Intro statistics · Estimation
Confidence interval practice problems
A confidence interval is an estimate ± a margin of error. The margin is a critical value times the standard error. The hard part on exams is the interpretation.
The two formulas you need
- Mean (σ unknown): x̄ ± t* · s/√n, with n − 1 degrees of freedom.
- Proportion: p̂ ± z* · √(p̂(1 − p̂)/n).
Common z* values: 1.645 for 90%, 1.96 for 95%, 2.576 for 99%.
Worked example
A sample of n = 25 exam scores has x̄ = 72 and s = 10. Build a 95% interval for the mean.
Standard error = 10/√25 = 2. With 24 degrees of freedom, t* = 2.064. Margin = 2.064 × 2 = 4.13.
Interval: 67.9 to 76.1.
Interpretation: we are 95% confident that the true mean score is between 67.9 and 76.1. "95% confident" means that the method captures the true mean in about 95% of samples. It does not mean there is a 95% chance this particular interval contains it.
Practice problems
1. In a sample of 400 students, 160 say they study on weekends. Build a 95% interval for the proportion.
Show answer
p̂ = 0.40. SE = √(0.40 × 0.60/400) = 0.0245. Margin = 1.96 × 0.0245 = 0.048. Interval: 0.352 to 0.448.
2. How does the interval change if you go from 95% to 99% confidence with the same data?
Show answer
It gets wider, because the critical value grows from 1.96 to 2.576.
3. How large must the sample be to cut the margin of error in half?
Show answer
About 4 times as large, because the margin shrinks with √n.
Frequently asked questions
Does a 95% interval contain 95% of the data?
No. It is about the population mean or proportion, not individual values.
When do I use t instead of z for a mean?
Use t when you estimate the spread with the sample standard deviation s, which is almost always the case in practice.
Reference: OpenStax Introductory Statistics 2e, chapter 8 (Confidence Intervals). t* from a standard t table.