The ultimate goal of this project is to model the increasing, decreasing, or stagnant change in the student population at Vanderbilt University over time. The general methodology by which this will be accomplished is to establish linear equations that model the population in the k+1 year of a given grade level, and look for eigenvalues. More specifically, this means setting the determinant of (A- λI) equal to 0 (where A is the matrix derived from the linear equations) in order to solve for λ. From there, the growth rate in the population of Vanderbilt Undergraduate students will be determined. Ultimately, the ratios of the populations of different grades will be concluded.
Coefficients to the population-in-year-k variables will be computed where the data is available, and derived from reasonable assumptions where it is not. For example, the data on transfers in and out of the university after the first year are readily available for the public to see (it is provided by Vanderbilt). However, the overall negative effect of transfers and dropouts on the number of incoming freshmen cannot be easily determined without extensive (and expensive) research. Therefore, those values will be assumed (the assumptions will be a source of error). A 7 by 7 matrix, A, will result from the linear equations derived for the various grades. This presents a few additional challenges in calculating the determinant, as Wolfram Alpha has a 5 by 5 limit. Therefore, Matlab, or some other computing software, will be employed to find the determinant and later put (A- λI) in RREF. The first four variables will be the four years at Vanderbilt in year k. The next variable will be the number of graduates who have just completed their 4 years. The last two variables will be the number of dropouts per year, and the net number of transfers per year.
It is understood that it is possible for “the numbers not to work out”. At the beginning of this proposal it was stated that we were going to see if the population is growing, declining, or remaining stagnant. However, if we actually find that it is declining, we know we have done something wrong (in terms of coefficients, not mathematics, most likely). Then, we will adjust our assumptions. Conveniently, we can also do a linear regression for the past year’s grade sizes (from released VU data) to get the actual student population growth rate. Our ultimate goal is for our numbers to produce those numbers.
Finally, this problem is useful to solve, as Vanderbilt needs to predict growth trends in its student population. This is so that the school may efficiently increase capacity of student housing, faculty size, and other amenities that are necessary for a college.