A machine is used to fill plastic bottles with bleach has mean of 3L ( = 3) and a standard deviation equal to 0.02L ( = 0.02). A sample of 25 bottles had a mean fill volume of 3.001L. The machine was then moved to another location. A sample of 20 bottles filled at the new location had a mean fill volume of 3.3L. It is believed that moving the machine may have changed the mean fill volume, but is unlikely to have changed the standard deviation. Assume that the sample come from approximately normal population.

(a) (1pt.) What is the frequentist approach estimate of the population mean, ?

(b) (1.5pts) What is the posterior distribution of the population mean, , using the Jeffrey's prior ?

Calculate the parameters of the posterior distribution using the Jeffrey's prior for the population

mean.

(c) (1pt.) Find a 99% credible set for the population mean? Do you think the population mean

changed after moving the machine?

(d) (1.5pts) Plot the posterior distribution under Jeffrey's prior along with the credible set limits.

(e) (1.5pts) We have discussed at the beginning of the semester that the Bayesian approach is a

sequential approach. Assume another sample of 25 is taken and found that the sample mean is

2.001. Find the new posterior distribution with its parameter values?

(f) (1pt.) Again, do you think the population mean changed after moving the machine based on the

new posterior distribution?