How do you estimate a population share with a range?
You will be able to: Construct a one-sample z-interval after checking its conditions.
How do you estimate a population share with a range?
In a simple random sample of 100 from 5000 club members, 60 support a weekend session. The estimate 60% is useful, but it should include sampling uncertainty.
A useful starting point: When is the normal approximation reasonable? →
Words and symbols before equations
- Confidence interval
- A data-based range of plausible parameter values from a specified procedure.
- Standard error SE
- Estimated standard deviation of the statistic’s sampling distribution.
- Critical value z*
- Standard-normal cutoff enclosing the chosen central confidence percentage.
- Margin of error ME
- z*×SE, half the interval width.
What this picture assumes
Synthetic study: a simple random sample without replacement from N=100000 independent units. The chosen sizes satisfy the 10% condition. Real studies require their own design checks. Success counts are rounded to integers from the requested percentage; the actual x/n is shown. The usual Wald interval requires at least 10 observed successes and failures. Mathematical endpoints are not silently clipped.
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 (0.504, 0.696). SE 0.04899; margin 0.09602. Coverage is approximate.
- 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 p as the proportion of all 5000 club members supporting the weekend session. A one-sample z-interval estimates this binary-response proportion.
Check the SRS, 100≤500, and observed counts 60 and 40, both at least 10. For an interval, use observed counts and p̂, because p is unknown.
The estimate is 0.60; SE=√(0.6×0.4/100)≈0.0490. At 95% confidence z*≈1.96, so ME≈0.0960 and the interval is (0.5040,0.6960). Explain the range in the population context.
A worked example, step by step
An SRS of 200 from 10000 students gives 80 yes responses. Build a 95% z-interval.
- Define p as the true yes-response proportion in those 10000 students; p̂=80/200=0.40.
- Random sampling is stated, 200≤1000, and counts 80 and 120 exceed 10.
- SE=√(0.4×0.6/200)≈0.03464; ME=1.96×SE≈0.06790.
- The interval is approximately (0.3321,0.4679). We are 95% confident it contains the population yes-response proportion.
An interval’s SE uses p̂. A test about a specified p₀ uses the null value instead.
What does a 0.096 margin mean here?
Compare with an explanation
9.6 percentage points on either side of the point estimate.
Predict. Change one thing. Explain.
Change the success count and confidence level. Read the conditions, center, SE and endpoints. Try a sparse count and explain why the normal interval is withheld.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
95% interval (0.504, 0.696). SE 0.04899; margin 0.09602. Coverage is approximate.
| Quantity | Value |
|---|---|
| Observed successes x | 60 |
| Observed failures n−x | 40 |
| Actual p̂=x/n | 0.6 |
Synthetic study: a simple random sample without replacement from N=100000 independent units. The chosen sizes satisfy the 10% condition. Real studies require their own design checks. Success counts are rounded to integers from the requested percentage; the actual x/n is shown. The usual Wald interval requires at least 10 observed successes and failures. Mathematical endpoints are not silently clipped.
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 questionAn SRS of 400 from N=20000 contains 200 yes responses. Construct a 95% interval and state its target.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: The target is the population yes-response proportion; p̂=.50.
- 1 point: SRS, 400≤2000 and 200 successes/200 failures satisfy conditions.
- 1 point: SE=.025 and ME≈1.96×.025=.049.
- 1 point: The interval (.451,.549) estimates the population share, not the fraction of future individuals inside a range.
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 SE estimate?
The sampling standard deviation of p̂.
RECALL 2What makes an interval wider?
A larger critical value or larger SE.
RECALL 3What population detail belongs in the answer?
The response of interest and the population whose proportion is estimated.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How do you estimate a population share with a range?
- p̂=x/n.
- SE=√[p̂(1−p̂)/n].
- Interval=p̂±z*SE; check observed successes and failures.
Remember: An interval’s SE uses p̂. A test about a specified p₀ uses the null value instead.
Conditions: Synthetic study: a simple random sample without replacement from N=100000 independent units. The chosen sizes satisfy the 10% condition. Real studies require their own design checks. Success counts are rounded to integers from the requested percentage; the actual x/n is shown. The usual Wald interval requires at least 10 observed successes and failures. Mathematical endpoints are not silently clipped.
Refresh Kid · AP Statistics Unit 3 · Objectives 3.3.A, 3.3.B, 3.3.C, 3.3.D · Review edition
Framework, scope and review status
Mapped to College Board, AP Statistics CED, Topic 3.3, objectives 3.3.A, 3.3.B, 3.3.C, 3.3.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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