A consumer testing organization obtained samples of size 12 from two brands of emergency ares and measured the burning times. They are:

(a) We will assume that the observations come from normal \(\left(A^{2}\right)\) and normal( \(\left(B^{2}\right)\), where \({ }^{2}=3^{2}\). Use independent normal \(\left(\mathrm{m} \mathrm{s}^{2}\right)\) prior distributions for \(A\) and \(B\), respectively, where \(m=20\) and \(s^{2}=8^{2}\). Find the posterior distributions of \(A\) and \({ }_{B}\), respectively.
(b) Find the posterior distribution of \(A \quad B\).
(c) Find a 95\% Bayesian credible interval for \({ }_{A}{ }_{B}\).
(d) Perform a Bayesian test of the hypothesis
\[
H_{0}: \quad A \quad B=0 \text { versus } H_{1}:{ }_{A} \quad{ }_{B}=0
\]
at the \(5 \%\) level of signi cance. What conclusion can we draw?

Transcribed Image Text:

Brand A Brand B 17.5 21.2 13.4 9.9 20.3 13.5 14.4 11.3 15.2 22.5 19.3 14.3 21.2 13.6 19.1 15.2 18.1 13.7 14.6 8.0 17.2 13.6 18.8 11.8