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Week 12 · Chapter

Chapter 12 — Confidence Intervals for a Proportion

Introduction to Statistics Free sample edition

The module's primary reading · ~40–50 minutes · Objective 6 (construct and correctly interpret confidence intervals for a population mean and a population proportion, including the meaning of the confidence level and the margin of error) · Read it first this week; the lecture, the tutorial, the quiz, and the Data Lab all build on it.

In plain language

Many of the numbers you hear about the world are shares, whether they come from 1,000 voters, 600 clinic patients, or 333 penguins. Think of the share of voters who back a plan, the share of patients who got a flu shot, or the share of penguins that are female. This week you learn to report a share from a sample as a confidence interval for a proportion (a range of believable values for the true share in the whole group, with a stated level of trust). You will build it by hand and with a spreadsheet, and you will learn to read the margin of error (the plus-or-minus part of the range) that comes with every news poll. You will also learn to plan how many people a survey needs, and what the plus-or-minus leaves out.

Why you already care about this

A news poll says 52% of residents support a new bus line, "with a margin of error of plus or minus 3 points." A campus survey says 38% of students skipped breakfast this morning. A clinic newsletter says 7 in 10 patients got their flu shot. Each one is a share, and each one comes from a sample, not from asking everyone.

You have probably wondered how 1,000 people on a phone can speak for a whole state. You may also have wondered why two polls taken the same week disagree by a few points. Both questions have the same answer, and it is this week's tool. A share from a sample wobbles, the wobble has a predictable size, and the plus-or-minus is that size, written down.

Field biologists face the same question with animals. A research team in Antarctica recorded the sex of 333 penguins and found 165 females. Is that a lopsided population, or a sample from an even split that happened to land a little low? By the end of the week you can answer that with one interval and one sentence.

By the end of this chapter you will be able to:

  • explain why the standard error of a sample proportion is √(p̂(1 − p̂) ⁄ n), and compute it;
  • check the conditions for a one-proportion z interval, including the large-counts rule of thumb, before you compute;
  • compute a 95% confidence interval for a proportion by hand and with a spreadsheet;
  • interpret the interval in one correct sentence and say which values it rules out;
  • read a poll's margin of error in percentage points and predict how it changes with the sample size;
  • choose a sample size for a target margin of error, rounding up;
  • name the kinds of error a margin of error does not cover.

Carry this sentence with you: the margin of error measures wobble — not bias.

Where we are

  • Already covered: Weeks 1 to 3 described one variable, and Week 4 related two. Weeks 5 to 7 built probability and the binomial, Week 8 added the z-score, Week 9 the normal model, Week 10 sampling distributions, and Week 11 the confidence interval for a mean.
  • This week: confidence intervals for a proportion: the standard error of p̂, the conditions, the one-proportion z interval, poll margins of error, choosing a sample size, and what the margin of error does not cover.
  • How they connect: Week 11 built an interval as an estimate plus or minus a critical value times a standard error; this week you use the same recipe with the sample proportion from Week 10.
  • Where we're going next: this week closes the estimation half of the inference unit. Next week asks a new question: how do you test a claim, such as "the split is even", with data?

Four earlier weeks matter most here. Week 1 gave you the line between a sample and a population, and it warned that a voluntary-response sample (people who choose to answer) and a convenience sample (people who are easy to reach) build bias into a study. Week 7 gave you the large-counts rule of thumb: a normal curve is a fair model for counts of successes when the expected numbers of successes and failures are both at least 10. Week 10 gave you the sample proportion p̂ (read "p hat", the share of a sample with a trait) and showed that it wobbles around the population proportion p (the share in the whole group) with a predictable spread. Week 11 gave you the margin of error, the critical value z* of 1.960 for 95%, and the careful meaning of "95% confident". This week those pieces snap together.

1. From a mean to a proportion: reuse the interval recipe

Think about how Week 11's interval was built. You had one number from a sample, the sample mean. You knew how much that number wobbles from sample to sample, its standard error. And you reached out a certain number of standard errors on each side, the critical value. The recipe was:

estimate ± critical value × standard error

Nothing in that recipe cares whether the estimate is a mean. It works for any statistic whose sampling distribution you understand. And in Week 10 you met a second one: the sample proportion.

A proportion is a share between 0 and 1. If 30 of 120 customers at a juice bar order a smoothie, the sample proportion is 30 ⁄ 120 = 0.25, or 25%. The population proportion p (the share among all customers) is the fixed number you would like to know. The sample proportion p̂ (the share in your sample) is the number you actually have.

Week 10 described how p̂ behaves across many random samples of the same size n (the sample size):

  • its mean is p, so on average it lands on target;
  • its standard deviation is √(p(1 − p) ⁄ n), so bigger samples wobble less;
  • its shape is approximately normal when np and n(1 − p) are both at least 10.

Put the recipe and the sampling distribution together and you have an interval for a proportion: p̂ plus or minus a critical value times the standard deviation of p̂. Only one piece is missing, and the next section fills it in.

Worked example — where the penguin question fits the recipe. The question: what share of penguins like the ones in the Palmer field study are female?

  1. Name the parameter. The share of females among all penguins like these is p (the population proportion). Nobody knows it.
  2. Name the sample. The 333 penguins with a recorded sex are a sample, and the share of females among them is a statistic.
  3. Name the estimate. That statistic is p̂, and it plays the role x̄ played last week.

In words: the penguin question is last week's question with a share in place of an average. A biologist who wants to know whether the colony is lopsided needs the same two things as before: a best guess and its wobble.

Misconception check — "a proportion needs a whole new method". Students often treat each new kind of interval as a new formula to memorize. It is tempting, because the symbols change. But every interval in this course has the same skeleton: estimate plus or minus critical value times standard error. The cure: when you meet a new interval, ask three questions. What is the estimate? What is its standard error? Which critical value do I use?

Memory hook: same skeleton — new estimate.

2. The standard error of p̂: plug in what you know

There is a catch in Week 10's formula. The spread of p̂ is √(p(1 − p) ⁄ n), and that formula needs p (the population proportion), the very number you are trying to estimate. Last week had the same problem with σ, and the fix there was to plug in the sample's own value. The fix here is the same: plug in p̂ wherever p appears.

  • The standard error of the sample proportion is the estimated standard deviation of p̂ from sample to sample, √(p̂(1 − p̂) ⁄ n); it uses the sample proportion p̂ in place of the unknown population proportion p.

Read the formula in words. Multiply the share with the trait by the share without it. Divide by the sample size. Take the square root.

Worked example — the standard error for the penguins. Use the outline's counts, and follow each step in words.

  1. The counts. 165 of the 333 sexed penguins are female and 168 are male, because 165 + 168 = 333. So n (the sample size) is 333.
  2. The point estimate. Divide 165 by 333 and round to four decimals: p̂ is 0.4955. It is your single best guess for p.
  3. The complement. The given share of males is 1 − 0.4955, which is 0.5045.
  4. Multiply. The given shares 0.4955 and 0.5045 multiply to about 0.25.
  5. Divide and take the root. Divide by 333 and take the square root. You land on the supplied SE of 0.0274.

In words: a share of females computed from 333 birds typically wobbles by about 0.0274, close to 3 percentage points, from one sample to the next. A field team deciding whether 0.4955 is "basically half" now knows how big a gap counts as small.

Why is there no t distribution this time? For a mean, the sample standard deviation s was a separate, extra guess, and the t distribution paid for that guess. For a proportion, the spread comes from p̂ itself, so there is no separate spread to estimate. The course uses the critical value z* from the standard normal curve: 1.645 for 90%, 1.960 for 95%, and 2.576 for 99%.

Honesty note: plugging in p̂ for p is an approximation. It works well when the sample is large enough, which is exactly what the next section checks.

Misconception check — using p̂ instead of √(p̂(1 − p̂) ⁄ n). Students often multiply z* by p̂ itself, or by p̂(1 − p̂) without dividing by n, because the formula has several layers. The interval comes out far too wide: for the penguins, 1.96 × 0.4955 is nearly a full unit on a scale that only runs from 0 to 1. The cure: compute the standard error on its own line first, and check that it is small, a few hundredths for a sample of hundreds.

Memory hook: share with, times share without, over n — then the root.

3. Conditions: check the counts before you compute

A one-proportion interval leans on a normal model for p̂. Before you trust it, check three conditions out loud.

  1. Random or representative sample — a sample chosen by chance, or one you can defend as typical. The individuals should come from a random sample, or at least a sample you can defend as representative of the population.
  2. Independence — no individual's value depends on another's. One person's answer should not set another person's.
  3. Large counts — enough of each outcome for a normal model. The sample needs at least 10 successes and at least 10 failures. This is Week 7's large-counts rule of thumb, applied now to the counts you actually observed.

The third condition is the one people skip. When a sample has only a handful of successes or failures, the sampling distribution of p̂ is lumpy and lopsided, and a normal curve misdescribes it.

Worked example — checking the penguin sample. Run the three checks in order.

  1. Random or representative. The penguins were studied in an observational field study, not drawn at random, so you treat them as representative of penguins in that region and say so.
  2. Independence. Each bird's sex is recorded separately, so one bird does not set another.
  3. Large counts. The sample has 165 successes (females) and 168 failures (males), both at least 10, so the large-counts rule of thumb holds and a normal model for p̂ is reasonable.

In words: the penguin sample passes all three checks, with one honest caveat about how the birds were chosen. A biologist reporting the interval should state that caveat beside it.

Worked example — a check that fails. Suppose you roll a die 12 times and get 2 sixes.

  1. Count successes. There are 2 successes (sixes).
  2. Count failures. There are 12 − 2 = 10 failures.
  3. Compare with 10. There are 2 successes and 10 failures, and 2 is below 10, so the one-proportion z interval is not appropriate for the share of sixes.

In words: with only 2 sixes, a normal curve would badly misdescribe how the share of sixes behaves. A player who wants to judge a die needs many more rolls first.

Caveat: some courses use 5 instead of 10 in this rule. It is a working rule of thumb, not a law of nature, and this course uses 10.

Misconception check — checking the conditions after the answer. Students often compute the interval first and check the conditions only if there is time, because the arithmetic feels like the real work. But an interval built on a failed condition can look perfectly precise and still be wrong. The cure: write the three checks before any arithmetic, every time, and stop if one fails.

Memory hook: ten and ten before you begin — not after.

4. The one-proportion z interval: build it by hand and by spreadsheet

Now put the pieces together.

  • A one-proportion z interval is the confidence interval for a population proportion p: p̂ plus or minus z* times the square root of p̂(1 − p̂) divided by n. It is used after checking for a random or representative sample, independent observations, and at least 10 successes and 10 failures in the sample.

In one line: p̂ ± z* · √(p̂(1 − p̂) ⁄ n). Read it from the inside out. The square root is the standard error. The critical value times the standard error is the margin of error. The estimate plus or minus that margin gives the two ends.

Worked example — the 95% interval for the share of female penguins. Use the outline's values: 333 birds, 165 females, p̂ of 0.4955, and SE of 0.0274.

  1. Critical value. The standard normal critical value z* for 95% is 1.960.
  2. Margin of error. Multiply the critical value 1.96 by the supplied SE of 0.0274, and you get about 0.0537. That is about 5.4 percentage points.
  3. Step down and up. Step 0.0537 down from 0.4955 and 0.0537 up from it.
  4. Report. The 95% one-proportion z interval runs from 0.4418 to 0.5492, about 44.2% to 54.9%.

In words: the true share of females is pinned down to a window about 11 percentage points wide. A biologist planning a breeding study can now decide whether that is precise enough or whether more birds need to be sexed.

Check the width. The interval is two margins wide, 2 × 5.4 = 10.8 percentage points. It is always twice the margin of error wide, because it reaches the same distance on each side.

In practice (your tools). In Google Sheets or Excel, open the penguin file and find the sex column, which holds the words female, male, or NA. =COUNTIF(G2:G345, "female") returns 165 and =COUNTIF(G2:G345, "male") returns 168; add them for n. Divide the first count by n for p̂. =SQRT(p*(1-p)/n), with your cell names in place of p and n, gives the standard error, and =NORM.S.INV(0.975) returns the critical value, 1.96 to two decimals. Multiply them for the margin of error, then subtract and add it.

Honesty note: the spreadsheet keeps every digit of p̂ and z*, so its bounds round to the same 0.4418 and 0.5492. If your hand answer differs in the fourth decimal, check whether you rounded too early; round only at the end.

Misconception check — reporting the interval in the wrong units. Students often compute 0.0537 and then write the interval as 49.55 ± 0.0537, mixing a percentage with a proportion. It is tempting because both look like "the share". The cure: pick one scale and stay on it. On the proportion scale the interval is 0.4955 ± 0.0537; on the percentage scale it is 49.55% ± 5.37 points.

Memory hook: p̂, then SE, then margin, then two ends — never skip a step.

5. Interpretation: say what the interval means

The template from Week 11 carries over word for word, with "proportion" in place of "mean".

Worked example — the interpretation sentence. Fill in the template with the penguin interval.

  1. The template. "We are [level] confident that the population proportion [of what, in context] is between [lower] and [upper]."
  2. Filled in. We are 95% confident that the population proportion of female penguins (among penguins like those studied) is between 0.4418 and 0.5492.

Every part of that sentence does a job. "95% confident" names the method's track record from Week 11: about 95% of intervals built this way capture p. "Population proportion" says the target is p, not p̂. The two ends say where p plausibly sits.

An interval also tells you which values are plausible and which are not. The value 0.5, an even split, sits inside the interval, so the data are consistent with an even split between females and males. Values such as 0.40 or 0.60 sit outside it, so a claim that only 40% of these penguins are female does not fit the data well.

Caveat: "consistent with an even split" is not the same as "proves an even split". The interval also contains 0.46 and 0.54. Next week you will learn a formal way to weigh a claim like "the split is even", and the interval you built here will give the same verdict.

Misconception check — "there is a 95% chance that p is between 0.4418 and 0.5492". Students often say this, because it sounds so close to right. But p is a fixed number, and once the interval is computed it either contains p or it does not. The 95% belongs to the procedure, not to this one interval. The cure: use the template, and if someone asks what "confident" means, describe the method's long-run capture rate.

Memory hook: inside means plausible; outside means doubtful — never proven.

6. Read a poll's margin of error

Now turn to the most common place you will meet this interval: a news poll. A pollster has not collected any data yet when they plan a poll, so they cannot use p̂ in the standard error. They plan instead with the conservative p̂ of 0.5.

  • The conservative p̂ is the planning value p̂ = 0.5, used when the true proportion is unknown. Because 0.5 × 0.5 = 0.25 is the largest value p̂(1 − p̂) can take, it gives the widest margin of error and the largest sample size for a given confidence level.

Try a few values to see why 0.5 is the worst case. For a share of 0.1, the product is 0.1 × 0.9 = 0.09. For 0.3, it is 0.3 × 0.7 = 0.21. For 0.5, it is 0.25, the biggest. A margin of error computed with 0.5 is never too small, whatever the true share turns out to be.

With the conservative p̂, the 95% margin of error is 1.96 × √(0.25 ⁄ n). Only n (the poll's sample size) is left to choose.

The 95% margin of error for a sample proportion near 0.5, by sample size, as a proportion and in percentage points
Sample size Margin of error (proportion) Margin of error (percentage points)
100 0.098 about 9.8
400 0.049 about 4.9
1,000 0.031 about 3.1
1,500 0.0253 about 2.5
2,500 0.0196 about 2.0

Worked example — quadruple to halve. Read the first two rows.

  1. Compare the sizes. 400 is 4 × 100.
  2. Compare the margins. The margin goes from 9.8 points to 4.9 points, half as wide, because √4 = 2.
  3. A bigger jump. Going from 100 to 2,500 is 25 times the sample and cuts the margin to one-fifth, 9.8 ⁄ 5 = 1.96 points, because √25 = 5.

In words: precision gets expensive. A pollster who wants to halve the margin must pay for four times as many interviews, and a fivefold improvement costs 25 times the interviews. That is why most national polls stop near 1,000 people: going from about 3 points to about 2 points would mean roughly 2.5 times the cost.

Percentage points, not percent. A margin of error is reported in percentage points.

  • A percentage point is one hundredth on the proportion scale, the unit in which a poll's margin of error is reported. A share of 52% with a margin of 3 percentage points runs from 49% to 55%, which is not the same as 3% of 52%.

So "52% support, plus or minus 3 points" means the interval runs from 49% to 55%. It does not mean 52% plus or minus 3% of 52%, which would be about 1.6 points.

The confidence level is a lever too. Go back to the penguins, with their standard error of 2.74 percentage points. At 90% the critical value is 1.645 and the margin is about 1.645 × 2.74 ≈ 4.5 percentage points. At 99% the critical value is 2.576 and the margin is about 2.576 × 2.74 ≈ 7.1 percentage points. More confidence costs a wider interval, here 7.1 points versus 4.5 points, a ratio of about 1.6 to 1 between the 99% and 90% margins.

Misconception check — "double the sample, halve the margin". Students often expect twice as many people to give half the margin of error, because it sounds fair. But the margin divides by √n, and doubling n divides it by only about 1.41. The ladder shows it: going from 1,000 to 2,500 people, two and a half times the sample, only moves the margin from about 3.1 points to about 2.0 points. The cure: to halve the margin, multiply the sample by four.

Memory hook: four times the people — half the wobble.

7. Choose a sample size before you collect data

A study is cheapest to fix before it starts. If you know how precise you need to be, you can turn the margin-of-error formula around and solve for the sample size.

  • The sample-size formula for a proportion is the rule n = (z* ⁄ ME)² × p̂(1 − p̂), which gives how many individuals a study needs for a target margin of error ME at a chosen confidence level. The result is always rounded up to the next whole number.

Here n (the number of people to survey) is what you solve for, and ME is the target margin of error, written as a proportion: 0.03 for ±3 points. When you have no earlier estimate of the share, plan with the conservative p̂ = 0.5, so p̂(1 − p̂) = 0.25.

Worked example — a campus survey with ±3 points. A student government wants to estimate the share of students who use the campus shuttle, to within ±3 points at 95%.

  1. Choose the inputs. The critical value is 1.96, the target ME is 0.03, and with no earlier estimate p̂(1 − p̂) is 0.25.
  2. Set up the formula. n = (1.96 ⁄ 0.03)² × 0.25.
  3. Compute. That is 1.96² × 0.25 ⁄ 0.03² = 1,067.1, which you round up to 1,068 people.
  4. Round up, always. You always round up, because rounding down would leave the margin a little above the target.

In words: the student government needs 1,068 completed responses. If it expects that only half the students it contacts will answer, it must plan to contact about twice that many, and it should worry about who the non-responders are (Section 8).

Worked example — a looser target. Suppose ±5 points is good enough.

  1. Compute. You need 1.96² × 0.25 ⁄ 0.05² = 384.16, which rounds up to 385 people.
  2. Compare. Loosening the margin from 3 points to 5 points cuts the sample from 1,068 to 385, about 2.8-fold fewer people.
  3. Count the cost. That is 1,068 interviews versus 385, about a 2.8-fold difference in cost.

In words: the survey team can choose between precision and cost. A tighter margin is not free, and the square in the formula makes it climb fast.

Misconception check — rounding to the nearest whole number. Students often round 1,067.1 down to 1,067 because the usual rounding rule says so. But 1,067 people give a margin a hair above 3 points, which misses the target. The cure: sample sizes always round up, even 384.16 to 385.

Memory hook: decide the margin first; then count the people, rounding up.

8. What the margin of error does not cover

Here is the most important limitation of the week. The margin of error measures sampling variability only. It describes how much p̂ wobbles because you happened to get this random sample rather than another one. It says nothing about whether the sample was drawn fairly or the question was asked fairly.

That leaves several kinds of error outside the plus-or-minus. You met the first two in Week 1.

  • Voluntary-response and convenience samples. People who choose to answer, or who are easy to reach, often differ from everyone else.
  • Undercoverage. Some groups have little or no chance of being included.
  • Non-response bias. Many people chosen for the sample never answer, and the ones who do answer differ from the ones who do not.
  • Question-wording bias. The wording or order of a question pushes people toward an answer.

The definitions, set off for reference:

  • Undercoverage is a form of bias that arises when some groups in the population have little or no chance of being included in the sample, for example people missing from the list a survey draws its sample from.
  • Non-response bias is bias that arises when many people chosen for a sample cannot be reached or will not answer, and those who respond differ in a systematic way from those who do not.
  • Question-wording bias is bias that arises when the wording or order of a survey question pushes respondents toward a particular answer, so the measured share differs from what people actually think.

Worked example — a huge poll that can still miss. A news site runs an online poll and 10,000 readers click an answer.

  1. Compute the margin anyway. A voluntary-response online poll of 10,000 readers would have a conservative margin of 1.96 × √(0.25 ⁄ 10,000), which is 1.96 × 0.5 ⁄ 100, a little under 1 percentage point.
  2. Ask who answered. Only readers of that site who cared enough to click took part.
  3. Conclude. The tiny margin is meaningless, because the people who choose to respond differ from those who do not.

In words: a careful random poll of 1,000 people, with a margin of about 3 points, can beat a self-selected poll of 10,000 with a margin under 1 point. A newsroom deciding which poll to report should choose the random one, because no number of clicks can remove the bias. Week 1 said it first: a bigger sample does not fix a biased method.

Question wording works the same way. "Do you support a small fee to keep the shuttle running?" and "Do you support a new tax on students?" can describe the same plan and draw very different shares. Both polls could report the same margin of error, and that margin would say nothing about the gap between them.

Misconception check — "the margin of error accounts for a biased sample". Students often read "±3 points" as a promise that the poll is within 3 points of the truth, no matter what. It is tempting, because the margin is the only error the poll reports. But the margin covers only random sampling wobble. Bias from who was asked, who answered, and how the question was worded sits outside it, and a bigger sample does not shrink it. The cure: when you read a poll, ask two separate questions. How big is the wobble? And who was missed, and how was it asked?

Pause and reflect — choose one. Pick either of the 2 prompts; they count the same.

  • (a) Think of a survey you have been asked to answer, such as a course evaluation, a store receipt survey, or an app pop-up. Who is most likely to answer it, and who is most likely to skip it? Explain in two or three sentences how that could push the result in one direction.
  • (b) Write two versions of one survey question about a campus or community issue you care about: one neutral, and one worded to push people toward "yes". Explain in two or three sentences why the margin of error would not reveal the difference.

Memory hook: the margin of error measures wobble — not bias.

Key terms

  • standard error of the sample proportion — The estimated standard deviation of p̂ from sample to sample, √(p̂(1 − p̂) ⁄ n); it uses the sample proportion p̂ in place of the unknown population proportion p. (first used: 2. The standard error of p̂: plug in what you know)
  • one-proportion z interval — The confidence interval for a population proportion p: p̂ plus or minus z* times the square root of p̂(1 − p̂) divided by n. It is used after checking for a random or representative sample, independent observations, and at least 10 successes and 10 failures in the sample. (first used: 4. The one-proportion z interval: build it by hand and by spreadsheet)
  • conservative p̂ — The planning value p̂ = 0.5, used when the true proportion is unknown. Because 0.5 × 0.5 = 0.25 is the largest value p̂(1 − p̂) can take, it gives the widest margin of error and the largest sample size for a given confidence level. (first used: 6. Read a poll's margin of error)
  • percentage point — One hundredth on the proportion scale, the unit in which a poll's margin of error is reported. A share of 52% with a margin of 3 percentage points runs from 49% to 55%, which is not the same as 3% of 52%. (first used: 6. Read a poll's margin of error)
  • sample-size formula for a proportion — The rule n = (z* ⁄ ME)² × p̂(1 − p̂), which gives how many individuals a study needs for a target margin of error ME at a chosen confidence level. The result is always rounded up to the next whole number. (first used: 7. Choose a sample size before you collect data)
  • undercoverage — A form of bias that arises when some groups in the population have little or no chance of being included in the sample, for example people missing from the list a survey draws its sample from. (first used: 8. What the margin of error does not cover)
  • non-response bias — Bias that arises when many people chosen for a sample cannot be reached or will not answer, and those who respond differ in a systematic way from those who do not. (first used: 8. What the margin of error does not cover)
  • question-wording bias — Bias that arises when the wording or order of a survey question pushes respondents toward a particular answer, so the measured share differs from what people actually think. (first used: 8. What the margin of error does not cover)

Summary

  • An interval for a proportion uses the same skeleton as every interval: estimate plus or minus critical value times standard error.
  • The standard error of the sample proportion is the square root of p̂ times one minus p̂, over n.
  • Check three conditions first: a random or representative sample, independence, and at least 10 successes and 10 failures.
  • The one-proportion z interval uses the standard normal critical value: 1.645, 1.960, or 2.576 for 90%, 95%, or 99%.
  • Interpret every interval as "we are 95% confident that the population proportion … is between … and …"; values inside are plausible, values outside are doubtful.
  • Poll margins are planned with the conservative p̂ of 0.5 and reported in percentage points; quadrupling the sample halves the margin.
  • To choose a sample size, solve the margin formula for n and round up; 1,068 people are enough for a 95% margin of 3 percentage points.
  • The margin of error covers sampling wobble only; voluntary response, undercoverage, non-response, and question wording sit outside it.

Check your understanding

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

  1. A clinic pulls a random sample of 625 patient records and finds that 125 patients missed a follow-up visit. Check the conditions, then find the 95% confidence interval for the share of all the clinic's patients who miss a follow-up, in percentage points, and write the interpretation sentence.
  2. A food truck asks 30 customers whether they would order a vegan option, and 4 say yes. The owner wants a 95% confidence interval for the share of all customers who would order it. Should the owner use the one-proportion z interval? Explain.
  3. A neighborhood association wants to estimate the share of households that would use a community garden, to within ±4 points at 95%, and has no earlier estimate. How many households should it survey?
  4. A TV station invites viewers to text "yes" or "no" to a question and gets 20,000 replies. It reports a margin of error of about 0.7 percentage points. What would you tell the station?

Answers (worked reasoning — read after attempting)

  1. The records are a random sample, each record is a separate patient, and there are 125 successes and 625 − 125 = 500 failures, both at least 10, so the conditions hold. The sample proportion is 125 ⁄ 625 = 0.20, or 20%. The standard error is √(0.20 × 0.80 ⁄ 625) = √(0.16 ⁄ 625), which is 0.4 ⁄ 25 = 0.016, or 1.6 percentage points. The margin of error is 1.96 × 1.6 ≈ 3.1 percentage points. The interval runs from 20 − 3.1 = 16.9% to 20 + 3.1 = 23.1%. We are 95% confident that the population proportion of the clinic's patients who miss a follow-up visit is between 16.9% and 23.1%. A clinic manager can decide whether roughly one patient in five missing a visit is worth a reminder program.
  2. No. With 4 successes and 30 − 4 = 26 failures, the successes fall below 10, so the large-counts rule of thumb fails and a normal model for p̂ is not reasonable. The owner needs a larger sample before using this interval; a sample with at least 10 yes answers and 10 no answers would pass the check.
  3. Use the sample-size formula, solving for n (the number of households), with the conservative p̂ = 0.5: n = (1.96 ⁄ 0.04)² × 0.25. Since 1.96 ⁄ 0.04 = 49, that is 49² × 0.25 = 2,401 × 0.25 = 600.25. Round up to 601 households. Rounding down to 600 would leave the margin a hair above 4 points.
  4. The arithmetic is right: 1.96 × √(0.25 ⁄ 20,000) comes to about 0.7 points. But the poll is a voluntary-response sample: only viewers who chose to text took part, and they can differ sharply from everyone else. The margin of error measures random sampling wobble only, so it says nothing about that bias. A random sample of about 1,000 people with a margin near 3 points would tell the station more.

Next up in this module: the Readings & Resources page, the lecture tutorial, the practice set, and the quiz — and next week the interval idea turns into a test of a claim about a population.

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