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Cluster AnalysisProblem 1: Hierarchical Cluster Analysis with the Football Bowl Subdivision (FBS)Which cluster has the largest average football stadium capacity?The cluster with the largest average football stadium capacity is Cluster 3, with an average of 87,615.80.Which school and cluster has the highest football stadium capacity?Michigan Wolverines had the highest stadium capacity, with 114,804. Furthermore, Cluster 5 had the highest amount of stadium capacity in total, with a sum of 1,921,325.How would you characterize the universities in this cluster (Cluster 5)?Cluster 5 has the greatest amount of schools within it, per the higher count of observations in comparison to all other clusters. Accordingly, Cluster 5 has the highest sum of overall stadium capacity. Cluster 5 also has the third largest average of stadium capacity out of all 10 clusters. These factors have critical implications for the universities within this cluster. The universities of Cluster 5 are mostly mid-tier in regards to many variables. These universities bring home a decent amount of athletic revenue, are in the middle of the pack in terms of stadium capacity, and have considerable (but not the highest) endowment and enrollment numbers. Cluster 5 doesn’t have any of the perennial football powerhouses like Ohio State and Michigan, or any of the colleges that boast an extremely high endowment like Stanford, but it contains formidable programs, such as NC State, Virginia Tech, and Ole Miss.What is the smallest cluster (the one with the fewest observations) and what makes it unique?The smallest cluster with the fewest observations is Cluster 10. This cluster is unique because only one observation (university) is present within it, compared to the other 9 clusters that contain multiple universities. The only university within Cluster 10 is Stanford University. Stanford commands its own cluster due to its significantly larger endowment in comparison to the other universities within the sample. Its endowment is so high in comparison, that its medium-range stadium capacity of 50,000 and medium-range enrollment do not drag it into any of the other clusters.What number of clusters seems to be the most natural fit based on the distance (after examining the dendrogram)?9 clusters seem to be the most natural fit based on the distance.Why should we rerun the cluster analysis using different variables or a different number of clusters (after counting the number of schools per cluster)?The cluster analysis should be reran for a different number of clusters. This is because one cluster only has one school, which would not be appropriate for the competitive environment of a football conference. Another cluster has upwards of 30 schools within it, which is much larger than the other 9 clusters and creates an imbalance within the Football Bowl Subdivision (FBS) landscape.Which is the better method (clustering with the variables above or clustering with just latitude and longitude as variables)?Clustering with stadium capacity, latitude, longitude, endowment, and enrollment is much more effective than clustering with just latitude and longitude. One of the reasons is that clustering with only latitude and longitude resulted in a cluster with 98 schools, while many other clusters only had 1 school. There is too much of a disparity between the count of schools for this clustering method to be viable. This method also does not effectively recognize the differences between each school.Problem 2: k-Means Cluster Analysis with the Football Bowl Subdivision (FBS)What is the smallest cluster?The cluster with the fewest observations is Cluster 8, with 1 observation.What is the least dense (most diverse) cluster?Cluster 10 is the least dense cluster, as it has the largest average distance in the cluster. The least dense cluster is the most diverse because a very dense cluster would constitute distances that are very close to each other. In our case, the least dense cluster contains observations with much farther distances from each other, which emphasizes their dissimilarity.What problems do you see with the plan of defining the school membership of the 10 conferences directly with these 10 clusters?Defining school membership under these 10 clusters would bring about a variety of issues. First of all, one cluster, Cluster 8, only has 1 observation, which is unfit for a football conference that inherently requires competition. Secondly, certain clusters contain way too many observations in comparison to others, like Cluster 2 with 22 observations in comparison to Cluster 3 with 5 observations. In accordance with FBS regulations in determining post-season bowl games, how will a committee be able to accurately measure up a school that has 21 similar opponents to face in the course of a regular season with a school that only has 4 other opponents.Problem 3: Both Types of Cluster Analysis with the Football Bowl Subdivision (FBS)What problems do you see with this plan? How could this approach be tweaked to solve the problem?By utilizing clusters by region and sub clusters by stadium capacity, endowment, and enrollment, a few problems become apparent. For example, the west region was broken up into two sub clusters. West Sub Cluster 1 has 17 schools within it, and West Sub Cluster 2 has 9 schools within it. It is our recommendation that for the west region, the number of sub clusters be optimized to have a more similar school amount across all the sub clusters. The east region shares this same uneven distribution of schools throughout its 3 sub clusters, with East Sub Cluster 1 having 24 schools within it, East Sub Cluster 2 having 29 schools, and East Sub Cluster 3 having only 4 schools. This disparity will cause an imbalance in the east region conferences and should be optimized through a more appropriate amount of sub clusters. The same school count disparities as described above are also present in the south region, which could also benefit from further optimization. ................
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