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An Effective Confidence Interval for the Mean With Samples of Size One and Two

It is counterintuitive that, with a sample of only one value from a normal distribution, one can construct a finite confidence interval of any size for the mean. It goes just as much against standard teaching that from a sample of size two such a CI might be shorter than that based on the t statistic. We refine an earlier version of this first result, and use it to prove the second. For samples of three and larger, we show that the t-based interval cannot be improved using this approach.