The Houhora site in Northland (top of North Island) New Zealand is one of the sites of early human inhabitation that New Zealand archeologists have investigated, in trying to determine when the Polynesian migration to New Zealand occurred and documenting local adaptations to New Zealand conditions. Petchey (2000) describe the Radiocarbon dating of well-preserved snapper Pagrus auratus bones found at the Houhora site. They obtained four acceptable samples which were analyzed by the Waikato University Carbon Dating Unit. Assume that the conventional radiocarbon age (CRA) of a sample follows the normal \(\left({ }^{2}\right.\) ) distribution where the standard deviation \(=40\) is known. The observations are:

(a) Use a normal \(\left(1000200^{2}\right)\) prior for . Find the posterior distribution \(g\left(\begin{array}{ccc}y_{1} & y_{4}\end{array}\right)\).
(b) Find a \(95 \%\) credible interval for .
(c) To nd the , the calibrated date, the Stuiver, Reimer, Braziunas marine curve (Stuiver et al., 1998) was used. We will approximate this curve with the linear function
\[
=2203 \quad 835
\]
Find the posterior distribution of given \(y_{1} \quad y_{4}\).
(d) Find a \(95 \%\) credible interval for , the calibrated date.

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Observation CRA 1 21 3 4 1010 1000 950 1050