What does zero tell you about a difference interval?
You will be able to: Interpret signed intervals and judge whether zero is plausible.
What does zero tell you about a difference interval?
An interval for school A minus school B is (.03,.15). Every included value is positive, so the interval supports higher population support in A.
A useful starting point: How do you estimate the gap between two populations? →
Words and symbols before equations
- Zero difference
- Equal population proportions.
- Signed interval
- An interval whose direction depends on the chosen group order.
- Compatible claim
- A parameter value lying within the plausible interval range.
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
For A minus B, (.03,.15) means A’s share is estimated to be 3 to 15 percentage points higher at the stated confidence level. Reversing the order gives (−.15,−.03).
An interval (−.04,.10) includes zero and both directions, so it does not establish a population difference. It also does not prove equality.
Confidence refers to approximate repeated coverage of the fixed pA−pB by the method. Do not say 95% of students have individual differences in this interval. An unpooled Wald interval and a pooled z-test are not exact inversions, so their borderline decisions can differ.
A worked example, step by step
Interpret a 95% interval (−.12,−.02) for pA−pB and state the interval for pB−pA.
- Define both population proportions and their shared binary response.
- All values are negative, supporting a lower share in A.
- We are 95% confident A’s share is 2 to 12 percentage points lower than B’s.
- Reverse and negate the endpoints to obtain (.02,.12) for B minus A.
An interval including zero means insufficient evidence of a difference from that interval, not demonstrated equality.
Can a positive sample gap have an interval containing zero?
Compare with an explanation
Yes. Sampling uncertainty can include zero and negative population gaps.
Predict. Change one thing. Explain.
Change the group counts until the interval crosses zero. Explain how the appropriate claim changes while the sample difference can remain positive.
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 questionA 90% interval for p₁−p₂ is (.01,.07). Interpret it and say what repeated confidence means.
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Compare with the answer and four-point rubric
- 1 point: The target is the ordered population response-proportion difference.
- 1 point: We are 90% confident group 1’s share is 1 to 7 percentage points higher.
- 1 point: Zero is excluded, supporting a positive population difference by this interval.
- 1 point: About 90% of intervals from repeated samples would capture the fixed difference under a well-performing procedure.
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 zero represent?
Equal population proportions.
RECALL 2What happens to endpoints when order reverses?
They are negated and swapped.
RECALL 3Does an interval establish causation?
Only an appropriate experimental design can support a causal claim; the interval alone cannot.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
What does zero tell you about a difference interval?
- All positive → group 1 higher; all negative → group 1 lower.
- Zero inside → difference not established by the interval.
- Reverse (a,b) to (−b,−a) when reversing group order.
Remember: An interval including zero means insufficient evidence of a difference from that interval, not demonstrated equality.
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 3.11.A, 3.11.B · Review edition
Framework, scope and review status
Mapped to College Board, AP Statistics CED, Topic 3.11, objectives 3.11.A, 3.11.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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