Suppose the model for certain data is a parabola y = Bo+B1x+B2x² with observations (1, 2.2), (2, 6.9), (3, 16.1), (4, 28.7), (5, 46.1). Describe the design matrix, the observation vector, and the parameter vector. Using these write down the system of equations to be approximated Xẞ = y, the parts of the normal equations XTX and XTy and write down the normal equations. Solve and determine the residual vector €. You may use a calculator for the computations but show the steps as described above.

. This problem will yield a standard formula given in elementary statistics for a least squares line, making use of the normal equations. (a) Given pairs of data points (x1, Y1), (x2, Y2), ..., (xn, Yn) consider approximating lines of the form y = mx+b. The error e; for the ith pair is the distance between y; and the height (y value) of the line at xi. This is ei = Yi — (mxi + b). If we consider the equations b + x;m = Yi for i n in the variables b and m we can = 1,2, = think of this as a system of equations Ax = 6 where A 1 x1 x2 = : [m] Хп Y1 Уп numbers. Here, note that the variables are m and b and the xi, Yi are given The least squares approximation for this system (which gives the intercept b and slope m of the best least squares line for the data) is the solution to the normal equations AT Ax = ATb. Determine ATA (a 2×2 matrix) and AT (a 2×1 matrix). The entries will be sums of terms involving the x; and y₁. Write these, first using Σ notation and then simplify the notation using…

= a) Recall that the formula for the projection p of vector & onto vector a is p = ±ªã. The plane through the origin in R³ given by ax+by+cz = 0 for real numbers a, b, c, is the set of all points (viewed as vectors) orthogonal to the normal n = (a, b, c) to the plane. To find the distance from a point w (xo, Yo, zo) to the plane ax+by+cz = 0 we can find the projection of w onto the normal ñ and then find the length of this projection. Do this to derive a generic formula for the distance in terms of xo, Yo, Zo, a, b, c. First find the length squared and then take a root. b) To find the distance from a point w (xo, Yo, zo) to a plane ax + by + cz = d, not necessarily not through the origin, one approach is to shift all z coordinates down by d. That is, replace (x, y, z) by (x, y, z — d). The relationship between the plane and w is unchanged so we can find the distance by finding the distance between (xo, Yo, zo - d) and the plane ax + by+cz = 0 through the origin. Do this to derive a…