Confidence Interval (Mean/Proportion)
Statistics
Intro: We compute the standard error and use z or t as appropriate.
Worked example
- $x̄=12.4, σ=3.0, n=36, level=0.95$
- SE $=\sigma/\sqrt{n}=3/6=0.5$, $z^*\approx1.96$.
- CI: $12.4\pm1.96(0.5)=12.4\pm0.98$.
- Answer: $\boxed{(11.42,\,13.38)}$.
- $x=56, n=120, level=0.95$
- $\hat{p}=56/120=0.4667,$ $\;z^*\approx1.96$.
- $\text{SE}=\sqrt{\hat{p}(1-\hat{p})/n}\approx0.0455$.
- CI: $0.4667\pm1.96\cdot0.0455=0.4667\pm0.0892$.
- Answer: $\boxed{(0.3775,\,0.5559)}$.
FAQs
Wald vs. better intervals?
For small n or extreme p̂, Agresti–Coull or Wilson may perform better than Wald.
Why choose MathGPT?
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How this calculator works
- Type or paste your function (LaTeX like
\sin,\lnworks too). - Press Generate a practice question button to generate the derivative and the full reasoning.
- Review each step to understand which rule was applied and why.
- Practice with similar problems to lock in the method.