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 sample estimates average wait time as 18 minutes with a margin of error of 3 minutes. Which interval is correct?
- Worked Example: First identify exactly what the question is asking: A sample estimates average wait time as 18 minutes with a margin of error of 3 minutes. Which interval is correct?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Subtract and add the margin of error.
- 18 - 3 = 15 and 18 + 3 = 21.
Answer: 15 to 21 minutes
Practice problems
1. Practice case A: An interval for average commute time estimates:
Choices: a population proportion · one person's exact value · a treatment label · a population mean
Show solution
- Warm-up: First identify exactly what the question is asking: An interval for average commute time estimates:
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Means are averages.
- The interval estimates the population average.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: a population mean
2. Practice case B: A sample mean is 73 with margin of error 4. Which interval is correct?
Choices: 69 to 77 · 73 to 77 · 4 to 73 · 77 to 81
Show solution
- Warm-up: First identify exactly what the question is asking: A sample mean is 73 with margin of error 4. Which interval is correct?
- For data questions, identify what each statistic measures before calculating so the result matches the question.
- Use sample mean plus or minus margin of error.
- That gives the lower and upper endpoints.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: 69 to 77
3. Practice case C: An interval from 28 to 36 minutes estimates:
Choices: a categorical proportion · the population average in context · the exact value for every individual · the sample size
Show solution
- Warm-up: First identify exactly what the question is asking: An interval from 28 to 36 minutes estimates:
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- A mean interval estimates an average.
- Units and context should stay attached.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: the population average in context
4. Practice case D: An interval from 45 to 53 has margin of error:
Choices: 12 · 20 · 4 · 8
Show solution
- Warm-up: First identify exactly what the question is asking: An interval from 45 to 53 has margin of error:
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Margin of error is half the width.
- Each interval has width 8, so the margin is 4.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: 4
5. Practice case E: Which situation gives less precise mean estimates?
Choices: a larger sample size · clearer units · a neutral survey question · more variability in the data
Show solution
- Warm-up: First identify exactly what the question is asking: Which situation gives less precise mean estimates?
- For data questions, identify what each statistic measures before calculating so the result matches the question.
- More variability makes the sample mean less stable.
- That usually widens the interval.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: more variability in the data
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