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

We regard these as independent samples from a normal distribution with mean $\mu$ and variance $\sigma^{2}=\tau^{-1}$ .  

Find a 95% posterior hpd interval for $\mu$ under each of the following two conditions.  

(a) The value of $\tau$ is known to be 0.1 and our prior distribution for $\mu$ is normal with mean 60.0 and standard deviation 20.0.  

(b) The value of $\tau$ is unknown. Our prior distribution for $\tau$ is a gamma distribution with mean 0.1 and standard deviation 0.05. Our conditional prior distribution for $\mu$ given $\tau$ is normal with mean 60.0 and precision $0.025\tau$ (that is, standard deviation $\sqrt{40}\tau^{-1/2}$ ).  