When is the normal approximation reasonable?
You will be able to: Check random sampling, the 10% condition and expected counts.
When is the normal approximation reasonable?
A school randomly samples 100 of its 3000 students. A normal model can help describe the sample share, but random selection and adequate counts must be checked first.
A useful starting point: How much should a sample percentage vary? →
Words and symbols before equations
- Randomization condition
- Data come from the specified random sampling process.
- 10% condition
- For sampling without replacement, n≤0.10N supports treating observations as approximately independent.
- Expected success count
- np under the population model.
- Expected failure count
- n(1−p).
What this picture assumes
Synthetic study: a simple random sample without replacement from N=100000 independent units. The chosen sizes satisfy the 10% condition. Real studies require their own design checks. The model uses known p; normal tail probabilities are approximate without continuity correction.
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.
- Mean 0.4; SD 0.04899. Approximate P(p̂ ≥ 0.5) = 0.02061; no continuity correction.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
Random selection supports population inference. A large convenience sample still may systematically miss part of the population.
Without replacement, check n≤0.10N. This makes ignoring finite-population dependence reasonable for the simple SD formula; it does not create random selection.
For a normal approximation to p̂, require np≥10 and n(1−p)≥10. For n=100,p=0.04 there are only 4 expected successes, so the normal approximation is not justified by this rule even though n is 100.
A worked example, step by step
An SRS of 200 from N=5000 is modeled with p=0.08. Check the usual conditions.
- The stated SRS satisfies randomization.
- 200≤500 verifies the 10% condition.
- Expected successes=200×0.08=16; expected failures=184.
- Both exceed 10, supporting the approximate normal sampling model.
There is no universal “n≥30” shortcut for a proportion. Check expected successes and failures.
Does passing the 10% condition establish randomness?
Compare with an explanation
No. The selection process must separately be random.
Predict. Change one thing. Explain.
Reduce p or n until the expected-count check fails. Observe that the model stops reporting a normal probability rather than treating a bell curve as automatically valid.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Mean 0.4; SD 0.04899. Approximate P(p̂ ≥ 0.5) = 0.02061; no continuity correction.
| Quantity | Value |
|---|---|
| Population p | 0.4 |
| Estimator mean | 0.4 |
| Sampling SD | 0.04899 |
| Expected successes / failures | 40 / 60 |
| Normal-count check | Pass |
Synthetic study: a simple random sample without replacement from N=100000 independent units. The chosen sizes satisfy the 10% condition. Real studies require their own design checks. The model uses known p; normal tail probabilities are approximate without continuity correction.
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 questionA random sample of 120 from N=900 is modeled with p=0.50. Evaluate both numerical conditions.
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Compare with the answer and four-point rubric
- 1 point: Expected successes and failures are both 60.
- 1 point: Those counts satisfy the normal-count criterion.
- 1 point: But 120 exceeds 10% of 900, which is 90.
- 1 point: The simple independence approximation is not justified by the 10% rule; a finite-population treatment may be needed.
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 1What does random selection support?
Generalization to the sampled population.
RECALL 2Why check the 10% rule?
To limit dependence when sampling without replacement.
RECALL 3What are the two expected counts?
np and n(1−p).
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
When is the normal approximation reasonable?
- Random sample required for the population model.
- Without replacement: n≤0.10N.
- Normal approximation: np≥10 and n(1−p)≥10.
Remember: There is no universal “n≥30” shortcut for a proportion. Check expected successes and failures.
Conditions: Synthetic study: a simple random sample without replacement from N=100000 independent units. The chosen sizes satisfy the 10% condition. Real studies require their own design checks. The model uses known p; normal tail probabilities are approximate without continuity correction.
Refresh Kid · AP Statistics Unit 3 · Objectives 3.2.B · Review edition
Framework, scope and review status
Mapped to College Board, AP Statistics CED, Topic 3.2, objectives 3.2.B. 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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