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?
- Worked Example: First identify exactly what the question is asking: 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?
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- Compare p-value to alpha.
- 0.02 is less than 0.05, so the result is statistically significant.
Answer: reject the null hypothesis
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
- Warm-up: First identify exactly what the question is asking: In a one-proportion test, what does p represent?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- A proportion test is about a population percent or fraction.
- The parameter is the population proportion.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
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
- Warm-up: First identify exactly what the question is asking: A claim says 30% of families prefer evening events. Which null hypothesis fits?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- The null usually includes equality.
- For a proportion, use p.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
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
- Warm-up: First identify exactly what the question is asking: A researcher suspects the true proportion is greater than 0.50. Which alternative fits?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- The alternative matches the suspected direction.
- Use p for a population proportion.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
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
- Warm-up: First identify exactly what the question is asking: For alpha = 0.10 and p-value = 0.04, what is the decision?
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- Reject when p-value is less than alpha.
- Each p-value listed is below the significance level.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
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
- Warm-up: First identify exactly what the question is asking: For alpha = 0.01 and p-value = 0.04, what is the decision?
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- Fail to reject when p-value is greater than 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: fail to reject the null hypothesis
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