Table 1.

Behavior of the Agresti–Coull confidence interval width ${\stackrel{\sim }{W}}_{\text{AC}}$ as compared to the fixed value w₀ at ${\pi }_0=\frac{1}{2}$ and α = 0.05.

Sample size	w0	$P\left({\stackrel{\sim }{W}}_{\mathrm{AC}}<w_0|{\pi }_0=\frac{1}{2}\right)$	$E\left({\stackrel{\sim }{W}}_{\text{AC}}|{\pi }_0=\frac{1}{2}\right)$	n*	CPn	${\text{CP}}_{n^{\ast }}$
10	0.5268	0.7539	0.5125	9	0.9783	0.9621
11	0.5088	1.0000	0.4955	10	0.9342	0.9794
100	0.1923	0.9204	0.1914	99	0.9439	0.9564
101	0.1914	1.0000	0.1905	100	0.9545	0.9438
200	0.1373	0.9437	0.1369	199	0.9454	0.9536
201	0.1369	1.0000	0.1366	200	0.9518	0.9449
1000	0.0619	0.9748	0.0618	999	0.9458	0.9494
1001	0.0618	1.0000	0.0618	1000	0.9510	0.9466
2000	0.0438	0.9822	0.0438	1999	0.9478	0.9514
2001	0.0438	1.0000	0.0438	2000	0.9510	0.9488

Notes: n*, suggested sample size based on $E\left({\stackrel{\sim }{W}}_{\text{AC}}|{\pi }_0=\frac{1}{2}\right)>w_0$; CP_n, coverage probability for sample size n; ${\text{CP}}_{n^{\ast }}$, coverage probability for sample size n*.