It sounds like you are saying to model each event as a Poisson process with rate proportional to the population proportion of the event times the length of the event.
I don’t think this is quite what I want but let me see if I can summarize your above thoughts.
In the above, we are given the observed value r = 0.0405 and the knowledge of the population total and the duration of the event in minutes, say p and t, respectively. We know that 0.0405 * p * t = 40,500 * t is the sum of the minutes of attendance calculated in some way for the sample of n = 34 people. Let s_i be the total minutes of attendance for person i in the sample. Then we know that sum_{i=1}^34 s_i = 40,500 * t and we want to place uncertainty around the quantity sum_{i=1}^34 s_i.
Are you saying to model n ~ Poisson(p * r * t), or in pmyc3, n = pm.Poisson(‘n’, mu=p * r * t, observed=n)? Isn’t the rate of that Poisson distribution much much greater than the observed variable n?