Diagrams of Confidence Limits of Occurrence Probability

Abstract

The confidence limits (p₁, p₂) of occurrence probability p can be calulated form observed. relative frequenvy k/N by the following formulas. If we take ε=(100-2α)% as the confidence coeffcient, p₁=n₂/(n₂+n₁F₁),
where F₁ is the α% point of F-distrbution with the degrees of freedom n₁=2(N-k+1) and n₂=2k, and
p₂=n₁F₂/(n₂+n₁F₂),
where F₂ is the α% point of F-distribution with the degrees of freedom n₁=2(k+1) and n₂=2(N-k).
Two kinds of tables and diagtrams are prepared, the one is for ε=90%(α=5%) and the other for ε=98%(α=1%). The table of (p₁, p₂) in which the arguments are N and k are used for 1_??_N_??_20, and the diagrams of (p₁, p₂) in which the arguments are N and k/N are used for 20_??_N_??_1000.
Comparing with these tables and diagrams, the error of approximate confidence limits by binomial probability paper is negligible of N>50, N-5>k>5 or N>100, N>k>0.
In the construction of an objective forcasting scheme, we can objectively proceed by making use of confidence limits of occurrence probability. Our tables and diagrams are also applied to find the confidence limits of corrlation coefficient calculated from the sings of variates. The diaram of arcsine transformation can be utilized for the purpose of testing the difference of two occurrence probabilities.