Seed lot heterogeneity is often evaluated through the range between germination percentages of four seed samples, considering normal and binomial approximations for calculating the tolerated range (S). In this paper, an exact test for the germination count range (R) is derived based on the hypergeometric and the binomial probability model for germination count. Through Monte Carlo simulations, the empirical distribution of R is built to evaluate the quantiles of the exact distributions. Moreover, a power analysis is performed by simulation. Sample size and germination rate effects are evaluated. In lots with a high germination rate, the proposed test based on the hypergeometric model is about 20% more powerful than the test based on the S-value. A table containing the critical values is presented and recommended to be used in off-range heterogeneity testing.

中文翻译：

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用发芽数范围的精确概率分布检验种子样品的异质性

种子批次异质性通常通过四个种子样品的发芽百分比之间的范围来评估，考虑计算耐受范围的正态和二项式近似值（小号）。在本文中，对发芽计数范围（R) 是基于发芽计数的超几何和二项式概率模型推导出来的。通过蒙特卡罗模拟，经验分布R用于评估精确分布的分位数。此外，通过仿真进行功率分析。评估样本大小和发芽率的影响。在发芽率高的批次中，基于超几何模型的建议测试比基于小号-价值。提供了一个包含临界值的表格，并建议用于离开范围异质性检验。