If only the lower and upper bounds are known for response data, this is referred to as interval censored data. A large number of statistical tools for right censored data have been generalized to interval censored data. Some of the fundamental tools include:
NPMLE (generalization of Kaplan Meier curves, which themselves are a generalization of the Empirical Distribution Function): this is the most basic method for looking the CDF/survival function of a univariate distribution. This can be fit a number of ways in R, I'm going to recommend icenReg's
ic_np, since I'm the author.
Log-rank statistic: similar to the log-rank statistic for right censored data, this is one of the basic tools for examining statistical significant differences between two univariate distributions. Implementations of this are available in R's interval package. From the original question, it sounds like this might be the test that most directly answers the OP's question.
Regression models: several of the classic right censored regression models are available. The canonical model is the proportional hazards model, although it's this author's believe that this is due more to historic development of methodology. Similar to the right censored case, the baseline distribution is not necessary to specify. Alternatively, there exist the proportional odds model and accelerated failure time (AFT) model. Personally, I believe the AFT model should be the default unless it fits the data poorly, as the AFT model is easier to interpret. All the models listed can be fit with icenReg (either
ic_bayes). The AFT model for interval censored data can also be fit with survival's