The standard process for making a polymer has mean yield \(35 \%\). A chemical engineer has developed a modified process. He runs the process on 10 batches and measures the yield (in percent) for each batch. They are:

Assume that yield is normal \(\left({ }^{2}\right.\) ) where the standard deviation \(=3\) is known.
(a) Use a normal(30 \(\left.10^{2}\right)\) prior for . Find the posterior distribution.
(b) The engineer wants to know if the modified process increases the mean yield. Set this up as a hypothesis test stating clearly the null and alternative hypotheses.
(c) Perform the test at the \(5 \%\) level of significance.

38.7 40.4 37.2 36.6 35.9 34.7 37.6 35.1 37.5 35.6