MATH 235 Final, Fall 2000

AMERICAN UNIVERSITY OF BEIRUT MATHEMATICS 235 FINAL EXAMINATION Febrnary 1, 2001 ‘Time = 1 Hour and 30 Minutes 1. For a simple linear regression model which is given by the following: yi = Bo + Oye, + 6, where i = 1,...,n and ¢; are [IID with mean 0 and variance a2. Show that the least squares estimate of B,, G1 = Sry — #)(yi — 9)/ X(ei — #)*. is an unbiased estimator of 6 and find its variance. N Manatees ave larve creatures that live in shallow water along the coast of Florida. Many manatees are injured or killed each year by power boats. Here are data on manatees killed and powerboat registration (in thousands of boats) in Florida for the period 1977 to 1990: Powerboat Manatees Year registrations Killed 1977 447 B 1978 460 21 1979 481 24 - 1980 498 16 1981 513 24 1982 512 20 1983 526 15 Loa 559 34 1985 585 33 1986 614 33 1987 645 39 1988 675 43 1989 Tl 50 1990 719 47 Here is a partial output from the Minitab reeression command for these data. with boat registration as the explanatory variable and manatees killed as the response variable: meme . peers ONC Predictor — Coef Stdev — t-ratio p Constant -41.430 7.412 -5.59 0.000 Boats 0.12486 0.01290 9.68 0.000 s = 4276 and [?* = 88.6%. (a) Coustruct the ANOVA table for this problem. (b) What is the meaning of AR? here? (c) Explain how to find a 95% confidence interval for the number of killed manatees if the number of registered boats is 700. yy = (1) where i = |... aud e; are IID (0,07). o? is assumed to be an noiknown constant. (a) Porrmilate tle model in equation (1) in a matrix form, ie, ¥ = XO+e (b) Compute VX. (Hint: You may make use of the following iden- tities: ST) = a(n + 1/2, NP? = n(n + 1)Qn + 1/6, LV = [n(n + 1)/2)*. and Hi* = n(n + 1)(2n + 1)(8n? + 3n — 1)/30) (c) Can you compute 9, the least squares estimate of @ without in- verting the product matrix X?Y. If ves explain how?