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"You're at a local computer fair looking for some nice sleeved cables or your new system, a stall operator shows you his cables claiming they are the branded version. But the stall operator is not the usual person running the stall and actually does not know if they're the branded ones or not.

You measure the cable precisely with callipers and find the cable to have diameter of 13mm. You also know from research that the non-branded cables come in two forms, thin and thick, which have mean diameters of 31mm and 11mm, both with a standard deviation of 2mm. You also know that the branded ones have mean diameter of 23mm with a standard deviation of 7.5mm. Both the branded and non-branded sizes are normally distributed. In this part of the country you know 30% of these types of cables are the branded version and 70% are non-branded. The thin and thick variety of non-branded cables are just as popular as one another with the public.

What are the odds that the cables are the branded version? You will need to use the Bayesian approach to hypothesis testing."

The question is wordy and hard to wrap your head around but I completely understand what it is asking, I just have no idea how to go about it, or where to start. I have spent hours scratching my head on this one!

We have been taught two methods of Bayesian hypothesis testing, estimation (one prior) and model comparison (two priors).

My first though was to use model comparison, as that results in the odds of one model over the other, where the one model is that the hypothesis is that they are branded and the second model is the hypothesis that they are non-branded. I don't know if that is the correct approach or if it is, what to do, the fact there are two varieties of non-branded ones confuses the hell out of me too.

Any help appreciated.