The breaking strength of a type of cloth is to be measured for 20 specimen. The underlying distribution is normal with unknown mean $\displaystyle \theta$ but with a standard deviation of 3. Suppose that the unknown mean has a prior distribution that is normal with mean 200 and standard deviation 2. If the average breaking strength for this sample of 20 specimen is 182, determine the Bayes estimate of $\displaystyle \theta$ and an interval around this value that contains $\displaystyle \theta$ with probability .95.

Is this correct?

$\displaystyle E(\theta|data) = \frac{1/4}{1/9+1/4}(200) + \frac{1/9}{1/9+1/4}(182) = 194.46$

$\displaystyle Var(\theta|data) = \frac{1}{1/9+1/4} = 36/13$

$\displaystyle P\{-1.96<\frac{\theta-194.46}{\sqrt{36/13}}<1.96|data\} = .95$

$\displaystyle \theta \in (191.20, 197.72) \mbox{ with .95 probability}$

How do I determine the Bayes estimate of $\displaystyle \theta$?