Question 2: (15 Total Marks) The Sydney Swans are an AF L team. As I write this question, they have 5 games left in the regular season. Let 5' represent the Swan's score in a game, 0 their opponent's score and D = S 0 the game's score difference. Suppose 0 ~ N(75.1,18.7), D ~ N(12.4,31.3) and Corr(D, 0) = 0.6795 a) Find the distribution of S. Also, nd the correlation between .S' and 0. [3 marks] b) Given an opponent scored 100 points, what is the chance the Swans won the game? [3 marks] c) Assuming games are independent, what is the chance that the Swans aggregate score in their remaining 5 games is more than double their opponents aggregate? In other words, calculate 5 5 5 13%: 592: 05)=Pr{ (Si20.)>0} i=1 i=1 i=1 where S; and 0,- are the ith game scores of the Swans and their opponent, respectively. [3 marks] Lance \"Buddy\" Franklin, a player on the Swans, is one of the all-time top goal scorers in the history of the AFL. Before this year, he had scored an average of 3.15 goals per game over his career. (1) All sports stars, no matter how great, eventually age and slow down. In his 13 games this year, Buddy has scored 2.85 goals per game, with a standard deviation of 1.68 goals. Can we conclude, at the a! = 0.05 level of signicance, that his goals per game average has decreased? [3 marks] e) In the 13 games so far this year Buddy has played, the Swans averaged 12.23 goals with a standard deviation of 3.49 goals. In the other 4 games, they averaged 14.75 goals with a standard deviation of 4.35 goals. Construct a 95% condence interval for the true difference in goals per game when Buddy plays versus when he doesn't. Is Buddy a drawback to the Swans this year? [3 marks]