How many responses are needed for a target margin?
You will be able to: Plan and round up a sample size for a desired margin of error.
How many responses are needed for a target margin?
A student council wants a poll with a margin near 5 percentage points at 95% confidence. Planning should happen before the poll is collected.
A useful starting point: How do you estimate a population share with a range? →
Words and symbols before equations
- Planning value p*
- A plausible proportion from prior information used to choose n.
- Target margin m
- Desired half-width, written as a proportion.
- Conservative planning
- Using p*=0.50 when no credible prior estimate is available.
- Round up
- Choose the next integer so the target bound is not exceeded.
What this picture assumes
This plans the normal interval using a chosen prior proportion. Use .50 without a defensible prior estimate. Round up; separately account for nonresponse, population size and design.
Read the picture in three steps
- Read the axes and labels first. Identify what each symbol and line represents. Read the units and fixed conditions before comparing quantities.
- Plan at least 385 completed random observations for 95% confidence and ±5 percentage points using p*=0.5. This does not account for nonresponse or design bias.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
Rearrange m=z*√[p*(1−p*)/n] to n≥(z*)²p*(1−p*)/m². This plans the usual normal interval; it does not guarantee the realized sample result or cure design bias.
At 95% with m=.05 and p*=.50, n≥1.96²×.25/.05²=384.16, so choose 385. A margin of 5 percentage points is .05, not 5.
The product p*(1−p*) is largest at .50, making that choice conservative. Halving the target margin requires about four times as many observations, all else fixed. Check the eventual design and finite population too.
A worked example, step by step
Plan for 95% confidence, margin .04 and prior p*=.30, using z*=1.96.
- Convert 4 percentage points to .04.
- Compute n≥1.96²×.30×.70/.04².
- The result is 504.21.
- Round up to 505 completed observations; plan separately for nonresponse and check population size.
Rounding down may miss the requested margin. More invitations are not the same as more completed random observations.
Why choose .50 without prior information?
Compare with an explanation
It maximizes p*(1−p*) and therefore gives the largest planned sample size.
Predict. Change one thing. Explain.
Halve the margin with the confidence level and planning value fixed. Then compare p*=.50 with .30. Explain the change in required n.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Plan at least 385 completed random observations for 95% confidence and ±5 percentage points using p*=0.5. This does not account for nonresponse or design bias.
| Planning quantity | Value |
|---|---|
| Critical z* | 1.95996 |
| Unrounded n | 384.1454 |
| Round up to completed observations | 385 |
| Population needed for 10% condition | 3850 |
This plans the normal interval using a chosen prior proportion. Use .50 without a defensible prior estimate. Round up; separately account for nonresponse, population size and design.
Explain what you noticed: Answer the investigation prompt above. State one observation and explain it using the probability values, reference groups, graph scales or model assumptions. Identify what the representation cannot tell you.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionUsing z*=2 and p*=.50, plan n for margin .05 and then .025.
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Compare with the answer and four-point rubric
- 1 point: For .05: n≥4×.25/.0025=400.
- 1 point: For .025: n≥4×.25/.000625=1600.
- 1 point: Halving the margin quadruples n.
- 1 point: These are planning calculations assuming appropriate random collection and the normal-interval conditions.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1Which scale should m use?
A proportion, such as .05 for 5 percentage points.
RECALL 2How should n be rounded?
Up to an integer.
RECALL 3Does planned precision remove nonresponse bias?
No.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How many responses are needed for a target margin?
- n≥(z*)²p*(1−p*)/m²; round upward.
- Without a planning estimate use p*=.50.
- Half the margin requires about four times n.
Remember: Rounding down may miss the requested margin. More invitations are not the same as more completed random observations.
Conditions: This plans the normal interval using a chosen prior proportion. Use .50 without a defensible prior estimate. Round up; separately account for nonresponse, population size and design.
Refresh Kid · AP Statistics Unit 3 · Objectives 3.3.D · Review edition
Framework, scope and review status
Mapped to College Board, AP Statistics CED, Topic 3.3, objectives 3.3.D. Framework effective Fall 2026, checked September 17, 2026. Unit 3 includes inference for one and two population proportions, errors and power, and chi-square homogeneity/independence tests; it is part of the revised five-unit course.
Examples and datasets are synthetic, independently authored teaching material. Inference requires a justified design and appropriate counts. Intervals use observed proportions; null tests use their reference-model proportions. The revised CED states chi-square expected counts should be greater than 5; this unit follows that wording even though some companion texts use at least 5. Normal and chi-square inference are approximate. The chi-square goodness-of-fit test is not included in this unit’s official scope.
The Organic Chemistry Tutor companion title and destination were located; the full video was not reviewed. Khan Academy’s destination was checked, but its lesson content was not fully readable by the research tool. OpenStax provides optional reference reading. No provider scripts, questions or graphics were copied. Refresh Kid is not affiliated with these providers.
GitHub’s 3D website collection informed optional spatial inspection. Our original categorical count grid uses self-hosted Three.js with its MIT license. Two rows and three columns organize synthetic people into categorical cells. Each stacked block represents one person. Camera rotation changes only the view; exact counts, expectations and contributions are always available in the 2D table. Inference curves and intervals remain 2D; use the exact table rather than apparent 3D size for comparisons. Complete labeled diagrams, count tables and explanations remain available without 3D.
Independent teacher review and observation of students remain pending. Technical checks do not certify statistical accuracy, accessibility or learning effectiveness. This is a review edition.
Released AP Statistics questions and scoring guides are optional. Older exams use the earlier framework, so check alignment before selecting parts. All practice on this page is original, not official AP material.
Learn → Explore → Practice → Review is informed by the IES learning guide. This implementation has not yet been evaluated with learners.
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