Suppose that you are interested in investigating how variable the performance of schoolchildren on a new mathematics test, and that you begin by trying this test out on children in 12 similar schools. It turns out that the average standard deviation is about 10 marks. You then want to try the test on a thirteenth school, which is fairly similar to those you have already investigated, and you reckon that the data on the other schools gives you a prior for the variance in this new school which has a mean of 100 and is worth eight direct observations on the school. What is the posterior distribution for the variance if you then observe a sample of size 30 from the school of which the standard deviation is 13.2? Give an interval in which the variance lies with 90% posterior probability.