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Confidence Interval

With known σ, build a confidence interval and margin of error for μ.

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x̄ ± z·σ/√n
e.g. 90 / 95 / 99

📖 Tutorial | Confidence Interval

1. Definition
Confidence interval, when σ is known, estimates μ by x̄±z·σ/√n; z comes from the confidence level (95% → 1.96).
2. Symbols
SymbolMeaning
x̄sample mean
σknown population sd
zstandard-normal quantile
marginz·σ/√n
3. How it works
  • Find z from the level (95%→1.96);
  • Margin E=z·σ/√n;
  • Interval [x̄−E, x̄+E];
  • If σ unknown, use a t interval.
4. Steps
  1. Enter x̄, σ, n and level;
  2. Click Calculate;
  3. Read z, margin and interval;
  4. Larger n gives a narrower interval.
5. Example
Example: x̄=10, σ=2, n=36, 95%. z=1.96, E=1.96×2/6≈0.653, interval (9.347,10.653).
6. Pitfalls
σ must be known; otherwise use the t distribution;
Higher level gives a wider interval;
The interval is random; the level is the long-run coverage rate.

❓ FAQ | Confidence Interval

What does a 95% CI mean?
About 95% of intervals from repeated sampling contain the true value—not that this interval has 95% probability.
Why use z?
When σ is known, the standardized statistic is standard normal.
What if σ is unknown?
Use the sample sd s and the t distribution.
Why does larger n narrow the interval?
The standard error σ/√n shrinks with n.
Is a higher level always better?
Higher level ⇒ wider interval ⇒ less precise; trade-off.