Testing for positive evidence of equally likely outcomes

Goodness-of-fit tests allow one to conclude that k possible outcomes are not equally likely. In this paper, we develop an exact equivalence test that allows one to conclude that k possible outcomes are approximately equally likely. We show that the power properties of the test compare favorably to those of possible alternative tests, and we develop an associated simultaneous confidence interval procedure. We apply the test to data sets on the digits of π, winning roulette numbers, and winning numbers from the Pennsylvania Lottery.