Homework100B4

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

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

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