Homework100B4

Homework100B4 - weights (in the unit of kg) of the same...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
STAT 100B HW 4 Due Friday 5pm Problem 1: For the simplest regression model y i = βx i + ± i , i = 1 ,...,n , derive the least squares estimate of β . Problem 2: For the simple regression y i = β 0 + β 1 x i + ± i , i = 1 ,...,n , derive the least squares estimates of β 0 and β 1 . Show that the least squares problem for the simple regression can be transformed into the least squares problem for the simplest regression. Problem 3: Let X be a column vector whose i -th component is x i - ¯ x , and let Y be a column vector whose i -th component is y i - ¯ y , for i = 1 ,...,n . (1) Using geometric argument, show that the sample correlation between ( x i ,i = 1 ,...,n ) and ( y i ,i = 1 ,...,n ) is cos θ , where θ is the angle between X and Y . (2) Let ˆ β 1 be the least squares estimate of β 1 in Problem 2. Using geometric argument, show that ˆ β 1 = h X , Y i / | X | 2 = | Y | cos θ/ | X | . Problem 4: The following is a data set where the first column records the heights (in the unit of cm) of 10 people randomly sampled from the population, and the second column records the
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: weights (in the unit of kg) of the same people. Let x i be height, and y i be weight, for i = 1 ,..., 10. 170 76 165 63 186 83 142 39 176 62 183 81 181 82 179 82 171 85 180 77 (1) Fit the simplest regression y i = βx i + ± i by estimating β by least squares. (2) Fit the simple regression y i = β + β 1 x i + ± i by estimating β and β 1 by least squares. (3) Using the learned simple regression, predict the weight of a person whose height is 175. (4) Draw (with your hands, not computers or calculators) the scatter plot and the fitted regres-sion lines of the simplest regression and simple regression. (5) Calculate the covariance and correlation between the two variables. What are the units of the covariance and correlation? 1...
View Full Document

This note was uploaded on 03/30/2011 for the course STAT 100B taught by Professor Wu during the Spring '11 term at UCLA.

Ask a homework question - tutors are online