A key issue in science is assessing robustness to data analysis choices, while avoiding selective reporting and providing valid inference. Specification Curve Analysis is a tool intended to prevent selective reporting. Alas, when used for inference it can create severe biases and false positives, due to wrongly adjusting for covariates, and mask important treatment effect heterogeneity. As our motivating application, it led an influential study to conclude there is no relevant association between technology use and teenager mental well-being. We discuss these issues and propose a strategy for valid inference. Bayesian Specification Curve Analysis (BSCA) uses Bayesian Model Averaging to incorporate covariates and heterogeneous effects across treatments, outcomes and sub-populations. BSCA gives significantly different insights into teenager well-being. It provides strong evidence that technology has relevant associations with teenager well-being: (1) well-being is negatively associated with electronic device usage, (2) social media use is negatively associated with self-assessed well-being but positively associated with parent-assessed well-being, and (3) has a stronger negative association with self-assessed well-being for girls compared to boys.Bayesian model averaging, treatment effect inference, selective reporting, social media, adolescents, mental health