One of the advantages of Bayesian statistics is that evidence from different sources can be combined. In Exercise 6 and Exercise 14, we found posterior distributions of µd using data sets from two different experiments. In the first experiment, the two treatments were given to two sets of cows, and the measurements were independent. In the second experiment, the two treatments were given to a third set of cows at different times and the measurements were paired. When we want to find the posterior distribution given data sets from two independent experiments, we should use the posterior distribution after the first experiment as the prior distribution for the second.

(a) Explain why the two data sets can be considered independent.

(b) Find the posterior distribution of µd|data where the data include all of the measurements x8 ...,x13, y14 ...,y19, d1,...,d7.

(c) Find a 95 % credible interval for µd based on all the data.

(d) Test the hypothesis H0 : µd = 0 versus H1 : µd = 0 at the 5 % level of significance. Can we conclude that 13C octanic acid breath test is effective in detecting reticular groove contraction in cattle?