Newbie needs help: my research sample is one end of the bell curve only

#1
Hello forum members

I did the wonderful Princeton Statistics 1 Coursera MOOC which I really enjoyed. But I have hit a problem with a project which I can't find answers to in my course notes as this introductory course only briefly covered non-parametric statistics.

I have a normally distributed sample for a continuous variable (duration). All I am looking for is descriptive statistics, there is no experimental condition involved, I just want to see which categorical variables seem to be related to the duration in order to suggest which factors to try to influence in order to reduce duration overall.

I have been able to collect more data for a number of categorical variables for the end of the bell curve I am interested in (long duration). I don't have resources to do this for the full sample of durations.

So my detailed sample is not normal but I do know which bit of the bell curve it is.
Some of my categorical variables from the long duration sub-sample have only two states - yes/no whereas others have up to 7 (e.g. which service provider).

What tests might be valid, and in what conditions (what should I do to test the validity) to try to get some sense of which categorical variables are related to longer duration? The subsample for which I have categorical variables n=265 and this represents about a quarter of the total sample, so a reasonable number to work from.

Thank you for any advice, or for links to any other threads I should read on this topic.
I am using R and R Studio for the analysis.

Best wishes!