Confidence Intervals 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 confidence interval for a proportion estimates an unknown population proportion using a sample proportion and a margin of error. 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
- Build a confidence interval for a proportion
- Interpret confidence level
- Write a conclusion in context
Worked example
Problem. A poll estimates that 46% of students prefer later start times, with a margin of error of 4 percentage points. Which interval is correct?
- Worked Example: First identify exactly what the question is asking: A poll estimates that 46% of students prefer later start times, with a margin of error of 4 percentage points. Which interval is correct?
- For percents, convert the percent to a decimal or fraction and connect it to the base amount in the problem.
- Subtract and add the margin of error.
- 46% - 4% = 42% and 46% + 4% = 50%.
Answer: 42% to 50%
Practice problems
1. Practice case A: A poll confidence interval is trying to estimate which value?
Choices: a population mean · a treatment label · a population proportion · one person's response
Show solution
- Warm-up: First identify exactly what the question is asking: A poll confidence interval is trying to estimate which value?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- A proportion is a percent or fraction of a population.
- The interval estimates the unknown population proportion.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: a population proportion
2. Practice case B: A sample proportion is 0.67 with margin of error 0.04. What interval is correct?
Choices: 0.71 to 0.75 · 0.63 to 0.71 · 0.67 to 0.71 · 0.04 to 0.67
Show solution
- Warm-up: First identify exactly what the question is asking: A sample proportion is 0.67 with margin of error 0.04. What interval is correct?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- A confidence interval is estimate plus or minus margin of error.
- Subtract and add the margin.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: 0.63 to 0.71
3. Practice case C: A 95% confidence interval for a population proportion is 0.31 to 0.39. What is the best interpretation?
Choices: the population proportion is plausibly in that interval · 95% of individual people are in the interval · the null hypothesis is definitely true · the sample size is 95
Show solution
- Warm-up: First identify exactly what the question is asking: A 95% confidence interval for a population proportion is 0.31 to 0.39. What is the best interpretation?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Confidence intervals estimate population parameters.
- The interval gives plausible values for 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: the population proportion is plausibly in that interval
4. Practice case D: What does 95% confidence refer to?
Choices: the probability that one fixed parameter moves · the percent of answer choices that are correct · the exact sample proportion · how often the method captures the true parameter in the long run
Show solution
- Warm-up: First identify exactly what the question is asking: What does 95% confidence refer to?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- The method has long-run reliability.
- It does not mean the parameter changes randomly.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: how often the method captures the true parameter in the long run
5. Practice case E: What usually makes a confidence interval for a proportion more precise?
Choices: use a leading question · remove the context · use a larger random sample · use fewer people
Show solution
- Warm-up: First identify exactly what the question is asking: What usually makes a confidence interval for a proportion more precise?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Larger samples reduce standard error.
- That usually narrows the interval.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: use a larger random sample
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