Oh, it turns out that a fairly reasonable confidence interval for
these data is obtained by using the logit:
. logit outcome [fweight = freq]
------------------------------------------------------------------------------
outcome | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
_cons | -2.351375 .7400129 -3.18 0.001 -3.801774 -.9009767
------------------------------------------------------------------------------
Just transform this confidence interval with the inverse logit.
Oh, one more tidbit:
. ci outcome, binomial wald
-- Binomial Wald ---
Variable | Obs Mean Std. Err. [95% Conf. Interval]
-------------+---------------------------------------------------------------
outcome | 23 .0869565 .0587534 0 .202111*
(*) The Wald interval was clipped at the lower endpoint
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