What's up with the Degree 12 Polynomial Model?

bkirk272

In Notebook 4, students are prompted to create a Degree 3, Degree 5, and Degree 12 model to compare default rates to average SAT scores. The Degree 3 and Degree 5 code is posted and runs as expected, but the code for Degree 12 comes pre-defined in commented-out form, and removing the # to run the code produces an image that differs from the model that is included as an image.

Are students supposed to run the commented-out code and assume the image represents the model that is produced? Why does removing the comment and running the code properly produce a different model than that of the image?

dashiell-young-saver

Good question! For computational efficiency reasons, R's graphing functions use approximations of least square solutions rather than the "true" solutions (the approximation method is called QR decomposition, and you can find a very technical and thorough explanation of it here). In the vast majority of situations, the approximations are good enough. However, when you start raising numbers to the 12th power, even very small estimation errors become very large ones. Hence, the graph that R outputs will look "off."

So, we commented out the code and put an image of a proper graph in its place. You can tell students that they shouldn't run the code, due to computational issues. That's why we already have the graph there for them 🙂. Next year, we may just hide the code cell entirely, to avoid confusion.

dylarm

dashiell-young-saver I think some of us may be curious, how did you generate the proper graph? I had assumed it was a computation time limitation of the server hosting the notebooks.

dashiell-young-saver

dylarm I plotted the true equation in Desmos, then transposed the function on top of the raw data 🙂