A significance measure essentially due to Liebermeister and dating back to 1877 may be preferable to Fisher's Exact Test, a conservative but commonly applied test when the sample sizes are small. We show that Liebermeister's measure is less conservative than Fisher's P‐value and just as easy to calculate, while retaining the important features of a significance measure. We also compare Liebermeister's measure with Lancaster's mid‐P, which has gained increasing acceptance as a replacement for Fisher's P‐value. Application is made to a recent striking medical study on appendicitis symptoms for which the Fisher test does not give significance.

A special inequality between the tail probabilities of certain related hypergeometrics was shown by Seneta and Phipps [19] to suggest useful 'quasi-exact' alternatives to Fisher's [5] Exact Test. With this result as motivation, two inequalities of Hájek and Havránek [6] are investigated in this paper and are generalised to produce inequalities in the form required. A parallel inequality in binomial tail probabilities is also established.

Abstract. A special inequality between the tail probabilities of certain related hypergeometrics was shown by Seneta and Phipps [19] to suggest useful 'quasi-exact' alternatives to Fisher's [5] Exact Test. With this result as motivation, two inequalities of Hájek and Havránek [6] are investigated in this paper and are generalised to produce inequalities in the form required. A parallel inequality in binomial tail probabilities is also established.

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