What is a Confidence Interval?
A confidence interval gives a range of plausible values for a population parameter, built around a sample estimate. It reflects both the sample's precision and the chosen confidence level, most commonly 95%.
A confidence interval is calculated as CI = x̄ ± z·(σ/√n), giving a range that is likely, at the chosen confidence level, to contain the true population mean.
Try it: interactive calculator
Step-by-step worked examples
A sample of n=64 students has mean test score x̄=78, σ=12. Find the 95% confidence interval (z=1.96).
E = z·(σ/√n) = 1.96×(12/√64) = 1.96×(12/8) = 1.96×1.5 = 2.94 CI = 78 ± 2.94 = (75.06, 80.94)
A factory samples n=100 bolts with mean length x̄=50.2 mm, σ=1.5 mm. Find the 90% CI (z=1.645).
E = 1.645×(1.5/√100) = 1.645×(1.5/10) = 1.645×0.15 = 0.247 CI = 50.2 ± 0.247 = (49.953, 50.447)
A poll of n=400 voters finds x̄=52% support, σ=25%. Find the 99% CI (z=2.576).
E = 2.576×(25/√400) = 2.576×(25/20) = 2.576×1.25 = 3.22 CI = 52% ± 3.22% = (48.78%, 55.22%)
Flashcards
Quick quiz
Q1.A 95% confidence interval means…
Q2.Increasing the sample size n…
Q3.The z-score for a 99% confidence level is approximately…
Q4.Margin of error E is calculated as…
The full card deck, worked steps and AI-tutor support for “What is a Confidence Interval?” are in Notek — study by hand before your exam.
Common mistakes
Thinking a 95% CI means 95% of the data lies inside it. — Correct: It's the estimation method's long-run success rate for capturing the true parameter, not a data-coverage statement.
Believing a wider interval means more confidence in the exact mean value. — Correct: A wider interval reflects more uncertainty about the estimate's precision, not more confidence in one value.
Forgetting to use the correct z (or t) score for the chosen confidence level. — Correct: Each confidence level has its own critical value — 90%→1.645, 95%→1.96, 99%→2.576.
Ignoring sample size when interpreting interval width. — Correct: Small samples produce wide, less precise intervals — always check n.
FAQ
What is a confidence interval?
It's a range of plausible values for a population parameter, calculated from sample data at a chosen confidence level.
What is the formula for a confidence interval?
CI = x̄ ± z·(σ/√n), where z depends on the confidence level (e.g., 1.96 for 95%).
What are examples of confidence intervals?
Estimating the average test score of students, the mean weight of manufactured parts, or the percentage of voters supporting a candidate.
How do you calculate a confidence interval step by step?
Find the margin of error E = z·(σ/√n), then add and subtract it from the sample mean x̄.