The usual confidence interval, based on Student's t distribution, has conditional confidence that is larger than the nominal confidence level. Although this fact is known, along with the fact that increased conditional confidence can be used to improve a confidence assertion, the confidence assertion of Student's t interval has never been critically examined. We do so here, and construct a confidence estimator that allows uniformly higher confidence in the interval and is closer (than 1 - α) to the indicator of coverage.
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Goutis et al. (1992) studied this question.
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