Ten measurements are made using a scientific instrument. Given the unknown value of a quantity $\theta$ , the natural logarithms of the measurements are independent and normally distributed with mean $\log\theta$ and known standard deviation 0.05.  

Our prior distribution is such that $\log\theta$ has a normal distribution with mean 2.5 and standard deviation 0.5.  

The logarithms of the measurements are as follows.  

2.99 3.03 3.04 3.01 3.12 2.98 3.03 2.98 3.07 3.10  

(a) Find the posterior distribution of $\log\theta$ .  
(b) Find a symmetric 95% posterior interval for $\log\theta$ .  
(c) Find a symmetric $95\%$ posterior interval for $\theta$ .  
(d) Find the posterior probability that $\theta<20.0$ .  