Cattle being ruminants have multiple chambers in their stomaches. Stimulating specific receptors causes reflex contraction of the reticular groove and swallowed fluid then bypasses the reticulo-rumen and moves directly to the abomasum. Scientists wanted to develop a simple nonradioactive, noninvasive test to determine when this occurs. In a study to determine the fate of swallowed fluids in cattle, McLeay, Carruthers, and Neil (1997) investigate a carbon 13 (13C) octanoic acid breath test as a means of detecting a reticular groove contraction in cattle. Twelve adult cows were randomly assigned to two groups of 6 cows. The first group had 200 mg of 13C octanoic acid administered into the reticulum, and the second group had the same dose of 13C octanoic acid administered into the reticulo-osmasal orifice. Change in the enrichment of 13C in breath was measured for each cow 10 minutes later. The results are:

(a) Explain why the observations of variables x and y can be considered independent in this experiment.

(b) Suppose the change in the enrichment of 13C for cows administered in the reticulum is normal (µ1, σ2 1), where σ2 1 = 1.002. Use a normal (2, 22)
prior for µ1. Calculate the posterior distribution of µ1|x8 ...,x13.

(c) Suppose the change in the enrichment of 13C for cows administered in the reticulo-omasal orifice is normal (µ2, σ2 2), where σ2 2 = 1.402. Use a normal (2, 22) prior for µ2. Calculate the posterior distribution of µ1|y14 ...,y19.

(d) Calculate the posterior distribution of µd = µ1 − µ2, the difference between the means.

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

(f) Test the hypothesis H0 : µ1 − µ2 = 0 versus H1 : µ1 − µ2 = 0 at the 5% level of significance. What conclusion can be drawn.