---
title: "Confidence Intervals for Means and Proportions"
description: "Turn the sampling distribution around to produce an interval and a stated coverage. The t-distribution replaces the normal when the population standard deviation is estimated."
canonical: https://lightmysky.com/learn/mathematics/confidence-intervals-for-means-and-proportions-mt_Pg2fPDW1sn
source: https://lightmysky.com/learn/mathematics/confidence-intervals-for-means-and-proportions-mt_Pg2fPDW1sn.md
retrieved: 2026-09-12
---

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# Confidence Intervals for Means and Proportions

Turn the sampling distribution around to produce an interval and a stated coverage. The t-distribution replaces the normal when the population standard deviation is estimated.

Subject: Mathematics · Area: Data & Statistics · Ages 20 to 21
Page: https://lightmysky.com/learn/mathematics/confidence-intervals-for-means-and-proportions-mt_Pg2fPDW1sn

## Ready when they can

- Construct an interval for a mean with the t-distribution
- Construct an interval for a proportion and check the conditions
- State what the confidence level does and does not say about one interval

## Lesson: Intervals that trap the truth

A confidence interval centres on the sample mean and stretches by a multiplier times the standard error. The standard error is sigma over the square root of n, so bigger samples pull the endpoints tighter. With a multiplier near 2 for 95 percent, a sample of 25 with spread 10 gives a margin of 4.

In practice sigma is rarely known, so the sample spread s takes its place and extra uncertainty enters. The t distribution with n minus 1 degrees of freedom accounts for that substitution with heavier tails. The rule is simple: sigma known or huge sample, use z; sigma estimated from a modest sample, use t. With 16 observations that means t with 15 degrees of freedom.

Intervals for a proportion centre on the sample share p-hat, with spread built from p-hat itself. In 200 trials with 60 successes, p-hat is 0.3. The normal approximation needs enough of both outcomes: at least 10 successes and 10 failures. Polls quoting plus or minus three points are reporting exactly this construction.

**Tip.** The confidence level describes the method, not any single finished interval. If the whole recipe ran again and again, about 95 percent of the resulting intervals would trap the true mean. One computed interval either covers the mean or it does not, so never say there is a 95 percent chance the truth sits inside it.

**Recap.** Centre on the estimate, stretch by the standard error, use t when sigma is estimated, and read the level as a long run rate.

## Practice

14 questions on this page, each with its working shown.

## Needs first

- [Finding Probabilities and Values from a Normal Model](https://lightmysky.com/learn/mathematics/finding-probabilities-and-values-from-a-normal-model-mt_naXMRySvos)
- [Sampling Distributions and the Central Limit Theorem](https://lightmysky.com/learn/mathematics/sampling-distributions-and-the-central-limit-theorem-mt_nugaEeUSii)
- [Maximum Likelihood Estimation](https://lightmysky.com/learn/mathematics/maximum-likelihood-estimation-mt_ZRTrcy7-KV)

## Opens up

- [Hypothesis Tests for a Mean with the t-Distribution](https://lightmysky.com/learn/mathematics/hypothesis-tests-for-a-mean-with-the-t-distribution-mt_ig2o39hAwh)
- [Comparing Two Populations](https://lightmysky.com/learn/mathematics/comparing-two-populations-mt_v34rZkMfqr)
- [Clinical Trials: Phases, Endpoints and Why Most Candidates Fail](https://lightmysky.com/learn/science/clinical-trials-phases-endpoints-and-why-most-candidates-fail-mt_xKBKzSYxLh)