in real life
Show solution
- Warm-up: First identify exactly what the question is asking: A statistically significant result may still be:
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Statistical significance is about evidence, not size or importance.
- Context decides practical importance.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: important in real life
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