Hypothesis Tests for Proportions
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 test for a proportion compares a sample proportion to a claimed population proportion and measures how surprising the sample would be if the claim were true. 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
- Test a claim about a population proportion
- Use p-values to judge evidence
- Write a conclusion about a proportion in context
Worked example
Problem. A school claims 50% of students prefer online review, and a sample gives p-value 0.02 for the claim that the true proportion is higher. At alpha = 0.05, what decision fits?
- Compare p-value to alpha.
- 0.02 is less than 0.05, so the result is statistically significant.
Answer: reject the null hypothesis
Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (multiple choice). Counts include repeated prompts and variants.
Practice problems
1. Practice case A: In a one-proportion test, what does p represent?
Choices: standard deviation only · population proportion · population mean · sample size
Show solution
- A proportion test is about a population percent or fraction.
- The parameter is the population proportion.
Answer: population proportion
2. Practice case B: A claim says 30% of families prefer evening events. Which null hypothesis fits?
Choices: x = 0.30 · mean = 0.30 · p = 0.30 · p > 0.30
Show solution
- The null usually includes equality.
- For a proportion, use p.
Answer: p = 0.30
3. Practice case C: A researcher suspects the true proportion is greater than 0.50. Which alternative fits?
Choices: p = 0.50 · mean > 0.50 · p < 0.50 · p > 0.50
Show solution
- The alternative matches the suspected direction.
- Use p for a population proportion.
Answer: p > 0.50
4. Practice case D: For alpha = 0.10 and p-value = 0.04, what is the decision?
Choices: reject the null hypothesis · fail to reject the null hypothesis · prove the null hypothesis · increase alpha to 1
Show solution
- Reject when p-value is less than alpha.
- Each p-value listed is below the significance level.
Answer: reject the null hypothesis
5. Practice case E: For alpha = 0.01 and p-value = 0.04, what is the decision?
Choices: claim certainty · fail to reject the null hypothesis · reject the null hypothesis · prove the alternative
Show solution
- Fail to reject when p-value is greater than alpha.
- The evidence is not strong enough at that level.
Answer: fail to reject the null hypothesis
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