Studying: The files school1.dat, school2.dat and school3.dat contain data on the amount of time students from three high schools spent on studying or homework during an exam period. Analyze data from each of these schools separately, using the normal model with a conjugate prior distribution, in which {µ0 = 5, σ2 0 = 4, κ0 = 1, ν0 = 2} and compute or approximate the following: a) posterior means and 95% confidence intervals for the mean θ and standard deviation σ from each school; b) the posterior probability that θi < θj < θk for all six permutations {i, j, k} of {1, 2, 3}; c) the posterior probability that Y˜ i < Y˜ j < Y˜ k for all six permutations {i, j, k} of {1, 2, 3}, where Y˜ i is a sample from the posterior predictive distribution of school i. d) Compute the posterior probability that θ1 is bigger than both θ2 and θ3, and the posterior probability that Y˜ 1 is bigger than both Y˜ 2 and Y˜ 3.

Submarines control how much they float or sink in the ocean by changing the volume of air and water contained in large ballast tanks. when the tanks are completely full of water, the submarine increases its overall mass and sinks down to the bottom. when the tanks are completely full of air, the submarine reduces its overall mass and floats to the surface. depending on the density of the seawater surrounding the submarine, it will pump seawater in or out of the tanks in order to achieve the same overall density as the sea water and float neutrally in the water. the volume of the submarine never changes. when the tanks are completely full of water, a submarine with a volume of 7.8\times10^3\text{ m}^37.8×10 3 m 3 7, point, 8, times, 10, start superscript, 3, end superscript, space, m, start superscript, 3, end superscript has a total mass of 8\times10^6\text{ kg}8×10 6 kg8, times, 10, start superscript, 6, end superscript, space, k, g. the density of the seawater is 10^3\text{ kg/m}^310 3 kg/m 3 10, start superscript, 3, end superscript, space, k, g, slash, m, start superscript, 3, end superscript. to make that submarine float neutrally, and neither float nor sink in the ocean, what volume of water does that submarine need to subtract from its tanks?