3. A marketing firm 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: ith customer time ti (min) amount Yi ($) 1 5 10 2 10 22 3 6 9 4 11 25 5 6 25 18 48 40 The firm assumes a linear model of the form y=mt + b, which predicts that a customer who spends t minutes in the store will spend y dollars at checkout. The firm's goal is to use the data (t1, y), (t2, y2), ..., (t6, Y6) from the table to determine a good choice of the slope m and the intercept b for the linear model . (a) Is there a single line that passes through all of the data points (t1, y), (t2, y2),...,(t6, 46) in the above table? To answer, write down the appropriate matrix equation Ax = c for the line parameters x= and determine whether this linear system of equations has a solution. (b) Instead of solving the equation exactly, the marketing firm might hope to find a line of "best fit that passes as close as possible to each data point. The least-squares error between the predictions of the linear model and the actual amount of money spent by the customers is given by Em, b) = (mti + b - y1)+ (mt2 + b - y2) ... + (mte + b - Y6)2. Find two equations for the values of m and b that minimize E(m, b) by identifying the critical points with respect to the model parameters m and b. (c) Show that these two equations are equivalent to the 2 x 2 linear system AT Ax = ATC, where AT is the transpose of the matrix A from part (a). Solve the 2 x 2 linear system to determine the values of m and b that minimize E(m,b), i.e., find the line that minimizes the least-squares error. 3. A marketing firm wants to understand the relationship between the length of time a customer spends in a store and the amount of money they spend at checkout. Here's a small sample of the data they collected by observing customer behavior: ith customer time ti (min) amount Yi ($) 1 5 10 2 10 22 3 6 9 4 11 25 5 6 25 18 48 40 The firm assumes a linear model of the form y=mt + b, which predicts that a customer who spends t minutes in the store will spend y dollars at checkout. The firm's goal is to use the data (t1, y), (t2, y2), ..., (t6, Y6) from the table to determine a good choice of the slope m and the intercept b for the linear model . (a) Is there a single line that passes through all of the data points (t1, y), (t2, y2),...,(t6, 46) in the above table? To answer, write down the appropriate matrix equation Ax = c for the line parameters x= and determine whether this linear system of equations has a solution. (b) Instead of solving the equation exactly, the marketing firm might hope to find a line of "best fit that passes as close as possible to each data point. The least-squares error between the predictions of the linear model and the actual amount of money spent by the customers is given by Em, b) = (mti + b - y1)+ (mt2 + b - y2) ... + (mte + 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. (c) Show that these two equations are equivalent to the 2 x 2 linear system AT Ax = ATC, where AT is the transpose 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., find the line that minimizes the least-squares error
