Hi guys,
As always, Nicholas already brilliantly answered your question, but I just wanted to complement it and say that I wrote a NB showing how to prior and posterior predictive checks exactly on this kind of model. It’ll illustrate Nicholas’ point about your priors being too wide and show you a principled way of choosing them.
It extends the example you’ll find in Osvaldo Martin’s very good book, Bayesian Analysis with Python.
Hope this helps 