What does 95% confidence actually describe?
You will be able to: Interpret confidence, assess a claim and explain interval-width tradeoffs.
What does 95% confidence actually describe?
Imagine repeating the same random poll many times. Each sample produces a different interval, while the true population share stays fixed.
A useful starting point: How many responses are needed for a target margin? →
Words and symbols before equations
- Coverage
- Whether a calculated interval contains the fixed parameter.
- Confidence level
- Approximate long-run coverage rate of the interval procedure under its conditions.
- Plausible value
- A parameter value compatible with the calculated interval.
- Precision
- Narrowness of an estimate’s interval, conditional on its method and assumptions.
What this picture assumes
100 independently simulated samples from a Bernoulli population with p=.50. All calculated Wald intervals enter the displayed coverage; those failing observed-count conditions are flagged. This is an illustration of approximate coverage, not a proof of nominal accuracy.
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.
- 100/100 calculated intervals contain p=.50. 0 intervals fail observed-count checks and are flagged in the full table. Nominal confidence 95% is not a guaranteed finite-run coverage percentage.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
A 95% procedure captures the fixed population parameter in about 95% of repeated intervals when the method works well. A particular computed interval either covers p or does not; 95% is not a probability assigned to a random population parameter in this framework.
If a valid 95% interval for a support share is (.54,.66), all included values exceed .50, supporting a majority claim at that confidence level. If it is (.46,.58), a majority is not established by the interval, although it is possible.
For the same sample, raising confidence increases z* and width. With confidence and approximate share fixed, increasing n decreases width roughly as 1/√n. Neither wider confidence nor larger n removes bias. Wald z-interval coverage is approximate, especially near boundaries.
A worked example, step by step
Interpret a 90% interval (.42,.54) for the proportion of all district families wanting an earlier bus.
- Name the target: the district-family preference proportion.
- State 90% confidence that the interval .42 to .54 contains that share.
- Because .50 lies inside, this interval does not establish that a majority wants the change.
- In repetitions, about 90% of intervals from this method capture the parameter, subject to its conditions; it is not a statement about 90% of families.
Confidence is about the method’s repeated coverage, not the percentage of individual observations inside the interval.
Does an interval containing .50 prove p=.50?
Compare with an explanation
No. It includes .50 among other plausible values.
Predict. Change one thing. Explain.
Keep the seed fixed and compare 90%,95%,99% intervals for the same repeated samples. Count coverage and misses. Explain why 100 simulated intervals need not have exactly the nominal percentage.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
100/100 calculated intervals contain p=.50. 0 intervals fail observed-count checks and are flagged in the full table. Nominal confidence 95% is not a guaranteed finite-run coverage percentage.
| Run | Lower | Upper | Contains .50? | Observed-count check |
|---|---|---|---|---|
| 1 | 0.333 | 0.527 | Yes | Pass |
| 2 | 0.3427 | 0.5373 | Yes | Pass |
| 3 | 0.3821 | 0.5779 | Yes | Pass |
| 4 | 0.333 | 0.527 | Yes | Pass |
| 5 | 0.3722 | 0.5678 | Yes | Pass |
| 6 | 0.3525 | 0.5475 | Yes | Pass |
| 7 | 0.412 | 0.608 | Yes | Pass |
| 8 | 0.3821 | 0.5779 | Yes | Pass |
| 9 | 0.392 | 0.588 | Yes | Pass |
| 10 | 0.3233 | 0.5167 | Yes | Pass |
| 11 | 0.3821 | 0.5779 | Yes | Pass |
| 12 | 0.4627 | 0.6573 | Yes | Pass |
| 13 | 0.4221 | 0.6179 | Yes | Pass |
| 14 | 0.402 | 0.598 | Yes | Pass |
| 15 | 0.402 | 0.598 | Yes | Pass |
| 16 | 0.402 | 0.598 | Yes | Pass |
| 17 | 0.4525 | 0.6475 | Yes | Pass |
| 18 | 0.4627 | 0.6573 | Yes | Pass |
| 19 | 0.473 | 0.667 | Yes | Pass |
| 20 | 0.3722 | 0.5678 | Yes | Pass |
| 21 | 0.3623 | 0.5577 | Yes | Pass |
| 22 | 0.3821 | 0.5779 | Yes | Pass |
| 23 | 0.4525 | 0.6475 | Yes | Pass |
| 24 | 0.412 | 0.608 | Yes | Pass |
| 25 | 0.402 | 0.598 | Yes | Pass |
| 26 | 0.4423 | 0.6377 | Yes | Pass |
| 27 | 0.3623 | 0.5577 | Yes | Pass |
| 28 | 0.4423 | 0.6377 | Yes | Pass |
| 29 | 0.402 | 0.598 | Yes | Pass |
| 30 | 0.3623 | 0.5577 | Yes | Pass |
| 31 | 0.4423 | 0.6377 | Yes | Pass |
| 32 | 0.3821 | 0.5779 | Yes | Pass |
| 33 | 0.4423 | 0.6377 | Yes | Pass |
| 34 | 0.3427 | 0.5373 | Yes | Pass |
| 35 | 0.4221 | 0.6179 | Yes | Pass |
| 36 | 0.412 | 0.608 | Yes | Pass |
| 37 | 0.392 | 0.588 | Yes | Pass |
| 38 | 0.4627 | 0.6573 | Yes | Pass |
| 39 | 0.402 | 0.598 | Yes | Pass |
| 40 | 0.4833 | 0.6767 | Yes | Pass |
| 41 | 0.333 | 0.527 | Yes | Pass |
| 42 | 0.4936 | 0.6864 | Yes | Pass |
| 43 | 0.412 | 0.608 | Yes | Pass |
| 44 | 0.3722 | 0.5678 | Yes | Pass |
| 45 | 0.4423 | 0.6377 | Yes | Pass |
| 46 | 0.4833 | 0.6767 | Yes | Pass |
| 47 | 0.392 | 0.588 | Yes | Pass |
| 48 | 0.4423 | 0.6377 | Yes | Pass |
| 49 | 0.3722 | 0.5678 | Yes | Pass |
| 50 | 0.3623 | 0.5577 | Yes | Pass |
| 51 | 0.4322 | 0.6278 | Yes | Pass |
| 52 | 0.3821 | 0.5779 | Yes | Pass |
| 53 | 0.412 | 0.608 | Yes | Pass |
| 54 | 0.412 | 0.608 | Yes | Pass |
| 55 | 0.3821 | 0.5779 | Yes | Pass |
| 56 | 0.3722 | 0.5678 | Yes | Pass |
| 57 | 0.4833 | 0.6767 | Yes | Pass |
| 58 | 0.4221 | 0.6179 | Yes | Pass |
| 59 | 0.4525 | 0.6475 | Yes | Pass |
| 60 | 0.402 | 0.598 | Yes | Pass |
| 61 | 0.3722 | 0.5678 | Yes | Pass |
| 62 | 0.402 | 0.598 | Yes | Pass |
| 63 | 0.3722 | 0.5678 | Yes | Pass |
| 64 | 0.473 | 0.667 | Yes | Pass |
| 65 | 0.3427 | 0.5373 | Yes | Pass |
| 66 | 0.3623 | 0.5577 | Yes | Pass |
| 67 | 0.4423 | 0.6377 | Yes | Pass |
| 68 | 0.3233 | 0.5167 | Yes | Pass |
| 69 | 0.3525 | 0.5475 | Yes | Pass |
| 70 | 0.3233 | 0.5167 | Yes | Pass |
| 71 | 0.333 | 0.527 | Yes | Pass |
| 72 | 0.4627 | 0.6573 | Yes | Pass |
| 73 | 0.392 | 0.588 | Yes | Pass |
| 74 | 0.473 | 0.667 | Yes | Pass |
| 75 | 0.3821 | 0.5779 | Yes | Pass |
| 76 | 0.4423 | 0.6377 | Yes | Pass |
| 77 | 0.412 | 0.608 | Yes | Pass |
| 78 | 0.392 | 0.588 | Yes | Pass |
| 79 | 0.4221 | 0.6179 | Yes | Pass |
| 80 | 0.3136 | 0.5064 | Yes | Pass |
| 81 | 0.392 | 0.588 | Yes | Pass |
| 82 | 0.412 | 0.608 | Yes | Pass |
| 83 | 0.4322 | 0.6278 | Yes | Pass |
| 84 | 0.3623 | 0.5577 | Yes | Pass |
| 85 | 0.4627 | 0.6573 | Yes | Pass |
| 86 | 0.3525 | 0.5475 | Yes | Pass |
| 87 | 0.4423 | 0.6377 | Yes | Pass |
| 88 | 0.4423 | 0.6377 | Yes | Pass |
| 89 | 0.3722 | 0.5678 | Yes | Pass |
| 90 | 0.3623 | 0.5577 | Yes | Pass |
| 91 | 0.412 | 0.608 | Yes | Pass |
| 92 | 0.3427 | 0.5373 | Yes | Pass |
| 93 | 0.4423 | 0.6377 | Yes | Pass |
| 94 | 0.3821 | 0.5779 | Yes | Pass |
| 95 | 0.4423 | 0.6377 | Yes | Pass |
| 96 | 0.4322 | 0.6278 | Yes | Pass |
| 97 | 0.4322 | 0.6278 | Yes | Pass |
| 98 | 0.402 | 0.598 | Yes | Pass |
| 99 | 0.4322 | 0.6278 | Yes | Pass |
| 100 | 0.3427 | 0.5373 | Yes | Pass |
100 independently simulated samples from a Bernoulli population with p=.50. All calculated Wald intervals enter the displayed coverage; those failing observed-count conditions are flagged. This is an illustration of approximate coverage, not a proof of nominal accuracy.
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 questionExplain a 95% interval (.48,.59) for a city preference share, assess a majority claim, and describe quadrupling n.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: We are 95% confident the interval contains the city’s true preference share.
- 1 point: Repeated intervals from a well-performing method capture the fixed share about 95% of the time.
- 1 point: Since .50 is included, a majority is not established by this interval.
- 1 point: Quadrupling n roughly halves the margin at the same confidence and share, assuming an appropriate design.
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 varies in repeated confidence intervals?
The sampled data and interval endpoints.
RECALL 2What stays fixed?
The population parameter.
RECALL 3Does observed simulation coverage have to equal 95%?
No; it fluctuates and normal-method coverage is approximate.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
What does 95% confidence actually describe?
- Higher confidence → wider interval for fixed data.
- Larger n → narrower interval, other things equal.
- Check a claim against the whole plausible range.
Remember: Confidence is about the method’s repeated coverage, not the percentage of individual observations inside the interval.
Conditions: 100 independently simulated samples from a Bernoulli population with p=.50. All calculated Wald intervals enter the displayed coverage; those failing observed-count conditions are flagged. This is an illustration of approximate coverage, not a proof of nominal accuracy.
Refresh Kid · AP Statistics Unit 3 · Objectives 3.4.A, 3.4.B, 3.4.C · Review edition
Framework, scope and review status
Mapped to College Board, AP Statistics CED, Topic 3.4, objectives 3.4.A, 3.4.B, 3.4.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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