The experiment described in Exercise 6 was repeated on another set of 7 cows, McLeay, Carruthers, and Neil (1997). However, in this case, the second treatment was given to the same set of 7 cows that were given the first treatment, at a later time when the first dose of 13C had been eliminated from the cow.

The data are given below:

(a) Explain why the variables x and y cannot be considered independent in this experiment.

(b) Calculate the differences di = xi − yi for i = 1,..., 7.

(c) Assume that the differences come from a normal (µd, σ2

d) distribution, where σ2 d = 1. Use a normal (0, 32) prior for µd. Calculate the posterior for µd|d1,...,d7.

(d) Calculate a 95% Bayesian credible interval for µd.

(e) Test the hypothesis H0 : µd = 0 versus H1 : µd = 0 at the 5% level of significance. What conclusion can be drawn?