The quality manager of a dairy company is concerned whether the levels of butterfat in a product are equal at two dairy factories which produce the product. He obtains random samples of size 10 from each of the factories' output and measures the butterfat. The results are:

(a) We will assume that the observations come from \(\operatorname{normal}\left({ }_{1}{ }^{2}\right)\) and \(\operatorname{normal}\left({ }_{2}{ }^{2}\right)\), where \({ }^{2}=12^{2}\). Use independent \(\operatorname{normal}\left(m s^{2}\right)\) prior distributions for 1 and 2 , respectively, where \(m=15\) and \(s^{2}=4^{2}\). Find the posterior distributions of \({ }_{1}\) and \({ }_{2}\), respectively.
(b) Find the posterior distribution of \(1 \quad 2\).
(c) Find a 95\% Bayesian credible interval for \({ }_{1} \quad{ }_{2}\).
(d) Perform a Bayesian test of the hypothesis
\[
H_{0}: \quad{ }_{1} \quad 2=0 \text { versus } H_{1}:{ }_{1} \quad{ }_{2}=0
\]
at the \(5 \%\) level of signi cance. What conclusion can we draw?

Transcribed Image Text:

Factory 1 Factory 2 16.2 16.1 12.7 16.3 14.8 14.0 15.6 16.2 14.7 15.2 13.8 16.5 16.7 14.4 13.7 16.3 16.8 16.9 14.7 13.7