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 is89 21 71S= 21 5 1771 17 58Let y = c₁x1 + ₂x2 + c3x3 be an "index" of student performance with cf + c3 + c3 = 1 and c₁ ≥ 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.numbers correct to 2 decimals

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…

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter1: Functions

Section1.2: The Least Square Line

Problem 5E

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For the following table of data. x 1 2 3 4 5 6 7 8 9 10 y 0 0.5 1 2 2.5 3 3 4 4.5 5 a. draw a scatterplot. b. calculate the correlation coefficient. c. calculate the least squares line and graph it on the scatterplot. d. predict the y value when x is 11.
Find the second-order contact matrix PQ mentioned in the text.
An instructor has given a short quiz consisting of two parts. For a randomly selected student, let X = the number of points earned on the first part and Y = the number of points earned on the %3D second part. The accompanying table shows the number of students who obtained the indicated points for X (rows) and Y (column). The class is composed of 100 students. Y 10 15 2 6. 2 10 4 15 20 10 10 1 15 14 Compute the covariance between the scores of students from the two parts of the quiz.
The covariance matrix of an image with three spectral components is shown below. Let x₁, x₂, and x3 denote the spectral components of each pixel in the image. Find a new variable of the form y₁ = C₁X₁ + C₂X₂ + C3x3 that has maximum possible variance, subject to the constraint that c3 + c3 + c3 = 1. What percentage of the total variance in the data is explained by y₁? 27.22 19.55 6.00 S= 19.55 21.88 13.36 6.00 13.36 25.32 (...) Which of the following could be the new variable y₁? O A. Y₁ = -0.56x₁ +0.06x2 -0.83x3 O C. y₁ = 0.62x₁ +0.62x2 +0.62x3 O B. y₁ = -0.54x₁ -0.78x2 +0.32x3 ⒸD. y₁=0.54x₁1 -0.78x2 + 0.32x3 O F. y₁ = 0.62x₁ +0.63x2 +0.47x3 y₁ = 0.62x₁ +0.63x2 -0.47x3 O E. y₁=0.54x₁1 -0.78x2 -0.32x3 O G. y₁ = 0.56x₁ +0.06x2 -0.83x3 O H.
Q1) Test a correlation relationship by using Pearson Superman coefficients. 2 2 y 2 1 1 1 X 3 4 2 2 2 2
A fair dice with sides numbered with {1, 2, 3, 4, 5, 6} was rolled three times. Let X denote number of times 6 was rolled in the first two rolls and let Y denote the total number of 6 rolled. Find the covariance of X and Y.
b. Suppose that e, is zero mean white noise with var(et) = o. Consider the process: i. ii. iii. iv. Y₁ = 1+0.4Y-1 + et - 0.3e-1 0.15€t-2 Write the model using lag operator notation. Assess if the process is covariance stationary. Identify this model as an ARIMA (p, d, q) process; that is, specify p, d, and q. Find μ = E(Y).
2. Let Y = aX + b %3D (1) Find the covariance of X and Y
Let X have a mean of 13 and a variance of 1.1. Let Y have a mean of 9.1 and a variance of 0.75. The covariance of x and y is 0.35. Let Z=4X – 5Y+2. What is the mean and variance of Z?
Consider a set of three assets, one risk-free and two risky. The covariance matrix [0.36 0.07] for the risky assets is given by = 0.07 The risk-free rate is 0.03 0.07 0.29 and risk aversion is A = 4. Calculate the optimal portfolio for this individual.
3. Let X be the nxp data matrix, define the deviation vector as d; = (x1j-Tj, ..., Xnj T;)'. Let 0 be the angle between the jth and Ith deviation vectors. Calculate the lengths of ||d;|| and cos(0). Relate these quantities to the sample covariance and sample correlation.
(b) Let X₁, X₂, X3 be uncorrelated random variables, having the same variance ². Consider the linear transformations Y₁ = X₁ + X₂, Y₂ = X₁ + X3 and Y3 = X₂ + X3 . Find the correlations of Yi, Y; for i #j. (5 marks)

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