A marketing rm wants to understand the relationship between the length of time a customer spends in a store and the amount of money they spend at checkout. Here's a small sample of the data they collected by observing customer behavior: '5'\" customer 1 2 3 4 5 6 time ti min 5 10 6 11 25 18 amount 3' $ 10 22 9 25 48 40 The rm assumes a linear model of the form 3; = mt + b, which predicts that a customer who spends t minutes in the store will spend 3; dollars at checkout. The rm's goal is to use the data (131, gal), (t2, yg), . . . , (t6, ya) from the table to determine a good choice of the slope m and the intercept b for the linear model . (a) (b) (C) Is there a single line that passes through all of the data points (131, gal), (t2, yg), . . . , (t6, ya) in the above table? To answer, write down the appropriate matrix equation Ax = c for the line parameters 3: = [TE] and determine whether this linear system of equations has a solution. Instead of solving the equation exactly, the marketing rm might hope to nd a line of \"best t\" that passes as close as possible to each data point. The least-squams error between the predictions of the linear model and the actual amount of money spent by the customers is given by E(m,b)=(mt1 + b m)? + (mtg + b :92)? . . . + (mtg + b y6)2. Find two equations for the values of m and b that minimize E(m, b) by identifying the critical points with respect to the model parameters m and b. Show that these two equations are equivalent to the 2 x 2 linear system ATAx = ATc, where AT is the tmnspose of the matrix A from part (a). Solve the 2 X 2 linear system to determine the values of m and b that minimize E(m, b), i.e., nd the line that minimizes the leastsquares error