Matlab, Fourth Edition: A Practical Introduction to Programming and Problem Solving
4th Edition
ISBN: 9780128045251
Author: Stormy Attaway Ph.D. Boston University
Publisher: Elsevier Science
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THIS IS THE second question of exdin where you have to Snow compere work to get run points.
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Suppose the model for certain data is a parabola y
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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
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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…
Chapter 6 Solutions
Matlab, Fourth Edition: A Practical Introduction to Programming and Problem Solving
Chapter 6, Problem 6.1PChapter 6, Problem 6.2PChapter 6, Problem 6.3PChapter 6, Problem 6.4PChapter 6, Problem 6.5PChapter 6, Problem 6.6PChapter 6, Problem 6.7PChapter 6, Problem 1EChapter 6, Problem 2EChapter 6, Problem 3E
Chapter 6, Problem 4EChapter 6, Problem 5EChapter 6, Problem 6EChapter 6, Problem 7EChapter 6, Problem 8EChapter 6, Problem 9EChapter 6, Problem 10EChapter 6, Problem 11EChapter 6, Problem 12EChapter 6, Problem 13EChapter 6, Problem 14EChapter 6, Problem 15EChapter 6, Problem 16EChapter 6, Problem 17EChapter 6, Problem 18EChapter 6, Problem 19EChapter 6, Problem 20EChapter 6, Problem 21EChapter 6, Problem 22EChapter 6, Problem 23EChapter 6, Problem 24EChapter 6, Problem 25EChapter 6, Problem 26EChapter 6, Problem 27EChapter 6, Problem 28EChapter 6, Problem 29E
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- = 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…arrow_forwardUse 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.arrow_forward(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…arrow_forward
- (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.arrow_forwardGive 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 4arrow_forwardPlease could you provide solutions to the following questions. Thanksarrow_forward
- Please could you provide solutions to the following questions. Thanksarrow_forwardor Beer and blood alcohol content: Many people believe that gender, weight, drinking habits, and many other factors are much more important in predicting blood alcohol content (BAC) than simply considering the number of drinks a person consumed. Here we examine data from sixteen student volunteers at Ohio State University who each drank a randomly assigned number of cans of beer. These students were evenly divided between men and women, and they differed in weight and drinking habits. Thirty minutes later, a police officer measured their blood alcohol content (BAC) in grams of alcohol per deciliter of blood (Malkevitc and Lesser, 2008). The scatterplot and regression table summarize the findings. to search BAC (grams per deciliter) 0.15- 0.10- 0.05- Part of Ti Calculator Output Ti Calculator y = ax+b Be and p = 0 t = 7.48 P 0.0000 df = 14 a = 0.0180 b = -0.0127 Part of Excel Output 2 4 0 0 6 Cans of beer (Intercept) beers Estimate -0.0127 0.0180 Std. Error t value 0.0126 P(>|t|) -1.00…arrow_forwardProctor ts Ces rces arch Question 31 The data below was collected from manufacturer advertisements of their vehicles horsepower (x) and highway gas mileage (mpg=y). Use this data to answer the following questions. horsepower 157 250 340 350 390 190 220 mpg 33 28 15 17 11 35 42 1. Find the p-value to determine if there is a linear correlation between horsepower and highway gas mileage (mpg). Record the p-value below. Round to four decimal places. p-value= 2. Is there a linear correlation between horsepower and highway gas mileage (mpg)? ? 3. If there is a linear correlation, write the correlation coefficient below. Otherwise, leave it blank. Round your final answer to four decimal places. Be careful with your sign. T= 4. If there is a linear correlation, write the regression equation below. Otherwise, leave it blank. Round all numbers to four decimal places. ŷ 5. Using the data shown above, predict the the highway gas mileage (mpg) for a car that has a horsepower of 225. Round your…arrow_forward
- arch es For Question 24 < Annual high temperatures in a certain location have been tracked for several years. Let X represent the number of years after 2000 and Y the high temperature. Based on the data shown below, calculate the linear regression equation using technology (each number to three decimal places). X y 4 33.38 5 31 6 31.92 7 33.64 8 33.06 9 32.28 10 33.7 11 30.82 12 31.14 13 32.56 14 34.58 Click Download CSV to download csv file of data or copy/paste the data into Excel. After downloading the file, you may want to save it as an Excel Workbook. The fitted regression equation is ŷ = Interpret the y-intercept of the equation: O In 2000, the temperature was about 32.152. In 2014, the temperature was about 34.58. In 2004, the temperature was about 32.152. In 2004, the temperature was about 0.045. Question Help: Video Submit Question F2 F3 F4 144 UP T M F5 114 % L F6 144 DOLL F7 6 F8 F9 F10 &arrow_forwardMath 105Z Worksheet Week 8 Instructions: Answer questions about the following contexts. You'll be graded for accuracy on problems 3 and 4 but you should try to complete the entire worksheet for full credit (see the rubric on Canvas). You'll need to upload your work as a single PDF to the appropriate assignment in Canvas. 1. A random sampling of the speeds of 12 cars monitored on a small stretch of highway yielded the following data: 45 55 50 60 60 55 60 50 70 65 55 60 (a) Find the mean speed. (b) Find the median speed. (c) Find the mode speed. (d) Construct a histogram to represent the data. 2. The mean exam score for the 24 students in section M is 72, and the mean exam score for the 30 students of section N is 78. Find the mean score for all 54 students. Page 1arrow_forwardDon't give answer without proper knowledge of this statistics and probability question.arrow_forward
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