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Week 12 · Lecture outline

Week 12 — Lecture Outline · Confidence Intervals for Proportions

Introduction to Statistics · MATH 11 Fall 2026 · Prof. Rivera Fictional sample

Course: Introduction to Statistics (MATH 11) · Silver Oak University (fictional sample) · Prof. Rivera
Objectives covered: Objective 6 — Construct and interpret confidence intervals for proportions.
SLOs touched: A (reason quantitatively from data) · B (communicate results to a non-technical audience)
Meeting pattern: 2 sessions × 75 min = 150 min. Segment minutes below total ~150; scale to your own pattern.

Campus note: This is the last full instructional week before the Thanksgiving break (campus closed Thu–Fri, Nov 26–27). Our Tue/Thu pattern (Tue Nov 17 / Thu Nov 19) is unaffected, and all Week 12 work is due Sun Nov 22 — comfortably before the break. Say so out loud; students relax when they see the deadline lands before travel.

Week at a Glance
The week's big question "That poll said '45% approve, ±4 points.' Where does that ±4 come from when the data is a percentage, not an average — and how do pollsters choose a sample size to hit a target margin?"
By the end of the week, students can… (1) check the three conditions for a one-proportion z-interval (random, independent, large-counts); (2) construct a confidence interval for a proportion as p̂ ± z*·√(p̂(1−p̂)/n), using a supplied z* value; (3) compute and name the margin of error; (4) choose a sample size for a target margin of error using n = (z*/ME)²·p̂(1−p̂), with p̂ = 0.5 for the conservative worst case; (5) interpret a proportion interval correctly and catch the two classic misinterpretations.
Key vocabulary sample proportion p̂, population proportion p, standard error SE = √(p̂(1−p̂)/n), critical value z*, margin of error (ME), confidence level, confidence interval, large-counts / success–failure condition, conservative (worst-case) p̂ = 0.5, "95% confident"
Materials slides (Deck 12), the week's readings + video links, a spreadsheet (Google Sheets or Excel), one approved chatbot (Gemini / Claude / ChatGPT) for the AI-critique moment and the tutorial, and the embedded z* table below (we supply every critical value — no tables to look up)
Timing note 8 segments, ~150 min total. Session 1 = Segments 1–4 (~75). Session 2 = Segments 5–8 (~75).

The z* table we use all week (embedded — never look it up)

Rule for this course: every confidence-interval and sample-size problem you see is engineered to use one of these exact z* values. The materials hand you z*; you never recall or look it up. (Proportions use z* — there are no degrees of freedom here; that was last week's t-interval for a mean.)

Confidence level z* critical value
90% 1.645
95% 1.960
99% 2.576

Read it as the punchline of Segment 3: more confidence buys a bigger z*, which buys a wider margin. 95% (z* = 1.960) is the workhorse.

Segment 1 — Hook & the Promise (8 min) · Session 1 opens

Hook. Put a real headline on the board: "A new poll finds 45% approve of the new policy — margin of error ±4 percentage points, based on 600 adults."
- "Last week we built a '±' for an average — a mean, like sleep hours. This week we build the '±' you actually see most often in the news: the one on a percentage. '45% approve, ±4 points' is a confidence interval for a proportion, and by Friday you'll build one yourself and choose the sample size behind it."
- "Two weeks ago a sample mean x̄ bounced around μ. The exact same idea holds for a percentage: a sample proportion p̂ bounces around the true proportion p from sample to sample. This week we wrap an honest range around p̂ and say how confident we are."

The promise (write it on the board): "By Friday you can take a poll's percentage and sample size — say 45% of 600 people — and produce the '±4 points' yourself: an interval for the true proportion, plus an honest statement of how confident you are. You'll also be able to flip it around: tell a pollster how many people they must survey to hit a target margin."

Why it matters line (memory hook): "A percentage from a sample is a guess at one point. A confidence interval is an honest guess — it admits it might be off, and says by how much."

Segment 2 — From Means to Proportions: the One-Proportion z-Interval (24 min)

Plain language first — same shape, new pieces.
- A proportion is just a percentage written as a decimal: "45% approve" is p̂ = 0.45. The hat (^) means it came from a sample; the bare p is the unknown truth for the whole population.
- Every interval we build still has the same anatomy as last week: a center (our best guess) and a reach (the margin of error). Only the formula for the pieces changes.
- Center = the point estimate, the sample proportion p̂ (the poll's percentage, as a decimal).
- Reach = the margin of error = a critical value times the standard error.

Why z, not t (head off the obvious question).
- Last week (means) we used t with degrees of freedom, because we had to estimate the spread from the sample's s. For a proportion, the spread is built right into p itself — there's no separate standard deviation to estimate — so we go back to the normal model and use z*. No degrees of freedom for proportions. (n = 600 doesn't give you "df = 599" here — that's a means-only idea.)

The formula (notation after the idea):

CI for a proportion: p̂ ± z* · √( p̂(1−p̂) / n )
- √( p̂(1−p̂)/n ) is the standard error (SE) of the sample proportion — the typical distance p̂ sits from the true p.
- z* · SE is the margin of error (ME).
- p̂ − ME is the lower endpoint; p̂ + ME is the upper endpoint.

One fully worked example — do every step out loud (numbers chosen so the arithmetic is clean):

A poll of n = 600 adults finds p̂ = 0.40 (40%) approve. Build a 95% confidence interval for the true proportion p.
1. Standard error: SE = √(p̂(1−p̂)/n) = √(0.40 × 0.60 / 600) = √(0.24 / 600) = √(0.0004) = 0.02.
2. Critical value: 95% confidence → from our table, z* = 1.960 (supplied — no lookup).
3. Margin of error: ME = z* · SE = 1.960 × 0.02 = 0.0392 ≈ 0.039 (about 3.9 percentage points).
4. Interval: p̂ ± ME = 0.40 ± 0.039 → lower 0.40 − 0.039 = 0.361, upper 0.40 + 0.039 = 0.439.
5. Answer: the 95% confidence interval is (0.361, 0.439) — i.e., we're 95% confident the true approval rate is between about 36.1% and 43.9%.

A second, even cleaner one (build it together, SE = 0.01):

A survey of n = 2500 voters finds p̂ = 0.50 support a measure, at 95%. SE = √(0.50 × 0.50 / 2500) = √(0.25/2500) = √(0.0001) = 0.01; z* = 1.960; ME = 1.960 × 0.01 = 0.0196 ≈ 0.02; interval 0.50 ± 0.02 = (0.48, 0.52) — about 48% to 52%, the classic "too close to call."

The four-step recipe (give them this card):
1. SE = √( p̂(1−p̂) ÷ n ). 2. z* = read it from our supplied table (your confidence level). 3. ME = z* × SE. 4. Interval = p̂ − ME to p̂ + ME.

Segment 3 — The Margin of Error, and What Moves It (18 min)

Plain language: the margin of error is the "give or take" — the half-width of the interval, ME = z* · √(p̂(1−p̂)/n). It is the single number a news story is reporting when it says "±4 points."

What makes the margin bigger or smaller (build intuition, not just formula):
- Bigger sample (n↑) → smaller ME. n is under the square root in the denominator, so more data tightens the interval. (To halve the margin you must roughly quadruple n — same square-root law as last week.)
- Higher confidence (90% → 95% → 99%) → bigger z* → bigger ME. Being more sure costs width. You buy confidence with vagueness.
- p̂ near 0.5 → bigger ME; p̂ near 0 or 1 → smaller ME. The product p̂(1−p̂) is largest at p̂ = 0.5 (it equals 0.25 there) and shrinks as you move toward 0 or 1. This is the seed of the sample-size worst case in Segment 4.

Worked contrast (same poll, three confidence levels — show the trade-off):

p̂ = 0.40, n = 600 (so SE = 0.02, from Segment 2).
- At 90% (z* = 1.645): ME = 1.645 × 0.02 = 0.0329 ≈ 0.033 → interval (0.367, 0.433).
- At 95% (z* = 1.960): ME = 1.960 × 0.02 = 0.0392 ≈ 0.039 → interval (0.361, 0.439).
- At 99% (z* = 2.576): ME = 2.576 × 0.02 = 0.0515 ≈ 0.052 → interval (0.348, 0.452).
Same poll, wider net for more confidence. Land it: there is no free lunch — precision and confidence trade against each other.

Quick interaction — Think-Pair-Share (~5 min): Show two reported margins for similar polls: "±2 points, n = 2400" and "±4 points, n = 600." "Which poll surveyed more people? Roughly how many times more?" Surface that cutting the margin in half (4 → 2 points) takes about four times the sample (600 → 2400) — the square-root law made concrete.

Segment 4 — Choosing a Sample Size for a Target Margin (20 min) · Session 1 closes (~75)

Plain language — flip the question around. So far we had a sample and found the margin. Pollsters do it backward: they decide on a target margin first (say "we want ±4 points") and ask "how many people must we survey?" Solve the margin-of-error equation for n.

The formula (notation after the idea):

n = ( z* / ME )² · p̂(1−p̂)
- Pick your confidence level → that's z* (from our table).
- Pick your target margin of error ME (e.g., 0.04 for ±4 points).
- You need a value for p̂ to plug in — but you haven't polled yet! Two cases:
- If you have a planning estimate (a prior poll, a reasonable guess), use it.
- If you have no idea, use the conservative / worst-case p̂ = 0.5, because p̂(1−p̂) is largest at 0.5 (= 0.25). That gives the biggest required n — so you're safe no matter what the true proportion turns out to be.
- Always round n UP to the next whole person. (You can't survey 600.25 people, and rounding down would miss your target.)

One fully worked example — worst-case (do every step out loud):

We want a 95% interval with a margin of error of ±0.04 (4 points) and have no prior estimate, so use p̂ = 0.5.
1. z* = 1.960 (95%, supplied); ME = 0.04; p̂(1−p̂) = 0.5 × 0.5 = 0.25.
2. (z*/ME)² = (1.960 / 0.04)² = (49)² = 2401.
3. n = 2401 × 0.25 = 600.25.
4. Round UP → n = 601. Survey at least 601 people to be 95% confident with a ±4-point margin.

A second worked example — with a planning estimate (build it together):

Same 95% and target ME = 0.0392 (≈ ±3.9 points), but a prior poll suggests p̂ ≈ 0.40.
- (z*/ME)² = (1.960 / 0.0392)² = (50)² = 2500; × p̂(1−p̂) = × (0.40 × 0.60 = 0.24) → n = 2500 × 0.24 = 600.0 → round up (already whole) → n = 600.
- Punchline: using a planning estimate of 0.40 instead of the worst-case 0.50 lowers the required sample (600 vs the 625 the worst case would demand here) — because 0.24 < 0.25. The worst case is a safe over-estimate; a good planning value can save you respondents.

The classic number to recognize (put it on a slide):

For 95% confidence and a ±5-point (0.05) margin, worst case: n = (1.960/0.05)² × 0.25 = 1536.64 × 0.25 = 384.16 → 385. "That's why national polls so often survey roughly a thousand people: a sample near 1,000 buys a margin around ±3 points."

Misconception + cure (the worst-case one):
- ❌ "I don't know p̂, so I can't compute a sample size."
✅ Cure: you don't need to know it — plug in p̂ = 0.5. It maximizes p̂(1−p̂), so it gives the largest n you could possibly need. Any true proportion will need that many or fewer. Worst-case planning is a feature, not a workaround.

Segment 5 — Checking the Conditions (16 min) · Session 2 opens

Hook back in: "Last session we built the interval and chose a sample size. Before we trust any of it, we owe three quick checks — the same spirit as last week's conditions, with one new twist for proportions."

Plain language — when are we allowed to do this? A one-proportion z-interval is trustworthy when all three hold:
- Random: the data come from a random sample (or randomized experiment) — so the sample represents the population. (Callback to Week 1: method beats size. A huge biased poll is still biased.)
- Independent (10% condition): observations don't influence each other; when sampling without replacement, the sample is less than 10% of the population (e.g., 600 adults out of a city of millions — fine).
- Large counts (success–failure): there are at least 10 expected "successes" and 10 expected "failures" — that is, n·p̂ ≥ 10 AND n(1−p̂) ≥ 10. This is what lets the normal (z) model approximate the sampling distribution of p̂. (This is the NEW condition this week — means didn't have it.)

One fully worked condition check (do it on the poll):

Poll: p̂ = 0.40, n = 600, sampled at random from a large city.
- Random? Yes — stated random sample. ✔
- Independent? 600 is far less than 10% of a city of millions. ✔
- Large counts? n·p̂ = 600 × 0.40 = 240 ≥ 10, and n(1−p̂) = 600 × 0.60 = 360 ≥ 10. ✔
- Conclusion: all three hold → the z-interval is trustworthy. (Show that 240 and 360 are both comfortably above 10 — most of our examples clear this easily; the condition mainly bites for small n or a very rare/very common trait.)

A "fails the check" example (so they see the condition has teeth):

p̂ = 0.02 (a rare trait), n = 200. n·p̂ = 200 × 0.02 = 4 < 10 → large-counts condition fails. The normal-model interval is not trustworthy here; you'd need a much larger sample (or a different method). Name it; don't compute an interval you can't trust.

Segment 6 — Interpreting a Proportion Interval CORRECTLY (24 min)

Hook back in: "We built it; now the part students lose the most points on, and the part that matters most in the real world: saying what the interval means — and refusing to say what it doesn't. This is identical to last week's interpretation skill, now about a percentage."

The one correct interpretation (make them memorize the template):

"We are 95% confident that the true population proportion lies between 0.361 and 0.439 (about 36% to 44%)."
What that sentence is really shorthand for: "This interval was produced by a method that captures the true proportion 95% of the time in the long run. We can't see whether this particular interval is one of the lucky 95% or the unlucky 5% — but we trust the method."

The picture that makes it click (draw it):

Imagine running 100 different polls and building 100 different 95% intervals. About 95 of them will contain the true p; about 5 will miss it. The "95%" is a property of the procedure across many samples, not of any one interval. The true p is a fixed number; it's the intervals that vary.

The TWO classic misinterpretations — name them, then cure each (the heart of the week):

  • ❌ Misinterpretation #1: "95% of people fall in the interval." / "±4 means almost everyone is within 4 points."
    ✅ Cure: No — a confidence interval is about the population proportion (one summary number), not the spread of individual people. Our interval (0.361, 0.439) is a range of plausible values for the overall approval rate, not a range that holds 95% of respondents. "A CI brackets the rate, not the people."

  • ❌ Misinterpretation #2: "There's a 95% chance the true proportion is in THIS interval."
    ✅ Cure: Once the interval is computed, the true proportion either is in it or isn't — there's no probability left to assign to this fixed interval. The 95% describes how often the method works before you poll, across many possible samples — not the odds for the one interval in your hand. "The method is 95% reliable; this interval is already decided."

A third, sneakier slip to flag:
- ❌ "95% of all polls' percentages fall in this interval." / "We're 95% sure the sample proportion is in here." ✅ Cure: the interval is about the unknown population proportion p, not about future poll results and certainly not about the p̂ we already have (p̂ is the center — it's definitely "in there").

Worked interpretation round (give them the interval, ask for the sentence):

Context: A random sample of n = 600 voters has p̂ = 0.40; the 95% interval is (0.361, 0.439).
- Correct sentence: "We're 95% confident the true proportion of all voters who approve is between about 36% and 44%."
- Spot the error A: "95% of voters approve by between 36% and 44%." → about individuals, and it's gibberish — the interval is for the one approval rate, not per-person. Wrong.
- Spot the error B: "There's a 95% probability the true approval rate is between 36% and 44%." → assigns probability to a fixed interval. Wrong; it's the method that's 95% reliable.

Memory hook (put it on a slide): "95% confident = the method catches the true rate 95% of the time. Not the people. Not a coin flip on this one interval."

Segment 7 — Technology Workflow + AI-Critique (20 min)

Technology workflow — build a proportion CI in a spreadsheet (exact steps):
1. Enter the poll's numbers in cells. Say p̂ in C1, n in C2. (If you only have a count of "yes" responses, get p̂ first: =count_of_yes / n.)
2. Standard error: =SQRT(C1*(1-C1)/C2) → SE (C3). (This is the proportion SE — note it uses p̂(1−p̂), not a separate standard deviation.)
3. Margin of error: =1.96*C3 → ME (C4), dropping in the supplied z* for 95%. (Use 1.645 for 90%, 2.576 for 99%.)
4. Endpoints: =C1 - C4 and =C1 + C4.
5. Sample size for a target margin: in another cell, =CEILING( (1.96/target_ME)^2 * 0.25, 1 ) for the worst case (p̂ = 0.5). Swap 0.25 for phat*(1-phat) if you have a planning estimate, and CEILING(…,1) rounds up to a whole person.
- Sanity check: if you plug the SE-built ME back into the sample-size formula, you should recover (about) your original n. And the large-counts check =C2*C1 and =C2*(1-C1) should both read ≥ 10 before you trust the interval.

AI-critique moment (students verify, not consume):

Paste this to an approved chatbot: "A poll finds 40% approve, margin of error ±4 points. A 95% confidence interval for the true approval rate is (0.361, 0.439). Explain what this means."
Then audit the answer against today's two cures. Chatbots very often produce one of the banned sentences — "there's a 95% probability the true proportion is in this interval" or "95% of people are within 4 points." Your job: catch it, and rewrite it as the correct long-run-method statement. The tool drafts; you judge. Also ask it to "find the sample size for a 95% interval with a ±3-point margin, worst case" and check whether it used p̂ = 0.5 and rounded UP — models sometimes forget the worst-case value or round down. (For reference: (1.960/0.03)² × 0.25 = 1067.1 → 1068.)

Segment 8 — Callback, Tease & Hand-off (12 min) · Session 2 closes (~75)

Callback: "Week 11 built a '±' for a mean with t. This week we built the more familiar '±' — the one on a percentage — with z, learned to choose the sample size behind a target margin, and learned to say what the interval means without overclaiming. Same honest-range idea, new formula."

Tease next week: "We've now estimated both a mean and a proportion with intervals — saying what's plausible. Next week we flip to hypothesis testing: instead of asking 'what range holds the truth?', we ask 'is this specific claim — say, that approval is exactly 50% — believable given our data?' That's the p-value, Type I and II errors, and significance."

Hand-off (the week's graded work):
- Lecture Tutorial 12 (AI tutor, share-link submission) — conditions, building a proportion CI, the margin of error, choosing a sample size, and the two interpretation traps.
- Quiz 12 (end of week) — 10 items on the one-proportion interval, the margin of error, choosing a sample size, and correct interpretation.
- Discussion 12 — find a real reported poll percentage with its margin of error (e.g., "45% approve, ±4 points") and reason, with your chatbot, about what the interval means and how the sample size drives the margin.
- Assignment 12 — four problems: check the conditions; construct a proportion CI; compute a required sample size; explain a proportion interval to a non-expert.

Instructor FAQ — Common Stumbles

Student says / does Quick cure
Uses t and degrees of freedom for a proportion. Proportions use z* (from our table) — no df. The t-with-df machinery was last week's interval for a mean, where we estimated the spread with s. For a proportion the spread comes from p itself.
"What's the standard error for a proportion?" SE = √(p̂(1−p̂)/n). It uses the sample proportion p̂, not a separate standard deviation. (For p̂ = 0.40, n = 600: √(0.24/600) = √0.0004 = 0.02.)
Forgets the large-counts condition. Before trusting the z-interval, check n·p̂ ≥ 10 AND n(1−p̂) ≥ 10 (at least 10 expected successes and 10 expected failures). It mainly bites for small n or a very rare/common trait.
In the sample-size problem, "I don't know p̂, so I'm stuck." Use the worst case p̂ = 0.5 — it maximizes p̂(1−p̂) = 0.25, giving the largest (safest) n. A planning estimate (if you have one) can only lower the required n.
Rounds the sample size DOWN (or leaves a decimal). Always round n UP to the next whole person. 600.25 → 601. Rounding down would miss the target margin.
Reports the interval or margin as a percentage incorrectly. Keep it consistent: 0.039 as a margin = 3.9 percentage points; the interval (0.361, 0.439) = 36.1% to 43.9%. Decide on decimals or percents and stay there.
"95% of people are in the interval." No — a CI brackets the population proportion (one rate), not individual people. It's a range for the overall percentage, not a range that holds 95% of respondents.
"There's a 95% chance the true proportion is in this interval." The interval is already fixed; p is either in it or not. The 95% describes the method over many samples (about 95 of 100 such intervals capture p), not the odds for this one.
Reports the interval as a single ± number with no center. Always state both: center (p̂) and margin (ME), or the two endpoints. "±0.039" alone is meaningless without the 0.40.
Thinks a 99% interval is "more accurate." Higher confidence → wider interval (bigger z*), i.e., less precise. More confidence is bought with more vagueness; 95% is the usual compromise.

Scope flag

This outline stays within Objective 6 (confidence intervals for proportions). The large-counts condition in Segment 5 and the spreadsheet CEILING step in Segment 7 are added rigor/context (helpful, not strictly demanded by the objective) — keep them for a complete treatment, or trim Segment 5's "fails the check" example for a leaner 60-minute version. Confidence intervals for means were Week 11; hypothesis testing begins Week 13.

~ Prof. Rivera's edition · Fall 2026 · built with thecoursemaker.com