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Week 12 · Chapter

Chapter 12 — Confidence Intervals for a Proportion

Introduction to Statistics Generic evergreen edition

Course: Introduction to Statistics (18-week generic edition)
The module's primary reading · ~30–40 minutes · Objective 6 · Read this before (or alongside) the week's lecture — everything else in the module builds on it.

Why you already care about this

Sometime recently, a headline in your feed ended with the same eight words of fine print: "…with a margin of error of ±3 percentage points." You have scrolled past that sentence a hundred times. It is quietly one of the most useful sentences in public life — and after this chapter, it belongs to you. You will know exactly where the 3 comes from, you will be able to recompute it from the article's own numbers, and you will know the two things it does not protect you from.

Here is the week in one sentence: last week you built a confidence interval for a mean; this week the same machine — estimate ± multiplier × standard error — estimates a percent, the most public statistic in the world. Percents are how the world talks about yes/no questions: what fraction of subscribers will renew, what fraction of households recycle, what fraction of orders come back, what fraction of voters support a plan. By the end of this chapter you will be able to: compute a sample proportion p̂ and its standard error; check the conditions that make the method legitimate; build and correctly interpret a one-proportion z-interval at 90%, 95%, or 99% confidence; compute the sample size a target margin of error requires (and round it the right direction); and audit a media poll's margin of error from nothing but its own reported numbers.

And one line worth memorizing now, because it is the week's moral: a percent without a margin of error is a number with its fine print torn off.

Where we are

The course's inference half is two weeks old. Week 10 supplied the engine: a statistic computed from a random sample varies from sample to sample, but in a predictable, bell-shaped way. Week 11 used that engine to build intervals for a mean, using the t-multiplier. This week the parameter is a proportion — Week 1's p, the truth about everyone, estimated by p̂ ("p-hat"), the sample's answer. Everything you learned about interpretation last week carries over word for word; what changes is the standard-error formula and the multiplier.

1. From one sample to an honest range

Start with the plain idea. You ask a random sample of people a yes/no question and compute the fraction who said yes. That fraction is the sample proportion p̂ — your point estimate of the population proportion p. It is almost certainly not exactly equal to p, and everyone knows it. So instead of reporting a bare number, an honest analyst reports a range: the estimate, plus and minus a cushion wide enough that the truth is probably inside.

How wide is wide enough? That is not a matter of taste — Week 10 answered it. Across all possible random samples, the p̂'s pile up in a bell shape centered at the true p, and the typical distance between a sample's p̂ and the truth is the standard error:

SE = √( p̂(1 − p̂) ⁄ n ) — "how far a sample's percent typically sits from the truth."

If sample percents typically stray about one SE from the truth — and almost always stray less than a couple of SEs — then reaching a couple of SEs out from your p̂ will usually reach back and capture p. That reach is the whole idea of a confidence interval. The anatomy, worth saying as a chant:

estimate ± multiplier × standard error, or for this week: p̂ ± z*·SE. The reach, z*·SE, is called the margin of error (ME). Center ± reach.

The multiplier z* is set by how confident you want the method to be. This course uses exactly three values, embedded here and in the tutorial and assignment coach — any other confidence level, technology supplies.

The table below lists the confidence multiplier z* for each confidence level used in this course.

Confidence level z*
90% 1.645
95% 1.96
99% 2.576

One question deserves an answer before it distracts you: last week used t* — why z* now? For a mean, the spread had to be estimated by a second, separate statistic (s), and the t-multiplier pays a tax for that extra uncertainty. For a proportion there is no separate spread number: p̂ itself feeds both the center and the SE formula. No second estimate, no tax. The one-line version: means → t, proportions → z. (You have met these z's already — Week 11's t-table columns sink toward exactly these values as the degrees of freedom grow.)

Misconception check. "The survey found 60%, so 60% of the population has the app." Week 1's hat is still on duty: p̂ is measured, p is true, and they essentially never coincide. The entire reason this chapter exists is the gap between them — the interval is how we report the gap honestly.

2. Building the interval: the password, then the build

Not every sample earns an interval. Before constructing anything, check three conditions — think of them as the method's password:

  1. Random. The data come from a random sample of the population. No formula repairs a biased sample (Week 1's lesson never retires).
  2. Large counts. The sample contains at least 10 successes and 10 failures — count them from the data. This is what makes the bell shape reliable enough to use z*.
  3. 10% condition. When sampling without replacement, the population should be at least 10 × n, so the draws are effectively independent.

Worked example (the week's anchor — every step shown). A campus tech office wants to know what proportion of the campus's 18,000 students have installed the campus app. It draws a random sample of 100 students from the enrollment database; 60 have the app.

  • Point estimate: p̂ = 60 ÷ 100 = 0.60.
  • Conditions: random ✓ (drawn from the full enrollment list) · successes 60 ≥ 10 and failures 40 ≥ 10 ✓ · 18,000 ≥ 10 × 100 ✓.
  • Standard error: SE = √(0.60 × 0.40 ⁄ 100) = √0.0024 ≈ 0.049.
  • Margin of error at 95%: ME = 1.96 × 0.049 ≈ 0.096.
  • The interval: 0.60 ± 0.096 → (0.504, 0.696) — in words, "somewhere between about 50.4% and 69.6%."
  • The sentence (memorize its shape): "We are 95% confident that the interval from 0.504 to 0.696 captures the true proportion of all 18,000 students who have installed the app."

Misconception check. "SE = √(60 × 40 ⁄ 100)…" The formula eats proportions, never counts: 0.60 and 0.40, not 60 and 40. A quick self-audit: for a proportion, an SE bigger than 1 is impossible — if you get one, you fed the formula counts.

In practice (your tools). The whole build is six spreadsheet cells. A1: successes 60; A2: sample size 100; A3: =A1/A2 → 0.6; A4: =SQRT(A3*(1-A3)/A2) → 0.04899; A5: =1.96*A4 → 0.09602; A6 and A7: =A3-A5 and =A3+A5 → 0.50398 and 0.69602 — matching the hand build. Change A1 and A2 and the sheet re-audits any survey on earth. Graphing tools and Desmos-class statistics tools package the same computation as 1-PropZInt.

3. What "95% confident" means — and what it doesn't

The phrase "95% confident" is the most misread phrase in statistics, so let's be exact about what it promises.

The 95% describes the method, not your one interval. If many teams each drew their own random sample of 100 students and each built an interval by this recipe, about 19 of every 20 of those intervals would capture the true p — and about 1 in 20, through honest bad luck, would miss. Your interval is one of that crowd. Once it is built, there is no probability left in it: p is a fixed number, your endpoints are fixed numbers, and the interval either captured p or it didn't — you just don't get to know which. The confidence lives in the recipe, not in one batch.

Two famous wrong sentences, and why each dies:

Misconception check. "There's a 95% chance that p is between 0.504 and 0.696." This puts the probability on p, as if the truth were hopping around. It isn't — p sits still; the interval is what varies from sample to sample. Say instead: intervals built this way capture the truth 95% of the time.

Misconception check. "95% of the students are between 0.504 and 0.696." The interval brackets a proportion, not people. Individual students are yes/no data points — nobody is "0.53." An interval for p says nothing about where individuals sit; it says where the population's percent plausibly sits.

The confidence dial. Same sample, same SE — only z* changes:

  • 90%: 1.645 × 0.049 ≈ 0.081 → (0.519, 0.681) — narrower, but the method misses 1 sample in 10.
  • 95%: 1.96 × 0.049 ≈ 0.096 → (0.504, 0.696).
  • 99%: 2.576 × 0.049 ≈ 0.126 → (0.474, 0.726) — surer, but wide enough to say much less.

The trade-off is a law, not a setting you can game: at a fixed sample size, surer means wider. The only way to be surer and narrower is more data — which brings us to the price tag.

4. Choosing the sample size: buying precision

Real studies run the machine backwards. Before any data exist, someone declares the precision they need — "we want the answer within ±3 points, at 95% confidence" — and the statistician answers with a bill. Solve the margin-of-error formula for n:

n = p*(1 − p*) · (z* ⁄ ME)², where p* is a planning value for the proportion.

Two choices for p*:

  • No prior idea? Use the conservative value p* = 0.5. The product p*(1 − p*) is largest there (0.25), so the n it demands is always enough, whatever the truth turns out to be.
  • Have a pilot estimate? Use it. A p* far from 0.5 shrinks the bill.

Worked example (every step). A city sustainability office wants to estimate the proportion of households that set out curbside recycling, with a 95% interval and a margin of error of at most ±0.03, and no prior estimate.

  • n = 0.25 × (1.96 ⁄ 0.03)² = 0.25 × 4268.44 = 1067.11 → survey 1,068 households.
  • With a pilot estimate p* = 0.8 (a neighboring city measured about 80%): n = 0.8 × 0.2 × 4268.44 = 0.16 × 4268.44 = 682.95 → 683 households. Prior knowledge cut the bill by about a third.

Misconception check. "1067.11 rounds to 1067." Never. The computed n is the minimum that meets the promised margin; rounding down breaks the promise by a hair. Always round up — the ceiling is a guarantee, not an approximation.

One more law hides in the formula's square: because ME shrinks like 1 ⁄ √n, halving the margin costs four times the sample, and a tenth of the margin costs a hundred times. Precision is bought on a square-law budget. That, in one line, is why professional polls stop near ±3: the next digit of precision costs more than headlines pay.

5. Polls and the margin of error in the media

Now take the machine outdoors. A well-reported poll tells you: who was sampled (the population it claims to speak for), how many (n), how they were reached, the margin of error, and the confidence level (almost always 95%, often unstated). Here is how to audit one from its own numbers.

Worked example (the audit — every step). A national outlet reports: "52% of 1,067 randomly sampled adults support expanding passenger rail service; margin of error ±3 percentage points."

  • SE = √(0.52 × 0.48 ⁄ 1067) ≈ 0.0153.
  • ME = 1.96 × 0.0153 ≈ 0.030 → ±3.0 points. The fine print checks out.
  • The napkin rule: at 95% with the conservative p* = 0.5, ME = 0.98 ⁄ √n ≈ 1 ⁄ √n. For n = 1,067, that is about 0.031 — "±3" on a napkin. This is why so many professional polls have n near 1,000: it is the ±3 price point.
  • Read the interval, not the point. 52 ± 3 → 49% to 55%. The interval dips below 50%, so "majority support" is plausible but not established. A headline announcing "majority backs rail" is over-reading its own data. (When an interval includes 50%, the honest word is undecided, not tied.)

And now the fine print's fine print — the part every headline omits. The margin of error prices exactly one thing: random sampling luck — the chance that an honest random sample landed a little high or low. Question wording, nonresponse, undercoverage — Week 1's entire bias catalog — ride outside the ±, invisible, and often bigger than it. The forever-example: the 1936 Literary Digest poll collected 2.4 million responses, which makes a microscopic computed margin of error — and still called the wrong winner, because the method was biased. A margin of error on a biased sample is a precise measurement of the wrong thing.

So carry a three-question checklist past every poll you meet: ① Who was actually sampled, and how — randomly, or opt-in? ② What is the whole interval (point estimate ± ME), and does the story's claim survive all of it? ③ What could bias this that the ± doesn't cover?

In practice (your tools). Keep your six-cell spreadsheet from Section 2 open when you read the news. Type in the poll's percent and its n, and the sheet reports the implied margin in seconds. If the article's ± is far from yours, the pollster likely used the conservative p* = 0.5 version (slightly larger) — or something odd is going on, which is worth noticing. This audit is exactly this week's Data Lab.

Key terms

  • Population proportion (p) — the true fraction of the population answering "yes"; a parameter, never directly seen.
  • Sample proportion (p̂) — successes ÷ n in the sample; the point estimate of p.
  • Point estimate — a single-number best guess at a parameter.
  • Standard error of p̂ — SE = √(p̂(1 − p̂) ⁄ n); how far sample proportions typically stray from p.
  • Critical value (z*) — the confidence multiplier: 1.645 (90%), 1.96 (95%), 2.576 (99%).
  • Margin of error (ME) — z* × SE; the interval's reach on each side of p̂.
  • Confidence interval — p̂ ± z*·SE; the range of plausible values for p.
  • Confidence level — the method's long-run capture rate: the percent of all random samples whose intervals would capture p.
  • One-proportion z-interval — this week's procedure: the confidence interval for a single population proportion.
  • Large-counts (success/failure) condition — at least 10 successes and 10 failures in the sample.
  • 10% condition — the population is at least 10 × n, keeping draws effectively independent.
  • Conservative planning value (p* = 0.5) — the sample-size planning value that maximizes p*(1 − p*), guaranteeing enough data.
  • Sample-size formula — n = p*(1 − p*)(z* ⁄ ME)², always rounded up.

Summary

  • A sample percent is an estimate, not the truth: report p̂ ± z*·SE — center ± reach — where SE = √(p̂(1 − p̂) ⁄ n) and z* = 1.645 / 1.96 / 2.576 for 90 / 95 / 99%. Means → t, proportions → z.
  • Check the password first: random sample · at least 10 successes and 10 failures · population ≥ 10n. Feed the SE formula proportions, never counts.
  • "95% confident" describes the method — about 19 of every 20 random samples yield a capturing interval. It is never a probability about your one interval, and never a statement about individuals.
  • Surer means wider. To be surer and narrower, buy data: n = p*(1 − p*)(z* ⁄ ME)², with p* = 0.5 when you know nothing — and always round up (1067.11 → 1,068).
  • The margin shrinks like 1 ⁄ √n: half the margin costs four times the sample. And a poll's ± covers sampling luck only — bias rides outside it. Read the whole interval, then ask what the ± doesn't cover.

Check your understanding

Work these before looking at the answers — they're ungraded, and the answers (with reasoning) are at the bottom of the page.

  1. A museum samples 400 of its visitors at random from ticket records; 220 used the museum's mobile audio guide. Build the 95% confidence interval for the proportion of all visitors who used the guide: give p̂, check the conditions, compute the SE and margin of error, and state the interval with a correct interpretation sentence.
  2. Using your interval from Question 1, explain in one or two sentences each why these two rewrites are wrong: (a) "95% of visitors used the guide between those two rates," and (b) "there is a 95% chance the true proportion is inside the interval."
  3. An online craft marketplace emails a survey to the 45 sellers who attended its pricing webinar; 43 of the 45 respondents say the platform's fees are fair. Name every reason a one-proportion z-interval for "the proportion of all sellers who consider the fees fair" is not legitimate here.
  4. A neighborhood association wants to estimate the proportion of households interested in a fiber-internet buildout, to within ±0.06 at 95% confidence, with no prior estimate. How many households must it sample — and why is rounding your formula's answer down never allowed?

Answers (worked reasoning — read after attempting)

  1. p̂ = 220 ÷ 400 = 0.55. Conditions: random ✓ (drawn from ticket records); successes 220 ≥ 10 and failures 180 ≥ 10 ✓; the population of all visitors is far more than 10 × 400 ✓. SE = √(0.55 × 0.45 ⁄ 400) = √0.000619 ≈ 0.0249. ME = 1.96 × 0.0249 ≈ 0.049. Interval: 0.55 ± 0.049 → (0.501, 0.599). Interpretation: "We are 95% confident that the interval from 0.501 to 0.599 captures the true proportion of all the museum's visitors who used the mobile guide." (Notice the interval barely clears 0.50 — "a majority used it" is plausible but only just.)
  2. (a) The interval brackets the population proportion, not individual visitors — each visitor either used the guide or didn't; no visitor is "0.55." (b) The 95% belongs to the method: about 19 of every 20 random samples produce a capturing interval. The true p is a fixed number and your built interval is fixed too — it either captured p or missed; there is no 95% chance left to assign to it.
  3. Three failures, any of which is disqualifying. Not random: webinar attendees are a self-selected, unusually engaged slice of sellers, and the survey's respondents are self-selected again (voluntary response). Large counts fails: failures = 45 − 43 = 2, far below 10, so the z machinery isn't reliable even on its own terms. Wrong population: at best this measures webinar-attending respondents, not "all sellers" — and no margin of error repairs any of this, because a ± prices sampling luck only, not bias.
  4. n = 0.25 × (1.96 ⁄ 0.06)² = 0.25 × 1067.11 = 266.78 → sample 267 households. The formula's output is the minimum n that achieves the promised margin; rounding down to 266 would leave the margin slightly larger than promised. The ceiling is a guarantee — always round up.

Next up in this module: the slide deck if you want the lecture's visual walk-through, then the Readings & Resources page for the go-deeper links — and when you're ready to practice, Lecture Tutorial 12 turns your chatbot into your personal Week 12 tutor.