The Football Bowl Subdivision (FBS) level of the National Collegiate Athletic Association (NCAA) consists of over 100 schools. Most of these schools belong to one of several conferences, or collections of schools, that compete with each other on a regular basis in collegiate sports. Suppose the NCAA has commissioned a study that will propose the formation of conferences based on the similarities of the constituent schools. The file fbs contains data on schools that belong to the Football Bowl Subdivision. Each row in this file contains information on a school.

The variables include football stadium capacity, latitude, longitude, endowment, and undergraduate enrollment.

a. Apply k-means clustering with k = 10 using football stadium capacity, latitude, longitude, endowment, and enrollment as variables. Standardize the input variables to adjust for the different magnitudes of the variables. Consider the cluster containing the most schools and characterize it with respect to its cluster centroid.

b. What problems do you see with the plan for defining the school membership of the 10 conferences directly with the 10 clusters in part (a)?

c. Repeat part (a), but this time do not standardize the values of the input variables.
What value of k results in the largest silhouette score? To understand how the clustering differs from part (a), identify the dominating factor(s) in the formation of these new clusters.