How do I explore?

1. Pick a sample size and a confidence level.
2. Press Sample (or Sample 100) to draw samples from the population. Each sample builds one confidence interval for the proportion on the right.
3. Press Clear plot to start over, or New population for a new true proportion.

What should I notice?

• About what fraction of the intervals capture the true proportion (the blue line)? How does it compare to the confidence level you chose?
• What happens to the width of the intervals when you make the sample size bigger? When you raise the confidence level?

Where is the data?

This sim is self-contained — the data is right here on the plot. Each confidence interval is one line, coloured by whether it captured the true proportion (the blue line). The running “X of N capture (Z%)” at the top is the capture rate.

Primary standard

• HSS-IC.B.4 / AP Statistics (UNC-4) — interpret a confidence level as the long-run capture rate of the interval procedure, here for a categorical (success/failure) variable.

Design intent

• The true proportion is known and fixed, so students can see which intervals miss — something impossible with a single real sample. The confidence level sets the long-run capture rate, not the certainty of any one interval.
• Interval width responds to sample size (∝ 1/√n) and to the confidence level, so students feel the precision-versus-confidence trade-off directly.

Discussion prompts

• If 95% confidence gives about 95% capture, why can't we say a single interval has a 95% chance of containing p?
• Why do higher-confidence intervals get wider, not narrower?

Suggested activity flow

• Observe first: draw a few samples one at a time; predict, then check whether each captures p.
• Then Sample 100 and read the capture rate. Change the sample size and the confidence level and predict the effect before you test it.

Assumptions and simplifications

• Each sample draws n success/failure outcomes; the interval is the one-proportion z (Wald) interval p̂ ± z*·√(p̂(1−p̂)/n) with z* for the chosen level.
• The Wald interval is the AP-standard method but slightly under-covers — 95% confidence often captures ~92–94%, most visibly at small n or p near 0 or 1. That gap is itself worth discussing: the guarantee for proportions is approximate.