How do sample size and confidence affect precision?
You will be able to: Explain interval width through the margin of error.
How do sample size and confidence affect precision?
A club can survey 25 or 100 randomly selected students. With the same observed spread, the larger sample gives a more precise estimate of average homework time.
A useful starting point: What does confidence say about the population average? →
Words and symbols before equations
- Precision
- How narrow the estimate’s interval is, conditional on the model.
- Width
- Upper endpoint minus lower endpoint, equal to 2ME.
- Confidence tradeoff
- More coverage requires a wider interval with fixed data.
- Planning spread
- A prior estimate of SD used to think about sample size.
What this picture assumes
Synthetic summaries, with positive sample SDs. Random samples without replacement use source populations of N=100000 each, satisfying 10% for these sizes. A randomized experiment does not require that sampling-fraction check. Design and shape are assumptions chosen here, not conclusions that summary statistics can verify. Strong skewness/outlier selection conservatively withholds inference at all sizes; real data require examination.
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 (19.9361, 24.0639) minutes; margin 2.0639 minutes. Chosen design, sampling fraction where applicable, and shape/size assumptions support the procedure.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
At fixed n and s, increasing confidence increases t*, ME and width. The extra coverage costs precision.
Holding s fixed, increasing n decreases s/√n. It also increases df and usually reduces t*. Thus width falls a little more than the square-root rule alone predicts for small samples.
The familiar “four times n halves the margin” is approximate for t intervals because t* also changes. Real new samples can have different s. More observations cannot repair a biased collection method.
A worked example, step by step
Compare SE for n=25 and n=100 with s=10 minutes.
- At n=25, SE=10/5=2 minutes.
- At n=100, SE=10/10=1 minute.
- The SE halves exactly when n quadruples and s is held fixed.
- At the same confidence, the t margin falls slightly more because df also increases.
A narrower biased estimate is still biased. Precision and trustworthy sampling are separate.
Does higher confidence make a fixed-data interval narrower?
Compare with an explanation
No. It makes it wider.
Predict. Change one thing. Explain.
Compare n=25 and n=100 at 95% confidence. Then hold n fixed and raise confidence. Use the readout to distinguish the SE effect from the critical-value effect.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
95% interval (19.9361, 24.0639) minutes; margin 2.0639 minutes. Chosen design, sampling fraction where applicable, and shape/size assumptions support the procedure.
| Quantity | Value |
|---|---|
| Estimate (minutes) | 22 |
| SE (minutes) | 1 |
| Degrees of freedom | 24 |
| t* | 2.0639 |
Synthetic summaries, with positive sample SDs. Random samples without replacement use source populations of N=100000 each, satisfying 10% for these sizes. A randomized experiment does not require that sampling-fraction check. Design and shape are assumptions chosen here, not conclusions that summary statistics can verify. Strong skewness/outlier selection conservatively withholds inference at all sizes; real data require examination.
Explain what you noticed: Answer the investigation prompt above. State one observation and explain it using the means, standard errors, pairing, 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 two ways to reduce ME and one reason a larger sample may not deliver the expected reduction.
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Compare with the answer and four-point rubric
- 1 point: Increase n, with other features held comparable.
- 1 point: Lower the confidence level, acknowledging reduced coverage.
- 1 point: ME depends on both SE and t*.
- 1 point: The new sample SD can differ; design problems also remain relevant.
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 1How is width related to ME?
Width=2ME.
RECALL 2Does mean shift alone change width?
No, holding s,n and confidence fixed.
RECALL 3Why is the t square-root margin rule approximate?
The critical value also changes with df.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How do sample size and confidence affect precision?
- Width=2t*s/√n.
- Higher confidence → larger t*.
- With s fixed, SE is proportional to 1/√n.
Remember: A narrower biased estimate is still biased. Precision and trustworthy sampling are separate.
Conditions: Synthetic summaries, with positive sample SDs. Random samples without replacement use source populations of N=100000 each, satisfying 10% for these sizes. A randomized experiment does not require that sampling-fraction check. Design and shape are assumptions chosen here, not conclusions that summary statistics can verify. Strong skewness/outlier selection conservatively withholds inference at all sizes; real data require examination.
Refresh Kid · AP Statistics Unit 4 · Objectives 4.3.C · Review edition
Framework, scope and review status
Mapped to College Board, AP Statistics CED, Topic 4.3, objectives 4.3.C. Framework effective Fall 2026, checked September 17, 2026. Unit 4 includes sampling distributions of means, one-sample and paired t inference, and independent two-sample t inference; it is part of the revised five-unit course.
Examples and datasets are synthetic, independently authored teaching material. Mean inference requires a justified design and suitable shape or sample size. Paired analysis uses one sample of differences. Independent two-sample inference uses separate variance estimates and technology-computed Welch degrees of freedom. Extreme skewness and influential observations need attention even in larger samples. This model conservatively withholds inference when those warnings are selected. Conclusions are limited by random sampling and/or assignment as appropriate.
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 paired-data display uses self-hosted Three.js with its MIT license. Two measurement columns are connected within each labeled student lane; horizontal position is before/after, vertical position is time, and depth separates identities rather than representing a numerical variable. Rotation can separate overlapping connectors. Exact values and differences always remain in the 2D table and labeled plot. The broken-matching option is an explicit counterexample, not a legitimate alternative analysis. No autoplay; complete teaching remains 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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