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.
For independent random observations, estimate a population mean with x-bar ± t* s/√n when the one-sample t conditions are reasonable. The estimated standard error is s/√n and df=n−1; use a t critical value for the chosen confidence level. For a small sample, check approximate normality and absence of strong skewness/outliers. Sampling without replacement also needs a small sampling fraction, commonly at most 10%. An interval estimates a mean, not the range of individual observations. For independent groups use SE=√(s1²/n1+s2²/n2); for paired measurements analyze one sample of within-person differences. The supplied critical values let you focus on the method; a table or calculator supplies them in independent work.
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 one-sample t interval uses 25 independent observations. What are its degrees of freedom?
- For this one-sample procedure, df=n−1=24.
Answer: 24
Lesson preview: this page shows 5 practice items. The interactive activity contains 8 practice items and 8 quiz items (typed response and multiple choice). Counts include repeated prompts and variants.
Practice problems
1. A random sample of 25 wait times has sample SD 5 minutes. Calculate the estimated standard error of the sample mean.
Show solution
- SE=s/√n=5/√25=5/5=1 minutes.
Answer: 1
2. For n=25, sample SD 5 and a 95% interval, use t*=2.064 (df=24). Calculate the margin of error to three decimals.
Show solution
- SE=1.
- Margin=t*×SE=2.064×1=2.064.
Answer: 2.064
3. The sample mean is 42, n=25 and sample SD is 5. Assume independent random sampling and an approximately normal population. Using t*=2.064, choose the 95% mean interval.
Choices: 39.936 to 44.064 · 41 to 43 · 39.936 to 42 · 42 to 44.064
Show solution
- SE=5/5=1.
- Margin=2.064×1=2.064.
- Subtract and add the margin to 42.
Answer: 39.936 to 44.064
4. A one-sample t interval uses 25 independent observations. What are its degrees of freedom?
Show solution
- For this one-sample procedure, df=n−1=24.
Answer: 24
5. A random sample of 25 has strong skewness and a severe outlier. Is the usual small-sample t interval automatically justified?
Choices: Yes; randomness removes skewness · Yes; the confidence level fixes outliers · No; the distribution conditions need attention · Yes; every n above 10 is sufficient
Show solution
- With a small sample, strong skewness and outliers can undermine the t procedure.
- Do not treat the formula as unconditional.
Answer: No; the distribution conditions need attention
Explain and apply
Complete these tasks on paper. Use the rubric to review your reasoning with a teacher or study partner; these tasks are not automatically scored.
Paired data: write and justify a complete interval
A researcher randomly selects 16 adults from a population of 1,000. Their after-minus-before task-time differences have mean -3 minutes and SD 4 minutes, with no strong skewness or outliers. Construct a 95% interval using t*=2.131. Explain the pairing, parameter, conditions and conclusion. The study does not randomly assign a treatment.
Review rubric and solution
- Define the population mean after-minus-before difference.
- Use a one-sample t interval on 16 differences, not two independent samples.
- Check random sampling, 16<=100, and the stated distribution condition.
- Calculate SE=1, margin=2.131, and interval (-5.131,-.869).
- Interpret a population mean decrease with confidence and avoid a causal claim.
- SE=4/√16=1 minute and df=15.
- Interval=-3±2.131=(-5.131,-.869) minutes.
- We are 95% confident the population mean change is a decrease of .869 to 5.131 minutes. Without random assignment, the interval alone does not establish the cause.
Practice this interactively with instant feedback and an AI tutor.
Practice Confidence Intervals for Means Take the free placement check