Suppose three tests are administered to a random sample of college students. Let X₁,...,Xy be observationvectors in R³ that list the three scores of each student, and for j = 1,2,3, let xj denote a student's score on the jthexam. Suppose the covariance matrix of the data is80 84 36S= 84 117 6936 69 50Let y = ₁x1 + ₂x2 + 3x3 be an "index" of student performance with cf + c3 + c3 = 1 and c₁ 2 0. Find theconstants C₁,C2,C3 so that the variance of y over the data set is as large as possible.Enter the values of c₁, c2,c3 into the answer box below, separated by commas.

As per the question we are given the covariance matrix of a students' data and the index : y = c1x1…

Answered: Suppose three tests are administered to…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Let A and B ben x n matrices. Suppose that det (AB2023) + 0. Show that Col(B) = R".
2. Given the Cartesian coordinates x = 300 mm with standard deviation o, = 0.04 mm, and y = 400 mm with standard deviation ay - 0.05 mm. Assume x and y are uncorrelated. Compute the polar coordinates r and 0 and their variance-covariance matrix
9) O Points: 0 of 1 Part 1 of 3 Save Calculate the correlation coefficient r, letting Row 1 represent the x-values and Row 2 the y-values. Then calculate it again, letting Row 2 represent the x-values and Row 1 the y-values. What effect does switching the variables have on r? Row 1 19 23 31 49 51 70 79 ID. Row 2 156 155 190 141 202 191 181 Calculate the correlation coefficient r, letting Row 1 represent the x-values and Row 2 the y-values. r= (Round to three decimal places as needed.)
Suppose that you have the following model and data to estimate the following equation Nobs Y X2 X3 1 1 1 3 2 2 2 2 3 4 4 5 4 3 3 4 5 2 5 1 a. Find the OLS estimators for the parameters using the matrices above. b. Obtain variance covariance matrix of , Var-Cov() and the standard errors. c. Based on the Var-Cov() assess the significance of the parameter of X2 and also conduct a test whether is less than -0.5 (α=0.05)
Please show step-by-step solution and do not skip steps. Explain your entire process in great detail.
. Let Y Z3+ e denote the standard form of the multivariate regression model where Y is a matrix of n observations on m dependent variables, Z is the nx (r + 1) design matrix, 3 is the matrix of regression coefficients and denotes the error matrix. Let Σ denote the m x m covariance matrix of any row of € and assume the rows are independent. The least square estimate for 3 is 3 (Z'Z)-¹Z'Y. The projection matrix is H = Z(Z'Z)-¹Z'. Let the ith column of 3 be denoted by 3, and the i, jth element of Σ by dij. Define the sample covariance matrix of the residuals by = Prove (a) E(3) = 3 (b) cov (3,3)= Σ (3₁3) = Oik (Z'Z)-¹ 1 n-p-1
A nutritionist is looking at the connection between hours of TV watched and choice of sugary snacks in children identified as at risk for obesity. He asks children to document the number of hours of TV they watch and the number of sugary snacks they eat each day as shown in the first three columns of the following table. In this table, fill in the missing values in the XY column, and then calculate ΣXY. Child Hours of TV Watched (X) Number of Sugary Snacks Eaten (Y) XY A 3 3 9 B 1 3 C 2 1 2 D 4 2 8 E 5 4 ΣXY =42 Fill in the missing values in the X² column of the following table, and then calculate ΣX². Child Hours of TV Watched (X) Number of Sugary Snacks Eaten (Y) X² A 3 3 9 B 1 3 1 C 2 1 D 4 2 16 E 5 4 ΣX² =55 Fill in the missing values in the X – 1 column and the (X – 1)² column of the following table, and then calculate Σ(X – 1)². Child Hours of TV Watched (X)…
A zero mean vector [B]2x1 is unitarily 2X1 1 13 transformed. Given A = 2 1 В, -1 3 1 and Ru: 0 < p < 1. р 1 Obtain covariance matrix RA. Also obtain `A' correlation between A(0) and A(1), if p=0.95.
Suppose three tests are administered to a random sample of college students. Let X₁, ... ; ,XN be observation vectors in R³ that list the three scores of each student, and for j = 1,2,3, let xj denote a student's score on the jth exam. Suppose the covariance matrix of the data is S = 52 26 52 26 52 13 26 26 65 Let y = C₁x1 + €2x2 + c3x3 be an "index" of student performance with c² + c3 + c3 = 1 and c₁ ≥ 0. Find the constants C₁, C2, C3 so that the variance of y over the data set is as large as possible.
6.7. Which of the following pairs of matrices are unitarily equivalent: 0 1 1 a) :) and 1 1 0 1 b) 1/2 ): 1/2 0 and 1

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