Confidence Interval for binomial test in Stata

I don't know if this question makes sense but I'll give it a try..

In Group A, 15 out of 40 patents have a good outcome (outcome can be either good or bad). The comparison group's "good" rate is 0.243.

So I did a binomial test to see if the difference is significant.

Per the one-sided test, it is:



. bitesti 40 15 0.243

N Observed k Expected k Assumed p Observed p
------------------------------------------------------------
40 15 9.72 0.24300 0.37500

Pr(k >= 15) = 0.043511 (one-sided test)
Pr(k <= 15) = 0.979716 (one-sided test)
Pr(k <= 4 or k >= 15) = 0.063801 (two-sided test)


But the 95% Conf. Interval is calculated as follows:


. cii 40 15

-- Binomial Exact --
Variable | Obs Mean Std. Err. [95% Conf. Interval]
-------------+---------------------------------------------------------------
| 40 .375 .0765466 .2272627 .5419852


Shouldn't it be the case that 0.243 should be outside of the 95% Conf. Interval of [.2272627, .5419852]?