2. (31 marks) A production manager is interested in the mean weight of items produced by a particular process. He feels the weight of items from the process is Normally distributed with mean u and that u is either 109.4,109.7,110,110.3 or 110.6. The production manager assesses the prior probabilities to be

P(u = 109.4) = 0.05, P(u = 109.7) = 0.20, P(u = 110.0) = 0.5, P(u = 110.3) = 0.20, P(u) = 110.6) = 0.05.

From past experience, he is willing to assume that the process has variance a2 = 4. He randomly selects a sample of four items from the process and weighs them determining the average weight to be 108 and another sample of size 9 with average weight of 112. Show your working out by hand but use R to code and calculate the following probabilities.

(a) (8 marks) Calculate P(Y = 108), where Y is the sample mean for the weight of the four items.

(b) (5 marks) Find the production manager's posterior distribution.

(c) (5 marks) Compute the means and the variances of the prior and posterior distri-butions. Use R to simulate the weights of the next four items.

(d) (5 marks) If the next sample collected is used next then calculate P(Y = 112) and this posterior distribution?

(e) (3 marks) If the prior probability of a hypothesis is set to 0 then what does this tell you about the posterior probability?

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