Samples are taken from twenty wagon loads of an industrial mineral and analysed. The amounts in ppm (parts per million) of an impurity are found to be as follows.

44.3 50.2 51.7 49.4 50.6 55.0 53.5 48.6 48.8 53.359.4 51.4 52.0 51.9 51.6 48.3 49.3 54.1 52.4 53.1

(maple mean = 51.445 and sample standard deviation = 3.095)

We regard these as independent samples from a normal distribution with mean and variance ^2. Our prior distribution for is normal with mean 60 and standard deviation 20.

Find the posterior distribution for .

Find the bayes estimate for under the square loss function.

Find a 95% posterior high probability density, or credibility interval, interval for under each of the following two conditions.