Confidence Intervals for Means
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 mean estimates an unknown population average using the sample mean, a measure of variability, 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 mean
- Interpret interval estimates
- Connect sample size and variability to margin of error
Worked example
Problem. A confidence interval for a mean is 37 to 45. What is the margin of error?
- Worked Example: First identify exactly what the question is asking: A confidence interval for a mean is 37 to 45. What is the margin of error?
- For data questions, identify what each statistic measures before calculating so the result matches the question.
- Margin of error is half the interval width.
- (45 - 37) / 2 = 4.
Answer: 4
Practice problems
1. Practice case A: A confidence interval for a mean is 37 to 45. What is the margin of error?
Show solution
- Warm-up: First identify exactly what the question is asking: A confidence interval for a mean is 37 to 45. What is the margin of error?
- For data questions, identify what each statistic measures before calculating so the result matches the question.
- Margin of error is half the interval width.
- (45 - 37) / 2 = 4.
- Check the result by substituting or estimating: the response should match 4 and make sense in the original problem.
Answer: 4
2. Practice case B: A confidence interval is used to estimate:
Choices: a population parameter · a residual only · the exact sample order · the treatment label
Show solution
- Warm-up: First identify exactly what the question is asking: A confidence interval is used to estimate:
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Intervals estimate unknown population values.
- They use sample data plus a margin of error.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: a population parameter
3. Practice case C: The null hypothesis usually represents:
Choices: the largest outlier · the default claim being tested · the sample size · the final proof of causation
Show solution
- Warm-up: First identify exactly what the question is asking: The null hypothesis usually represents:
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- A hypothesis test begins with a default claim.
- The data are judged against that null claim.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: the default claim being tested
4. Practice case D: A small p-value means the sample result is:
Choices: the same as the confidence level · always caused by bias · unusual if the null hypothesis is true · proof that the null is true
Show solution
- Warm-up: First identify exactly what the question is asking: A small p-value means the sample result is:
- For data questions, identify what each statistic measures before calculating so the result matches the question.
- The p-value assumes the null is true.
- Small p-values give evidence against that null claim.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: unusual if the null hypothesis is true
5. Practice case E: At alpha = 0.05, a p-value of 0.03 leads to:
Choices: fail to reject the null hypothesis · increase the sample size to 0.05 · ignore the alternative · reject the null hypothesis
Show solution
- Warm-up: First identify exactly what the question is asking: At alpha = 0.05, a p-value of 0.03 leads to:
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- Compare p-value to alpha.
- 0.03 is less than 0.05, so reject the null.
- 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
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