How do you estimate the gap between two populations?
You will be able to: Build a two-sample z-interval with an unpooled standard error.
How do you estimate the gap between two populations?
Independent SRSs of 100 students from each of two 5000-student schools find 60 and 40 supporters. The sample gap is .20, but its uncertainty uses both samples.
A useful starting point: How variable is the difference between two sample shares? →
Words and symbols before equations
- Parameters p₁,p₂
- Population support proportions for schools 1 and 2.
- Unpooled SE
- Spread estimated separately from each group’s sample share.
- Difference interval
- A confidence interval for p₁−p₂, not two unrelated single-share intervals.
What this picture assumes
Synthetic study: independent SRSs from two separate populations, each N=100000. The chosen sizes satisfy both 10% conditions. Paired responses require a different method. Actual integer success counts and resulting shares are displayed. This interval uses separate observed proportions (unpooled SE) and requires four observed counts of at least 10.
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.
- 95% interval (0.0642, 0.3358), unpooled SE 0.06928. All values are positive: evidence group 1 has the higher share.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
Name the target p₁−p₂ and keep that order. Check independent SRSs, 100≤500 in each school, and observed success/failure counts 60/40 and 40/60.
SE=√[.60×.40/100+.40×.60/100]≈.06928. The 95% margin is 1.96×SE≈.13579, giving (.0642,.3358).
We are 95% confident school 1’s population support share is about 6.42 to 33.58 percentage points higher than school 2’s. Do not pool the sample shares for this interval; equality has not been assumed. For randomized experiments, verify assignment and group independence and match the conclusion to that design.
A worked example, step by step
Two independent SRSs give 100/200 and 60/200 successes, from large populations. Construct a 95% difference interval.
- p̂₁=.50,p̂₂=.30, so D=.20.
- Assume randomization and 10% conditions are verified; counts 100/100 and 60/140 exceed 10.
- SE=√(.25/200+.21/200)=√.0023≈.04796; ME≈.0940.
- The interval is about (.1060,.2940), estimating p₁−p₂ in the stated order.
Pooling belongs to an equality-null test, not this usual difference interval.
Should a two-proportion interval use one pooled p̂?
Compare with an explanation
No. It estimates spread separately without assuming equal population proportions.
Predict. Change one thing. Explain.
Compare sample sizes and confidence levels. Read each group’s observed count checks and explain why both groups contribute to uncertainty.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
95% interval (0.0642, 0.3358), unpooled SE 0.06928. All values are positive: evidence group 1 has the higher share.
| Group | Successes | Failures | Actual share |
|---|---|---|---|
| 1 | 60 | 40 | 0.6 |
| 2 | 40 | 60 | 0.4 |
Synthetic study: independent SRSs from two separate populations, each N=100000. The chosen sizes satisfy both 10% conditions. Paired responses require a different method. Actual integer success counts and resulting shares are displayed. This interval uses separate observed proportions (unpooled SE) and requires four observed counts of at least 10.
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 questionIndependent suitable samples give p̂₁=.60,p̂₂=.40,n₁=n₂=100. Find a 95% interval and interpret its units.
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Compare with the answer and four-point rubric
- 1 point: D=.20.
- 1 point: Unpooled SE=√.0048≈.06928.
- 1 point: ME≈.13579, giving (.0642,.3358).
- 1 point: The estimated gap is between about 6.42 and 33.58 percentage points in the specified population order.
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 1Why is this SE unpooled?
The interval does not assume equal population proportions.
RECALL 2What are the normal-count checks?
At least 10 observed successes and failures in each group.
RECALL 3Do percentages and percentage points mean the same thing?
No; a difference in shares is expressed in percentage points.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How do you estimate the gap between two populations?
- D=p̂₁−p̂₂.
- SE=√[p̂₁(1−p̂₁)/n₁+p̂₂(1−p̂₂)/n₂].
- Interval=D±z*SE.
Remember: Pooling belongs to an equality-null test, not this usual difference interval.
Conditions: Synthetic study: independent SRSs from two separate populations, each N=100000. The chosen sizes satisfy both 10% conditions. Paired responses require a different method. Actual integer success counts and resulting shares are displayed. This interval uses separate observed proportions (unpooled SE) and requires four observed counts of at least 10.
Refresh Kid · AP Statistics Unit 3 · Objectives 3.10.A, 3.10.B, 3.10.C, 3.10.D · Review edition
Framework, scope and review status
Mapped to College Board, AP Statistics CED, Topic 3.10, objectives 3.10.A, 3.10.B, 3.10.C, 3.10.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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