Historically, to bound the mean for small sample sizes, practitioners have had to choose between using methods with unrealistic assumptions about the unknown distribution (e.g., Gaussianity) and methods like Hoeffdings inequality that use weaker assumptions but produce much looser (wider) intervals. In 1969, Anderson (1969) proposed a mean confidence interval strictly better than or equal to Hoeffdings whose only assumption is that the distributions support is contained in an interval $[a,b]$. For the first time since then, we present a new family of bounds that compares favorably to Andersons. We prove that each bound in the family has {em guaranteed coverage}, i.e., it holds with probability at least $1-alpha$ for all distributions on an interval $[a,b]$. Furthermore, one of the bounds is tighter than or equal to Andersons for all samples. In simulations, we show that for many distributions, the gain over Andersons bound is substantial.
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