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 how 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 Hours of Sleep 90 80 70 Mark on Text Hours of Sleep 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 fit" through the data as shown below. Marks and Hours of Sleep 90 70 Mark on Text 60 Hours of Sleep When you take MATH 2930, you will learn way more about the statistics involved here, but for now, let's focus on the linear algebra involved! To make it more appropriate for this class, let's work on a "cubic curve of best fit".Pick five distinct data points (21, y1), (x2, 12), (x3, 13), (14, y4), (x5, 1/5)- 1. Find a cubic polynomial p(x) = a + br + cr + de3 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' + do 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"