ce Problems x + canvas.pdx.edu/courses/106252/assignments/1102408?module_item_id=4856656 m Proctor ments THIS IS THE second question of exdin where you have to Snow compere work to get run points. If you use TI calculator functions, you may write your work along with functions and values you I used and answers on a paper or type on an empty Word document or Excel file. If you use Ti calculator app on your computer or phone, you may provide screen shots of those on a word document or Excel file. Clearly provide the function along with values for the work. If you use Excel, you may use the Excel calculator or Excel as a calculator on your computer to provide complete work and the final answer on it. Please provide all the answers to the following questions on the same Excel sheet. If you use formula, you may write all your steps with numbers on a paper and submit a photo of it. Otherwise, you may type all those steps on a Word document or an Excel file. Do NOT round in the middle steps…

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…

Use Gram-Schmidt orthogonalization to find an orthonormal set of vectors with the same span as = 1 2 -0-08-0 a2 = 3 a3 = -2 Find also a QR factorization A = QR with the columns of Q the orthonormal set found above and R upper triangular and invertible and A with columns ā₁, ā2, ā3.

(a) Let Q be an orthonormal matrix. That is, the columns of Q have unit length and are pairwise orthogonal. Prove that the length of Qã is equal to the length of for any column vector x. Note that you can do this by showing that the lengths squared are equal. (b) Prove that if A and B are symmetric and πT Ax = x² Bã for all & then A = B. Make use of coordinate vectors ē;, which are 0 except for a 1 in position i. Explain why, for matrix A, entries can be found by arj = Aej. First show that diagonal = = entries aii bir. Then consider what (ē¿ + ē¿)ªA(ē¿ + ē¿) in terms of entries of A (and similarly for B) and make use of symmetry aij = aji and similarly for B. (c) Prove the converse: If for all x, ||Q|| ||||| then Q is orthonormal. One ap- proach is to use part (b) to show that QTQ = I. Recall that QTQ is symmetric (and trivially I is symmetric). Recall also that, written in matrix notation, vector length (squared) is given by ||7||2 = (this is the dot product written in matrix…

(a) What is wrong with the following statement and proof? A is invertible if and only if ATA is invertible. 'Proof': Since det(A) = det(AT), det(ATA) = det(A)². Either both are 0 (and both A and ATA are not invertible) or both are nonzero (and both A and ATA are invertible). (b) Prove that Null(A) C Null(ATA). That is, prove that if Ax = 0 then AT Ax = 0. (c) Prove that Null(ATA) C Null(A). That is, prove that if AT Añ = 0 then Aã = Ổ. Recall that the length (squared) of a vector v is V, which written in matrix multiplication notation is Tv. (d) Parts (b) and (c) imply that Null(A) = • Nul(ATA). Use this to prove that the columns of A are linearly independent if and only if ATA has an inverse.

Give a 99% confidence interval, for μ1 M2 given the following information. n₁ = 20, 1 = 2.31, s₁ = 0.47 n2 = 40, 2 = 2.21, 82 = 0.55 + Use Technology Rounded to 2 decimal places. Hint Question Help: Video Submit Question ch Η UP T SINK O DELL F2 F3 F4 F5 F6 F7 F8 144 ►11 4

Please could you provide solutions to the following questions. Thanks