I am writing a project plan and I am struggling with the statistical analysis plan as I am in the early stages of learning. I was hoping someone could guide me with what stats tests I could conduct with the following primary endpoint - thank you.

Alternative hypothesis: treatment is superior to placebo in achieving a state of pain relief.

Primary endpoint: difference in time to initial meaningful pain relief comparing the treatment Vs placebo.

I am thinking of the following analysis: As time is continuous and I am are comparing between categorical data (treatment, placebo), then I need a t-test (or Mann-Whitney depending on what the distribution will look like). With a continuous outcome, I also need to look at linear regression modeling?

Thank you for your help - much appreciated.