The observed weights (in grams) of 20 pieces of candy randomly sampled from candy- making machines in a certain production area are as follows:

46 58 40 47 47 53 43 48 50 55 49 50 52 56 49 54 51 50 52 50 Assume that weights of this type of candy are known to follow a normal distribution, and that the mean weight of candies produced by machines in this area is known to be 51 g. We are trying to estimate the variance, which we will now call 2.

1. What is the conjugate family of prior distributions for a normal variance (not precision) when the mean is known?

2. Suppose previous experience suggests that the expected value of 2 is 12 and the variance of 2 is 4. What parameter values are needed for the prior distribution to match these moments?

3. What is the posterior distribution p(2|y) for these data under the prior from the previous step?

4. Find the posterior mean and variance of 2.