Ranked set sampling (RSS) is a practical and statistically efficient methodology, particularly advantageous when the direct measurement of selected units is costly, challenging, destructive, or time-intensive. This approach relies on the feasibility of ranking units easily through expert judgment or by utilizing an auxiliary variable. In this investigation, we assume that the pair (X,Y) follows a bivariate exponentiated inverted Weibull distribution, where Y denotes the primary variable of interest that is difficult to measure, while X serves as a correlated auxiliary variable that is readily observable. Our primary focus is on estimating the population mean of the study variable Y employing the ranked set sample mean. The performance of this estimator is subsequently compared with its counterpart derived from simple random sampling (SRS), specifically in terms of variance. Furthermore, we adopt the best linear unbiased estimator (BLUE) to estimate the mean of Y based on RSS observations. Beyond the conventional RSS approach, we investigate alternative designs, including unbalanced upper extreme ranked set sampling, lower extreme ranked set sampling, and multistage ranked set sampling. The efficiency of these proposed estimators is evaluated through numerical analysis and graphical comparisons. The practical relevance of the proposed methodology is demonstrated through an application to sustainability-related data, specifically the breaking strength of jute fibers, which is a natural and biodegradable material widely used in sustainable packaging and eco-friendly composites. This real-data application highlights the potential of the proposed estimators in supporting sustainable material research and quality control in green manufacturing.