Analysis of interval data from completely randomized designs

Data collected in many fields often take the form of intervals due to measurement uncertainty, aggregation, or data protection needs. Conventional methods for analyzing data from designed experiments focus on single-valued observations and cannot be directly applicable to interval-valued data. Recent studies and critiques have increasingly focused on neutrosophic and simulation-based approaches. The former extend classical point-based statistical analysis through indeterminacy representations, while the latter preserve conventional inferential frameworks by sampling within observed intervals according to distributional assumptions. However, neutrosophic methods may neglect dependence structures among statistical quantities, whereas simulation-based methods depend on within-interval distributional assumptions that are often unknown or debated. This study addresses these challenges by proposing an optimization-based framework that derives feasible bounds for classical test statistics without imposing within-interval probability models. For completely randomized designs, the F-bounds are obtained by reformulating interval ANOVA as a box-constrained quadratic fractional programming problem, thereby extending classical ANOVA to interval-valued significance testing while preserving its inferential structure. The effectiveness of the proposed method is validated through both simulation studies and a real-life case study. This research is among the very first to provide a rigorous extension of classical ANOVA for interval settings. It bridges mathematical programming techniques with statistical hypothesis testing while ensuring globally optimal solutions. Notably, the method naturally reduces to classical ANOVA when interval data collapse to point values, providing a foundation for further methodological advancements in this field.