Can you help me with 3 and 4.
Can you help me with this. Thank you.
Project 6A Ranking Sport Teams This project uses matrix algebra, linear systems, and least-squares technique application. One problem that has perplexed folks for quite some time is how to rank things, especially sports teams. In American sports this is a big issue when it comes to college football. At the highest level there are 130 teams that are competing for a National Championship. There is no reasonable way for all of the teams to play each other. So, ranking the teams is particularly tricky. In 1997, Kenneth Massey proposed a method of ranking in an undergraduate honors project. His idea was very simple: when team A wins against team B by a number of points p, this determines an equation ra - r8 = p, where ra and re are the rating values of the two teams. Any set of games and teams will develop a system of equations. However, the system will not have a solution - when team A and B play again, there will probably be a different result. Massey used the Least-Squares Method to develop a set of solutions that were the closest match to the original system. Since there was no unique solution, he imposed the condition that all of the ratings must add up to zero. In this solution, the team with the most positive rating was ranked first, and the team with the most negative rating was ranked last. In 1999, when Massey was a graduate student at Virginia Tech, his method was used as part of the formula for declaring a National Champion in college football. The method was included until 2013, when it was replaced by the current playoff system. One criticism of this method is that it uses the point difference p to help rank the teams. One could argue that there is no real difference in a team winning by 50 points and a team winning by 40 points, but there is a real difference in a team winning by 13 points and a team winning by 3 points. In 2002, Wesley Colley proposed a method that did not depend on point differences at all. The mathematics behind the creation of the method are a bit more involved than Massey's Method, but it still involves solving a system of equations. We would not say that Colley's Method is better or worse than Massey's Method, but that it is different. One thing to note about Massey's Method is that it was designed to be predictive, i.e. if Team A were to play Team F, then the outcome would be a certain way. Colley's Method was not designed to be predictive. In fact, Massey's Method can weight certain games more than others to take into account special circumstances star player out, early season game, win on the road. Mathematics majors at Davidson College use weighted Massey's Method to make very good predictions for the NCAA Basketball Tournament. Read the Sport Ranking Application handout. Then, use the games on the next page to rank the teams. Below are listed all of the Football Bowl Subdivision teams in North Carolina and their games against each other from 2019. 34 6 41 56 24 18 Home team NC State App. State Wake Forest UNC UNC Wake Forest Wake Forest NC State 31 20 44 39 10 Away Team East Carolina Charlotte UNC App. State Duke NC State Duke UNC 34 17 10 27 41 Use an online tool to do all computations. 1. Write the system of equations - one equation for each game in matrix equation form. 2. Compute the Massey matrix, M, and the vector of point differences, d. 3. Solve the ranking question using the Massey Method. 4. Interpret the solution from #3. 5. Write the Colley matrix C, and vector b. 6. Solve the ranking question using the Colley Method. 7. Interpret the solution from #6. Project 6A Ranking Sport Teams This project uses matrix algebra, linear systems, and least-squares technique application. One problem that has perplexed folks for quite some time is how to rank things, especially sports teams. In American sports this is a big issue when it comes to college football. At the highest level there are 130 teams that are competing for a National Championship. There is no reasonable way for all of the teams to play each other. So, ranking the teams is particularly tricky. In 1997, Kenneth Massey proposed a method of ranking in an undergraduate honors project. His idea was very simple: when team A wins against team B by a number of points p, this determines an equation ra - r8 = p, where ra and re are the rating values of the two teams. Any set of games and teams will develop a system of equations. However, the system will not have a solution - when team A and B play again, there will probably be a different result. Massey used the Least-Squares Method to develop a set of solutions that were the closest match to the original system. Since there was no unique solution, he imposed the condition that all of the ratings must add up to zero. In this solution, the team with the most positive rating was ranked first, and the team with the most negative rating was ranked last. In 1999, when Massey was a graduate student at Virginia Tech, his method was used as part of the formula for declaring a National Champion in college football. The method was included until 2013, when it was replaced by the current playoff system. One criticism of this method is that it uses the point difference p to help rank the teams. One could argue that there is no real difference in a team winning by 50 points and a team winning by 40 points, but there is a real difference in a team winning by 13 points and a team winning by 3 points. In 2002, Wesley Colley proposed a method that did not depend on point differences at all. The mathematics behind the creation of the method are a bit more involved than Massey's Method, but it still involves solving a system of equations. We would not say that Colley's Method is better or worse than Massey's Method, but that it is different. One thing to note about Massey's Method is that it was designed to be predictive, i.e. if Team A were to play Team F, then the outcome would be a certain way. Colley's Method was not designed to be predictive. In fact, Massey's Method can weight certain games more than others to take into account special circumstances star player out, early season game, win on the road. Mathematics majors at Davidson College use weighted Massey's Method to make very good predictions for the NCAA Basketball Tournament. Read the Sport Ranking Application handout. Then, use the games on the next page to rank the teams. Below are listed all of the Football Bowl Subdivision teams in North Carolina and their games against each other from 2019. 34 6 41 56 24 18 Home team NC State App. State Wake Forest UNC UNC Wake Forest Wake Forest NC State 31 20 44 39 10 Away Team East Carolina Charlotte UNC App. State Duke NC State Duke UNC 34 17 10 27 41 Use an online tool to do all computations. 1. Write the system of equations - one equation for each game in matrix equation form. 2. Compute the Massey matrix, M, and the vector of point differences, d. 3. Solve the ranking question using the Massey Method. 4. Interpret the solution from #3. 5. Write the Colley matrix C, and vector b. 6. Solve the ranking question using the Colley Method. 7. Interpret the solution from #6
