Are two groups really independent comparisons?
You will be able to: Define a two-proportion equality test and justify its design.
Are two groups really independent comparisons?
Two schools are sampled independently to compare preference shares. Asking the same students twice would instead create paired data.
A useful starting point: What does zero tell you about a difference interval? →
Words and symbols before equations
- Two-sample z-test
- A normal-approximation test comparing two independent population proportions.
- Equality null
- H₀:p₁−p₂=0.
- Pooled proportion p̂c
- Total successes divided by total observations under an equality assumption.
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. Equality null p₁=p₂. The normal test uses the pooled share and four pooled expected counts of at least 10. Zero estimated SE makes the z-test unavailable.
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.
- D=0.2; z=2.8284; p-value=0.002339. At α=0.05, reject equal population shares. Alternative p₁ > p₂.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
Define p₁ and p₂ in context. Choose H₀:p₁=p₂ and an alternative >,< or ≠ from the question before inspecting outcomes.
Require independent random samples or an appropriate randomized experiment. Check 10% separately for samples without replacement; that finite-population sampling check is not needed simply because treatments were randomly assigned in an experiment.
For the equality-null z-test, check expected counts using the pooled p̂c: n₁p̂c,n₁(1−p̂c),n₂p̂c,n₂(1−p̂c), all at least 10. Paired responses do not satisfy independent-group assumptions.
A worked example, step by step
Before independent samples, a district asks whether school A’s support exceeds B’s. State the hypotheses and design checks.
- Define pA and pB as each school’s population support share.
- H₀:pA−pB=0 and Hₐ:pA−pB>0.
- Verify independent random selection and the 10% condition for both sampled populations.
- Check four pooled expected counts after collecting data before using the usual normal approximation.
A large number of paired observations does not turn them into independent groups.
Does different sample size invalidate a two-proportion test?
Compare with an explanation
No. The groups may have different sizes if the design and count conditions hold.
Predict. Change one thing. Explain.
Switch the alternative while keeping school A first. Explain the direction of evidence and why changing the labels reverses the sign.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
D=0.2; z=2.8284; p-value=0.002339. At α=0.05, reject equal population shares. Alternative p₁ > p₂.
| Group | Successes | Failures | Actual share |
|---|---|---|---|
| 1 | 60 | 40 | 0.6 |
| 2 | 40 | 60 | 0.4 |
| Null quantity | Value |
|---|---|
| Pooled share | 0.5 |
| Four expected counts | 50, 50, 50, 50 |
| Pooled SE | 0.07071 |
Synthetic study: independent SRSs from two separate populations, each N=100000. The chosen sizes satisfy both 10% conditions. Paired responses require a different method. Equality null p₁=p₂. The normal test uses the pooled share and four pooled expected counts of at least 10. Zero estimated SE makes the z-test unavailable.
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 randomized experiment assigns different volunteers to A or B, with binary success. State an appropriate comparison and a limit on generalization.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: Define the success probabilities under the two treatments for the relevant participants or target model.
- 1 point: Test equality versus the pre-specified direction with a suitable two-proportion method.
- 1 point: Verify random assignment, independence and pooled expected counts.
- 1 point: Random assignment supports a causal comparison for the study; a volunteer sample alone does not ensure broad population representation.
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 specify group order?
It fixes the sign and tail direction.
RECALL 2What count check does the equality test use?
Expected successes and failures computed from the pooled share.
RECALL 3Does random assignment guarantee representativeness?
No.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Are two groups really independent comparisons?
- H₀:p₁=p₂, or p₁−p₂=0.
- Pre-specify Hₐ and group order.
- Use pooled expected counts for the equality test.
Remember: A large number of paired observations does not turn them into independent groups.
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. Equality null p₁=p₂. The normal test uses the pooled share and four pooled expected counts of at least 10. Zero estimated SE makes the z-test unavailable.
Refresh Kid · AP Statistics Unit 3 · Objectives 3.12.A, 3.12.B, 3.12.C · Review edition
Framework, scope and review status
Mapped to College Board, AP Statistics CED, Topic 3.12, objectives 3.12.A, 3.12.B, 3.12.C. 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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