At some point in high school, you probably worked on a problem that required a 'line of best fit". Many years ago._ I taught a small statistics class in the summer evenings at Brock University. My memory of this is very hazy it may not have been summer, it may not have been evenings. it may not have been Brock and it may not have been me. 'Whatever the circumstances, the following data were collected for students before a big test. I asked them to report hmv many hours of sleep they got before the test and then I recorded their grade. Each small circle on the graph below represents a student. Marks and Hum a! Shop Hours-dang: The goal was to try to find a relationship between sleep and academic performance on the test. In this case. we found a \"line of best t" through the data as shown below. Marks and Hum a! Shop 2 4 i a 1n Hours-dang: \"'hen you take I'L-IATH ease, you will learn way more about the statistics involved here. but for now._ let's focus on the linear algebra involvedl To make it more appropriate for this class, let's work on a \"cubic curve of best t". Pick five distinct data points (21, 31), (12, 2), (x3, y3), (TA, y4), (x5, 15). 1. Find a cubic polynomial p(x) = a + br + cx2 + dx3 that fits perfectly through the first three points. This is called 'polynomial interpolation'. It turns out this system of equations is always consistent (no matter which distinct points you pick). Is the polynomial unique? 2. Find a cubic polynomial p(x) = a + br + cr' + da' that fits perfectly through the first four points. Once again, it turns out this system of equations is always consistent (no matter which distinct points you pick). Is the polynomial unique? 3. Is it always possible to find a cubic polynomial that fits through all five points? 4. If AX = B is the system of equations that would need to be solved to find the solution to Question 3, solve instead the system of equations ATAX = ATB to find the "best fit cubic function". Grading Notes: Completing the questions listed above with reasonably well-written solutions that are mostly error-free will earn a grade of 3.5/5. This is a B grade and B is defined at York as being Good. Good, by definition, means that you did everything was asked of you and showed a reasonable understanding of the course materials. Grades of B+, A and A+ are reserved for work that goes above and beyond this standard. There are many ways you can go above and beyond the B standard and I'm leaving it up to you to determine what that means to you - I won't be defining it! I know this will be stressful for some people in the class and I acknowledge that stress