# Hypothesis testing — time between two actions

I am a complete newbie to statistics, and have gotten stuck on a difficult real-world problem.

What I'd like to do is demonstrate confidence that, for a set of < 50 observations of two actions, the time between the two actions was less than an hour. For the first action, I have a timezone-naive timestamp. For the second, I have a timezone-aware timestamp (UTC). The location of the actions is not known. Time between the two actions is known to be less than eight hours.

To visualize the problem, I have removed time zone information from the second action timestamps, subtracted the two, and taken the mod 60 value of the result, to represent the "minute within the hour" of the difference. This results in a histogram that looks like this: Just from a visual examination, I feel confident that the time between the actions was less than an hour (usually around 5 minutes, as opposed to an hour and five minutes, two hours and five minutes... etc) but I don't know how to express this confidence quantitatively. I'm having a bit of trouble determining what the null hypothesis would be, here. Any insight would be greatly appreciated!

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• One thing comes to mind is Kernel Density Estimator. You can fit a kernel density estimator to the distribition you showed above. After that you can calculate the probability time is less than or equal to 5 minutes. Lets say this gives the value of 0.95. Then you can say time is less than 5 minutes with 95% probability given the data at hand. Statsmodels has nice KDE implementations: statsmodels.org/stable/examples/notebooks/generated/… – CheeseBurger 2 days ago
• I am curious on what makes you confident that that time between actions was actually less than an hour, given you have taken a modulo 60 on the minute difference between the paired timestamps. It might as well be the distribution is dropping so quickly that it is implausible to go beyond 60 mins, but you have to spell it out. – B.Liu 2 days ago