Glass fragments found on a suspect's shoes or clothes are often used to connect the suspect to

Question:

Glass fragments found on a suspect's shoes or clothes are often used to connect the suspect to a crime scene. The index of refraction of the fragments are compared to the refractive index of the glass from the crime scene. To make this comparison rigorous, we need to know the variability the index of refraction is over a pane of glass. Bennett et al. (2003) analyzed the refractive index in a pane of oat glass, searching for any spatial pattern. Here are samples of the refractive index from the edge and from the middle of the pane.

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For these data,\(\quad y_{1}=151999 \quad y_{2}=152001 \quad s d_{1}=00002257 \quad\) and \(s d_{2}=00003075\).
(a) Suppose glass at the edge of the pane is normal \(\left(\begin{array}{ll}1 & 2 \\ 1\end{array}\right)\), where \({ }_{1}=\) 00003. Calculate the posterior distribution of 1 when you use a normal (1 \(520000001^{2}\) ) prior for 1 .
(b) Suppose glass in the middle of the pane is \(\operatorname{normal}\left(\begin{array}{ll}2 & 2 \\ 2\end{array}\right)\), where \(2=00003\). Calculate the posterior distribution of 2 when you use a normal \(\left(1520000001^{2}\right)\) prior for 2 .
(c) Find the posterior distribution of \(d=1 \quad 2\).
(d) Find a 95\% credible interval for \(d\).
(e) Perform a Bayesian test of the hypothesis
\[
H_{0}: \quad d=0 \quad \text { versus } \quad H_{1}: \quad d=0
\]
at the \(5 \%\) level of signi cance.