University of California, Los Angeles Department of Statistics Statistics 100C Instructor: Nicolas Christou Homework 3 EXERCISE 1 Data have been collected for 19 observations of two variables, y and x, in order to run a regression of y on x. You are given 19 that sy = 10, (y yi )2 = 180. i=1 i a. Compute the proportion of the variation in y that can be explained by x. [Ans. 0.90] b. Compute the standard error of the estimate (se ). [Ans. 10.59] EXERCISE 2 Data on y and x were collected to run a regression of y on x. The intercept is included. You are given the following: n n x = 76, y = 880, i=1 (xi x)2 = 6800, i=1 (xi x)(yi y ) = 14200, rxy = 0.72, se = 20.13. a. What is the value of 1 ? [Ans. 2.088] b. What is the value of 0 ? [Ans. 721.312] n (y i=1 i c. What is the value of y )2 ? [Ans. 57188 ] sh is ar stu ed d vi y re aC s ou ou rc rs e eH w er as o. co m d. What is the sample size n? [Ans. 70] EXERCISE 3 Observations on both X and Y are standardized, having estimated means of 0 and standard deviation 1. Show that the tted line equation has the form yi = rxi , where r is the correlation coecient between Y and X. EXERCISE 4 Show that for the model yi = 0 + 1 xi + i you can express SSR as n 2 SSR = 1 (xi x)2 i=1 EXERCISE 5 Let F be the F -statistic for the model yi = 0 + 1 xi + i . Show that the F -statistic can be expressed in terms of R2 as follows: F = R2 (n 2). 1 R2 EXERCISE 6 Show that for the model yi = 0 + 1 xi + i it is true that n 2 E(M SR) = 2 + 1 (xi x)2 . i=n EXERCISE 7 For the regression model yi = 0 + 1 xi + i , suppose we want to test H0 : 0 21 = 0 Ha : 0 21 = 0 Construct a t-test to test the above hypothesis. Th EXERCISE 8 Consider the regression model yi = 0 + 1 xi + https://www.coursehero.com/file/9636332/homework3/ Powered by TCPDF (www.tcpdf.org) i. Find var(ei ), where ei = Yi Yi