What does the univariate t test tell you that the bivariate correlation test alone does not? What does the bivariate test tell you that the univariate test does not tell you? Select one: Only the univariate comparison tells you that vanilla is preferred over chocolate on average; only the bivariate analysis tells you that people who prefer vanilla more also prefer chocolate more. Only the univariate comparison tells you how to predict chocolate preference from vanilla preference; only the bivariate analysis tells you that people who prefer vanilla more also prefer chocolate more. Only the univariate comparison tells you that people who prefer vanilla more also prefer chocolate more; only the bivariate analysis tells you that chocolate is preferred over vanilla on average Only the univariate comparison tells you that the preferences for chocolate and vanilla are not different; only the bivariate analysis tells you that people who prefer vanilla more also prefer chocolate more.Problem 7.4 (10 points) A Markov chain Xo, X1, X2, ... with state space S = {1, 2,3, 4) has the following transition graph: 0.5 0.5 0.5 1 0.5 0.5 0.5 2 0.5 0.5 (a) Provide the transition matrix for the Markov chain. (b) Determine all recurrent and all transient states. (c) Determine all communication classes. Is the Markov chain irreducible? (d) Find the stationary distribution. (e) Can you say something about the limiting distribution of this Markov chain?8. (10 points)(The Weak Law of Large Numbers) In order to estimate f, the true fraction of smokers in a large population, Alvin selects n people at random. His estimator M., is obtained by dividing S,, the number of smokers in his sample, by N, i.e., M. = S,. Alvin chooses the sample size n to be the smallest possible number for which the Chebyshev inequality yields a guarantee that P(IMn - f1 2 () So