Each sample produces one interval p̂ ± z* × SE. Teal bars capture the true p (dashed line); coral bars miss. Over many draws the capture rate converges to the confidence level — that level describes the method, not any one interval.
▬ captures p ▬ misses p ○ = p̂ (sample proportion) (newest on top, up to 120 shown)
Try: p = 0.5, n = 100, 95% — Draw 200 and the capture rate hovers near 95%. Switch to 99%: intervals widen (z* = 2.576) and capture rate climbs. The level is the long-run success rate of the procedure.