5. Discussion
As is common in the literature, we have discussed externally randomized and data-randomized intervals separately. It is
the author’s opinion that the distinction between these types of intervals is entirely artificial. The only difference between
an externally randomized split sample interval and its data-randomized counterpart is whether the order in which the trials
are recorded is determined after or before the experiment is concluded. Indeed, it is wrong to claim that data-randomization
only relies on randomness in the data, when in fact it relies on randomness that occurs when the experiment is planned.
Put differently, both types of randomizations are external in the sense that they rely on randomness not contained in the
sufficient statistic X. From both a practical and a mathematical point of view, there is no difference between recording
X1, . . . , Xn and recording X and Y or ν1, except that the latter saves space. For this reason, the Stevens interval can be
recommended over the competing randomized intervals: it offers exact coverage and perfect location, and for larger n the
randomization has a smaller impact on the bounds of the Stevens interval than it has on the split sample Wilson interval.
5. Discussion
As is common in the literature, we have discussed externally randomized and data-randomized intervals separately. It is
the author’s opinion that the distinction between these types of intervals is entirely artificial. The only difference between
an externally randomized split sample interval and its data-randomized counterpart is whether the order in which the trials
are recorded is determined after or before the experiment is concluded. Indeed, it is wrong to claim that data-randomization
only relies on randomness in the data, when in fact it relies on randomness that occurs when the experiment is planned.
Put differently, both types of randomizations are external in the sense that they rely on randomness not contained in the
sufficient statistic X. From both a practical and a mathematical point of view, there is no difference between recording
X1, . . . , Xn and recording X and Y or ν1, except that the latter saves space. For this reason, the Stevens interval can be
recommended over the competing randomized intervals: it offers exact coverage and perfect location, and for larger n the
randomization has a smaller impact on the bounds of the Stevens interval than it has on the split sample Wilson interval.
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