L-INF-4: Confidence Intervals for a Proportion

Module 4 · Statistical Inference

Section 1: Introduction

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A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

Polls in the news end with a line like this: “54% support the plan — margin of error ±3 percentage points, 19 times out of 20.” That last phrase is not a throwaway disclaimer. It describes how often the method behind the ±3 captures the truth in the long run. This lesson is about where the ±3 comes from, when it can be trusted, how many people a poll needs to earn it, and what it leaves out.

Start with a bet. It is not scored, and nothing you pick is saved or sent.

Three hypothetical polls each want a margin of error of ±3 points at 95% confidence: one of Quebec (about 9.0 million people), one of Canada (about 41.4 million) and one of the United States (about 342 million). Which poll needs to interview the most people? The Quebec poll The Canada poll The United States poll All three need about the same number

Noted. C7 settles the bet, and the Boss Fight comes back to it.

In L-INF-2 you built confidence intervals for a population mean with . This lesson keeps the same skeleton (point estimate ± margin of error) but swaps in a new statistic: the sample proportion . Proportions have their own standard error, and that formula hides a subtlety that trips up almost everyone the first time.

Here is the kind of question you will be able to answer by the end: A hypothetical poll contacts 1,000 adults and finds that 540 support a proposed bike-lane network. What range is supported for the population proportion, and how many people would a future poll need for greater precision? The numbers are an illustrative teaching scenario, not a result from any real polling firm.

After this lesson, you will be able to:

Distinguish from and verify the random-sample, independence, and success–failure conditions for a proportion interval
Construct a two-sided confidence interval for and report its margin of error in context
Interpret confidence as the long-run performance of a method rather than a probability assigned to the fixed parameter , and say what the margin of error does not cover
Determine the minimum sample size for a target margin of error, with or without a defensible prior estimate, and explain why it does not depend on the population’s size

If L-INF-2 gave you the blueprint, this lesson gives you a new set of tools for a different kind of question. The math is closely parallel, so most of what you already know transfers directly.

Section 2: Prerequisites

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A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

Confidence intervals for proportions reuse ideas from earlier lessons. Each prerequisite below names where the idea comes from, what to recall, and why it matters today.

From L-INF-2: confidence-interval structure and critical values. Recall point estimate ± margin of error and the course-rounded values for 90%, 95%, and 99%. We will keep that structure and those critical values while changing the statistic and standard error.
From L-INF-1: sampling distributions and standard error. A statistic varies across random samples, and its standard error describes that sample-to-sample spread. Today the statistic is , so we need its standard error rather than the one for .
From L-PR-5: binomial successes and failures. A binary response has success probability and failure probability . That variance structure produces and explains why both observed outcome counts must be large enough.
From course arithmetic: proportions, complements, and square roots. Write 54% as 0.54, compute , and take the square root of the entire variance expression. These operations are needed in every standard-error calculation.

Retrieval Checkpoint

A poll reports a 95% confidence interval for a proportion as . What is the margin of error () in this study?

Before continuing, check what you remember:

I can identify the point estimate and margin of error in “estimate ± margin.” I can explain that standard error describes how a statistic varies across random samples. I can convert a percentage to a decimal proportion and compute its complement.

What transfers—and what changes: Keep L-INF-2’s point estimate ± margin structure and the same values. Do not transfer the mean standard error . A proportion uses , and its gatekeeper is at least 10 observed successes and 10 observed failures, together with randomness and independence.

Section 3: Core Concepts

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A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

Navigation tip: Seven concepts live in this section. They build on each other in order — C1 through C4 give you the interval formula, C5 tells you when to use it, C6 tells you how to talk about it, and C7 adds the sample-size tool. If you already know L-INF-2 well, C1–C4 will go quickly.

C1 — The Sample Proportion

Suppose we want to know what fraction of Montréal adults have a library card. We can’t ask all 1.8 million — so we sample people and count how many () have one. The sample proportion is simply that fraction:

Sample Proportion

If individuals in a random sample of size have a characteristic of interest, the sample proportion is:

(read “p-hat”) estimates the unknown population proportion . must be a count (a whole number); is the sample size.

Notice the notation carefully: is the true proportion in the whole population — unknown, fixed, a parameter. is what we compute from our data — known, varies sample to sample, a statistic. This distinction is going to matter a great deal when we write the standard error formula.

and are not the same thing. is the true population proportion — it exists, but we don’t know its value (that’s why we’re building an interval). is our best guess from the data. We will always use in our calculations, never — because we don’t have .

The p̂ machine below makes this concrete. The true stays fixed (and hidden: you don’t get to know it) while each new sample produces a different . Pile up the samples and watch their shape: how far the pile spreads is exactly what C2’s standard error measures. Before you draw, make the figure’s prediction about versus .

Predict first: will samples from p = 0.1 spread out more or less than samples from p = 0.5?

Not scored.

Sample size n

True p

p is fixed but hidden. Make your prediction, then draw samples of n = 50.

Figure 1: The dashed blue line is the true proportion p: one fixed value that never moves (and is usually unknown, so it starts as p = ?). Each orange dot is one sample's p̂ = x/n; samples that give the same p̂ stack up, and the newest sample is ringed in green. After 30 samples the shaded band shows p ± 1 standard error. Compare the piles at p = 0.1, 0.5 and 0.9: the pile is widest at 0.5, and a larger n makes every pile narrower.

C2 — The Standard Error of

In L-INF-2, the standard error of was . Where did that come from? From the variance of a sum of random variables. The same idea applies here — but for a proportion.

Think first of independent Bernoulli trials: each sampled person is a success (has the characteristic) with probability or a failure with probability . Under independent sampling with a fixed success probability, the sample count is binomial. A simple random sample drawn without replacement from a finite population is not exactly binomial, but when the sample is no more than 10% of the population, the binomial model is a close approximation. This is why the independence check in C5 matters.

For the binomial model, recall from L-PR-5 that has variance . The proportion divides the count by , and dividing a random variable by divides its variance by :

Taking the square root gives the standard error. It is the spread you saw in the p̂ machine: widest at , narrower near 0 and 1, and narrower again as grows.

Standard Error of a Proportion

The theoretical standard error of is:

Since is unknown, we substitute to get the estimated standard error:

Notice: we estimate an estimate. We use to estimate for the SE formula itself. This works fine in practice, but it’s one more reason the conditions check (C5) matters — the approximation is only reliable when is large enough.

Never write in a calculation. You do not know . The formula that goes into actual computations always uses : . Only the theoretical expression uses .

Why no this time? In L-INF-3, estimating the spread cost a : the sample’s was a second, separate estimate of , and its own wobble had to be paid for with a wider interval. A yes/no variable has no separate spread to estimate: fixes both the centre and the standard error . So the proportion interval keeps . The honest caveat: because one number does both jobs, a small sample with an unlucky gets a wrong centre and a wrong spread at once. That is exactly why C5’s count condition matters.

C3 — Margin of Error

The margin of error is the ”±” part of the interval. It answers: “How far from do we need to reach to be confident we’ve captured ?”

Margin of Error for a Proportion

where is the critical value for the desired confidence level. This course uses the rounded values for 90%, for 95%, and for 99%.

The margin of error is the half-width of the interval — not the full width. A 95% CI with spans 0.06 units in total. When a poll reports “margin of error ±3 points,” that ±3 is exactly .

The margin of error is half the interval width. If , the interval is , which has total width 0.08, not 0.04. The ”±” already signals this — but it’s easy to report as the width when writing up results.

C4 — The Confidence Interval Formula

Two-Sided Confidence Interval for a Proportion

A confidence interval for the population proportion is:

which gives the interval .

Conditions must be checked before applying this formula (see C5).

The structure is identical to L-INF-2: point estimate ± (critical value × standard error). Only the point estimate and SE formula have changed. Everything else — the z* values, the interpretation logic, the sample size approach — carries over directly.

This lesson uses two-sided intervals, which report both a lower and an upper endpoint. A one-sided confidence bound reports only a lower bound or only an upper bound and uses a critical value chosen for that one-tail confidence target. You cannot create a valid one-sided bound by calculating a two-sided interval and discarding one endpoint; designing those bounds is outside this lesson’s procedure.

Figure 2: One sample's interval (green bar) on the proportion scale. The whole bar is the width, 2E; the half from p̂ to either end is the margin of error, E. Drag the sliders to see how n and p̂ change the width, and press a confidence level to see its z* and the width–confidence tradeoff. The dashed bar and the bottom row show what happens when you quadruple the sample size: E is halved.

C5 — Conditions for the z-Interval

The CI formula works because is approximately normally distributed when the sample is large enough (this follows from the CLT applied to a proportion). “Large enough” has a specific meaning here:

Conditions for the z-Interval for a Proportion

Before computing a CI for , verify both:

— at least 10 successes in the sample
— at least 10 failures in the sample

Also assume the data come from a random sample (or can be treated as one), and that of the population size (so observations are approximately independent).

Intuitively: if almost everyone in your sample is a “success” (say, with ), the distribution of is strongly skewed and a normal approximation can be poor. Having at least 10 observed outcomes on each side usually reduces skewness enough for the standard procedure to be workable. It does not make the sampling distribution exactly symmetric or guarantee the stated confidence level, so treat the count rule as a practical adequacy guideline rather than a theorem.

Figure 3: The exact sampling distribution of p̂ (orange bars) compared with the normal approximation (ink curve); the blue dashed line marks the true p. Shaded strips expose probability that the normal curve assigns to impossible p̂ values below 0 or above 1; arrowheads mark its continuing tails. The ≥10 success–failure guideline is a practical adequacy check, not a theorem about visual symmetry.

What does a poor fit cost? Figure 4 builds real 95% intervals at a setting where the count rule fails, and (only 3 expected successes), and counts how often they capture . The curve under the intervals gives the exact long-run capture rate for every possible : inside the shaded zones it falls well below the promised 95%.

Figure 4: Each bar is a 95% (or chosen-level) Wald interval built from a genuinely random sample of size n. Orange bars capture the fixed true p (blue dashed line); dashed red bars marked ✗ miss it. The curve below is the exact long-run capture rate for every true p at this n. Inside the shaded zones — fewer than 10 expected successes or failures — it drops well below the promised 95%: about 81% at p = 0.10, n = 30 and about 88% at p = 0.05, n = 100. Outside them it stays near 95% but zig-zags, which is why the lesson calls the ≥10 rule a guideline rather than a guarantee. The Wilson interval of Challenge 1 stays much closer to 95% everywhere.

Check conditions with , not . You’re checking whether your observed sample has enough successes and failures. Use and — these are literally the observed count of successes and failures in your sample (i.e., and ).

C6 — Correct Interpretation

This is the concept most students get wrong — even after getting the arithmetic right. The issue is subtle but important.

Imagine running 1,000 polls, each on a fresh random sample of the same size. Each poll gives a different , and therefore a different interval. A 95% procedure is designed so that about 95% of intervals capture the true in the long run when its assumptions and approximation are adequate. A finite run need not produce exactly 950 captures, and the actual coverage of this standard z-interval can differ somewhat from 95%, especially near the limits of its conditions.

Now you run your poll. You get one specific interval — say, (0.48, 0.56). Either is in that interval or it isn’t. There is no randomness left: is a fixed (unknown) number, and your interval is a fixed pair of numbers. Figure 4 shows this picture: the dashed line for never moves; only the intervals do.

Correct CI Interpretation

“We are 95% confident that the true population proportion lies between [lower] and [upper].”

This means: under repeated sampling in comparable conditions, the method targets a long-run capture rate of about 95%. It does not promise that any finite batch will hit exactly 95%.

“There is a 95% probability that is between [lower] and [upper].” This statement is incorrect. is not random — it is a fixed population parameter. It doesn’t have a probability of being in a range; it either is or isn’t. The 95% refers to the procedure, not to the probability that is in any one specific interval.

What “±3 points” does not cover. The margin of error measures one thing only: the luck of which people the random sample happened to pick. It says nothing about people who refused to answer, people who were never reachable (no phone, no address on the list), or a question worded so that it leans one way. Those errors do not shrink as grows. When researchers compared 4,221 published election polls with the actual results, the typical real error was about twice what the stated margins implied (Shirani-Mehr, Rothschild, Goel and Gelman, 2018). Read “±3 points” as “at least ±3 points”.

C7 — Sample Size Determination

Suppose you want to design a poll. You want your margin of error to be at most at a given confidence level. How large does your sample need to be? Solve the margin-of-error formula for :

Sample Size Formula for a Proportion

To plan for a margin of error of at most under a chosen planning value :

where is your best prior estimate for . A prior estimate must be defensible: recent, and from the same kind of population. If no defensible prior estimate is available, use — this maximizes and gives the largest (most conservative) sample size.

Always round up to the next whole number.

With a prior value other than 0.5, this is a conditional plan: the realized margin can be larger if the eventual sample proportion is closer to 0.5. The computed minimum is also relative to the course-rounded value used.

This is the course’s large-population planning formula. It assumes independent observations, or a simple random sample for which the rounded-up result satisfies . After computing , verify that population-size condition. If it fails, this formula should not be described as the exact minimum; a finite-population design requires an adjustment beyond this lesson’s scope.

Why does give the largest sample? Because is maximized at (you can verify: , while and ). Within this planning formula, using protects against undershooting the target for any possible true .

Back to the opening bet

Look at the formula again: , and . The population size appears nowhere in it. For ±3 points at 95% with no earlier estimate:

so the poll needs 1,068 people, for Quebec, for Canada and for the United States alike.

If you skipped the bet in the introduction: most people expect the bigger country to need the bigger poll.

You bet on Quebec. It needs the same 1,068 as the others.

You bet on Canada. It needs the same 1,068 as the others.

You bet on the United States, as most people do. It needs the same 1,068 as the others.

You bet that all three need about the same number. Right: exactly 1,068 each.

It feels as if a sample should cover a fair slice of the population. But a random sample’s precision comes from how many people are in it, not from what fraction of the population they are, in the same way that one spoonful tells you how salty a well-stirred pot of soup is whatever the size of the pot. Population size matters in one situation only: when the sample is a big slice of the population. That is the 10% check. Past it, observations are no longer close to independent, and an adjustment beyond this lesson actually makes the needed sample smaller. Figure 5 lets you check all of this, and its inset shows what a planning value other than 0.5 risks.

Margin of error, any large population Target ±3

Confidence

Population

Quebec, about 9.0 million people: for ±3 points at 95% you need 1,068 people.

If you have an earlier estimate p*

Earlier estimate p* = 0.50

With p* = 0.50: 1,068 people for ±3 at 95%.

No earlier estimate? p* = 0.5 never undershoots.

Figure 5: How the margin of error shrinks as a poll surveys more people (course formula, p = 0.5). Switching between Quebec, Canada and the United States leaves the curve exactly where it is: all three need 1,068 people for ±3 points at 95%, because n = (z*/E)²·p*(1 − p*) contains no population size N. The 2,000-student college is the only case where N matters: 1,068 people would be more than 10% of it, so the course formula's condition fails and a finite-population adjustment (beyond this lesson) takes over. The inset shows what an earlier estimate does: any p* other than 0.5 leaves a risk zone of true values where the margin ends up above ±3. The three polls are hypothetical. Population estimates, rounded: Quebec 9.0 million and Canada 41.4 million (Statistics Canada, preliminary estimates for April 1, 2026); United States 341.8 million (U.S. Census Bureau, Vintage 2025 estimate for July 1, 2025).

Always round sample size UP. If the formula gives 384.16, you need 385 people — not 384. Rounding down means you’ve committed to a margin of error larger than the target, which defeats the purpose of the calculation.

Which proportion goes in? So far, two different values have gone into : the observed when you build an interval from data, and a planning value (0.5 unless you have a defensible estimate) when you plan a survey not yet run. Never the unknown itself. Looking ahead: in L-INF-6 a third value appears, the hypothesized proportion , used to test a claim. Start noticing which one a problem calls for.

Section 4: Worked Examples

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A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

These four examples deliberately fade support. First, study a complete model. Then predict, choose, and finally diagnose more of the work before comparing your reasoning with a solution.

Example 1 — A Transit Policy Poll (Fully Worked)

Problem: In this hypothetical teaching example, a polling team takes a simple random sample of 600 Quebec adults from a sampling frame of Quebec adults. Of those sampled, 312 support a proposed transit line. Build and interpret a two-sided 95% confidence interval for the true proportion of Quebec adults who support the proposal.

Step 1: Identify the tool and the data.

I notice that the response is binary—support or do not support—and the target is a population proportion. So I choose a proportion interval rather than a mean interval.

(sample size)
(population size)
(number of supporters in the sample)
Confidence level = 95%, so

Step 2: Compute the sample proportion.

So 52% of the sample supports the proposal. This is the point estimate for .

Step 3: Check the conditions.

I notice that the poll is described as a simple random sample. The sample is no more than 10% of the stated population because:

So treating responses as approximately independent is reasonable. The observed outcome counts are:

Both counts clear 10, so the standard z-interval is an appropriate approximation here.

Step 4: Compute the standard error.

Step 5: Compute the margin of error.

Step 6: Build the interval.

So the interval is approximately .

Step 7: Interpret in context.

Using this method, we are 95% confident that the true proportion of Quebec adults who support the proposed line lies between 48.0% and 56.0%.

Reality check: The margin of error is about 4.0 percentage points, so the full interval spans about 8.0 points. With the same confidence level and a similar proportion, cutting that margin roughly in half would require about four times the sample size.

Example 2 — Predict the Effect of Confidence

Complete task: A quality-control inspector takes a random sample of 80 bottles from a production run of 5,000 and finds 12 with a filling defect. Before calculating, predict whether a two-sided 90% interval will be wider or narrower than a 95% interval from the same data. Then verify the conditions, construct the 90% interval, and interpret it.

Pause and write. Record your answer and reasoning on paper or in your notes before continuing. Even if you are unsure, committing your thinking helps you see what you understood and what you need to revisit.

Show Solution

Prediction: The 90% interval will be narrower because is smaller than the 95% critical value .

Conditions:

The sample is random, and , so the independence condition is reasonable.

Standard error:

Margin of error:

Endpoints:

Using this method, we are 90% confident that the true defect proportion for this production run lies between about 8.4% and 21.6%.

Example 3 — Choose the Conservative Planning Value

Complete task: A researcher wants to estimate the proportion of Quebec CEGEP students who work more than 15 hours per week during the semester. The target margin of error is at most 4 percentage points at 95% confidence, and no defensible prior estimate is available. For this planning exercise, assume the target population contains at least 10,000 students. Decide which planning value to use, calculate the required sample size, verify the population-size condition, and explain the rounding decision.

Show Solution

With no prior estimate, choose because it maximizes and therefore gives the conservative sample size.

Round up to . A planning result of 600.25 people is impossible, and rounding down would make the planned margin slightly larger than the target under the formula.

Because :

So the rounded-up plan also satisfies the course’s population-size condition.

Example 4 — Diagnose a Misleading Interval

Flawed analysis: A simple random sample of 60 patients from the clinic’s 2,000 eligible patients includes 6 who report a side effect. A student writes: “Because , the normal model is valid. The 95% interval is approximately , so there is a 95% probability that the true side-effect proportion lies in that range.”

Complete task: Identify every substantive error in the student’s reasoning. State what should be checked instead, whether the standard two-sided z-interval from this lesson should be reported, and how confidence would be interpreted if an appropriate interval method were used.

Show Solution

The sample is described as a simple random sample, and:

So the random-sampling and independence checks are reasonable. The remaining errors are:

Error 1 — the wrong condition: The shortcut concerns some mean procedures; it is not the success–failure condition for a proportion interval.

Only one observed outcome count clears 10, so this lesson’s standard z-interval should not be reported as a reliable approximation.

Repair: Collect more data, or use a method designed for sparse proportions, such as an exact or Wilson-style interval.

Error 2 — probability assigned to fixed : After the sample is observed, both and the computed interval are fixed. An appropriate interpretation would describe the interval procedure’s long-run capture performance, not a 95% probability that lies in this one range.

Section 5: Guided Practice

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A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

Build the procedure in order first; then choose the method at the decision points where an answer cannot be produced by calculation alone.

Problem 1 — Count, Then Decide (C1 + C5)

Some of these surveys can support this lesson’s interval and some cannot. Count first, then decide.

Problem 2 — Standard Error and Interval Construction (C2 + C3 + C4)

No step hands you the critical value: choose it from the confidence level.

Problem 3 — Which Proportion Goes In? (C2 + C7)

These two tasks use the same city and the same question. Only one fact differs. For each, choose the value that goes into .

Structural Difference

Which value goes into p(1 − p)?

Identify what key word or condition separates Case A from Case B, then select the method for each.

Scenario A

A random sample of 900 Montrealers finds that 387 cycle to work in summer (p̂ = 0.43). Build a 95% confidence interval for the proportion of all Montrealers who do.

Select method:

Scenario B

A city plans next year's survey of Montrealers who cycle to work in summer. No earlier estimate exists. Find the sample size for a margin of ±3 points at 95%.

Select method:

Problem 4 — Practical Interpretation of Confidence (C6)

A researcher surveys 500 adults and finds that 310 support stricter food labelling laws. The resulting 95% CI is (0.577, 0.663).

Which of the following is the correct interpretation?

Problem 5 — Determining Required Sample Size (C7)

Section 6: Independent Practice

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A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

These are cold retrieval problems: choose the quantity to produce, commit an answer, then use the feedback to diagnose your method. From here on, the numbers are not tidy: sample sizes and proportions are not round, so carry unrounded values through the calculation.

Problem 1 — Build a Confidence Interval (C1 + C2 + C4 + C5)

Problem 2 — Margin of Error and Interpretation (C3 + C6)

After checking your numerical result, state the interval’s interpretation on paper or in your notes. Name the generated target population and outcome, and describe the confidence level as the method’s approximate long-run capture rate under its conditions—not as a probability assigned to fixed .

Problem 3 — Can This Survey Support the Interval? (C5)

Sort into groups

Can this survey support the ± interval for p?

Each card describes a survey. Sort it by what this lesson's interval can do with it.

Drag an item into its group, or select it and then choose the group. On a keyboard: select an item and press the group’s number (1–3); 0 sends it back to Unsorted. Each item also has its own “move” buttons.

Unsorted

Problem 4 — Determine the Required Sample Size (C7)

Problem 5 — The Price of Precision (C7)

Complete the table

The price of precision

Fill in every empty cell, then check the whole table once. The cells depend on one another.

Problem 6 — Can the Headline Say “Majority”? (C6)

Build the Conclusion

Fact-check the headline

Choose a phrase for each blank so that the sentence states the correct conclusion.

Problem 7 — Diagnose a Survey Report (C2 + C3 + C5 + C6 + C7)

Problem 8 — Multi-Step Survey Decision (C1 + C2 + C3 + C4 + C5 + C6 + C7)

A public-health team selects a simple random sample of 500 adults from a population of 12,000. Of those sampled, 190 report meeting a weekly activity guideline.

(a) Verify all conditions for this lesson’s two-sided z-interval.

(b) Construct a 95% confidence interval for the population proportion.

(c) Interpret the interval and decide whether it supports the claim that fewer than 45% of adults meet the guideline.

(d) Using this sample proportion as a planning estimate, determine the minimum sample size for a future 95% interval with margin of error at most 0.03.

Show Solution

(a) Conditions

The sample is described as a simple random sample.

All conditions are satisfied.

(b) Interval

The 95% interval is approximately .

(c) Interpretation and claim

Using this method, we are 95% confident that the true proportion of adults who meet the guideline lies between about 33.8% and 42.3%. Because the entire interval is below 45%, the data support the “fewer than 45%” claim at the corresponding confidence level. They do not prove the claim with certainty.

(d) Future sample size

Round up to adults.

Mixed Review — Retrieval from Earlier Lessons

These final two items retrieve prerequisite ideas and do not count as new L-INF-4 main problems.

Section 7: Mastery Check

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A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

No pre-attempt hints and no forced steps. One cold attempt per idea: work independently first, then compare your reasoning with the feedback.

Question 1 — Feynman Test

A friend who missed this lesson asks you: “The formula for the standard error of a proportion involves — but you said is what we’re trying to estimate. So aren’t we using an unknown to find an unknown? Doesn’t that break the whole thing?”

On paper, in your course notes, or in a notes app, give a concise plain-language explanation. Address why replaces , why the success–failure condition matters, and what “95% confident” describes. Then compare your explanation with the model.

Compare with a model answer

The theoretical spread of depends on the unknown , so we plug in the observed to estimate that spread. When the sample is random, observations are reasonably independent, and both observed outcome counts are at least 10, the standard z procedure is usually a workable approximation. If those conditions fail, substituting can help produce a misleading interval, so we use another method or collect more data.

“95% confident” describes the method: across many comparable random samples, it aims to capture the fixed population proportion about 95% of the time. It is not a 95% probability statement about after this one interval has been calculated.

Compare: Did your explanation distinguish the fixed from the sample-dependent , state the role of the conditions, and attach 95% to the method rather than to the parameter?

Question 2 — May the Interval Be Built?

Question 3 — Cold Interval

Question 4 — Plan the Survey

Question 5 — Fact-Check the Headline

Self-Assessment

How confident are you with the concepts from this lesson?

Still confusedReady for the Boss Fight

If your confidence is below 60%, revisit Section 3 and compare your work with the four examples before the Boss Fight. The next scenarios expect you to decide what the evidence can support.

Section 8: Boss Fight

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A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

Choose one of two equally demanding missions. Neither tells you which method to use: choosing the tool is part of the fight. Each starts with your own complete attempt; open the steps only if you need to find where it went wrong. Lose, and you get a fresh rematch with new numbers.

📰 The Newsroom

Tonight’s poll is in and the editor already has a headline. Does the poll back it up, and what about the young-voters story?

📋 The Pollster

A client wants a precise poll on a fixed budget, and has a theory about why Quebec needs fewer people.

📰 The Newsroom

Task 1. The editor needs the range first.

Task 2. The next night’s poll arrives with its interval already computed, and with a second story attached. Write the note to the editor.

Build the Conclusion

The note to the editor

Choose a phrase for each blank so that the sentence states the correct conclusion.

Then, on paper: draft the two sentences that run under the headline: what the poll supports, and what its “±” does not cover.

Compare with a model answer

“A random poll of 1,068 adults puts support between about 50% and 56%, a margin of ±3 points 19 times out of 20, so the poll narrowly backs a majority. The margin counts only the luck of who was sampled; people who refused, people who could not be reached and the wording of the question can add error it does not include.” (Use your own poll’s numbers. If 0.50 is inside your interval, the honest sentence is that the poll cannot tell whether a majority supports the plan.)

📋 The Pollster

Task 1. How many people does the client’s target need?

Task 2. The client will not raise the budget. What can it buy?

Task 3. Another client, another budget, and a theory about Quebec. Complete your reply.

Build the Conclusion

Reply to the client

Choose a phrase for each blank so that the sentence states the correct conclusion.

Then, on paper: go back to the bet you made in the Introduction. In two or three sentences, explain to the client why a poll of Quebec needs as many people as a poll of the United States, and name the one situation in which the population’s size would matter.

Compare with a model answer

The margin of error depends on how many people are interviewed, , through ; the population’s size does not appear. So ±3 points at 95% needs 1,068 random interviews whether the population is 9 million or 342 million. Size matters only when the sample is a large share of the population (more than about 10%), as it would be for a small college; then fewer interviews are needed, not more.

Section 9: Challenge Problems

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A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

Optional stretch problems — these go beyond the lesson objectives. They’re here for students who want to push further. C1 previews a more robust method; C2 builds deep intuition; C3 requires creative multi-step reasoning.

Challenge 1 — The Wilson Score Interval

The standard z-interval can fail near 0 or 1 or with a small sample. This optional extension compares it with the Wilson score interval; its solution remains deliberately ungraded because Wilson intervals are outside this lesson’s core outcomes. Figure 4’s curve shows why the extension exists: inside the shaded zones the standard interval captures far less than 95% of the time, while the Wilson interval stays much closer to its promise.

Challenge 2 — The Shape of the Margin of Error

For a fixed and confidence level of 95%, the margin of error is:

This is a function of .

(a) Compute for .

(b) At what value of is maximized? Why does this make sense?

Show Solution

(a) Using :


0.1	0.09
0.2	0.16
0.3	0.21
0.4	0.24
0.5	0.25
0.6	0.24
0.7	0.21
0.8	0.16
0.9	0.09

(b) is maximized at , where . This makes sense: is maximized at 0.5 because that’s where a Bernoulli random variable has maximum variance. A proportion near 0.5 means successes and failures are equally unpredictable — maximum uncertainty.

(c) Incorrect. Using gives , which is not the worst case. The worst case is always (giving 0.25). For every true strictly between 0.3 and 0.7, the product is larger than 0.21 (up to 0.25 at ), so a plan built on would underestimate the required sample size for any true in that whole range, including values below 0.5 such as 0.4. The claim “safe for any ” fails for all of . Safe means conservative: use unless you’re very confident the true proportion is far from 0.5.

Challenge 3 — Two Polls, One Question

Overlapping intervals are a classic trap: two 95% intervals can overlap even when the two proportions really differ, so overlap by itself is inconclusive. (A related poll-reading fact: the lead between two options in one poll has a margin about twice the poll’s stated margin.)

Section 10: Solutions Reference

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A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

Complete, step-by-step solutions for all problems in Sections 5–9 are available on the solutions page. Solutions include worked arithmetic, common mistakes to watch for, and interpretation guidance.

View Full Solutions →

If you’re stuck: For interval construction, revisit C1–C5 and Example 1. For interpretation, compare C6 with Example 4. For planning, pair C7 with Example 3. The solutions page shows the reasoning behind each step, not just the final answer.

Quick-Reference Formulas

Sample Proportion:

Estimated Standard Error of :

Two-Sided Confidence Interval for :

The stated confidence level describes the method’s approximate long-run capture rate under its conditions. A one-sided bound uses a different one-tail critical value; do not create one by discarding an endpoint from this two-sided interval.

Required Sample Size: (Use for a conservative estimate if no prior estimate is available. Always round up to the next whole number, then verify .)

Condition	Rule to check
Randomness	Was the sample randomly selected?
Independence	Is of the population?
Success/Failure	Are both and ?