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Unformatted text preview: UNIVERSITY OF TORONTO Faculty of Arts and Science DECEMBER EXAMINATIONS 2005 STA 302 H1F / STA 1001 H1F Duration  3 hours Aids Allowed: Calculator LAST NAME: SOLUTIONS FIRST NAME: STUDENT NUMBER: • There are 17 pages including this page. • The last page is a table of formulae that may be useful. For all questions you can assume that the results on the formula page are known. • Tables of the t distribution can be found on page 14 and tables of the F distribution can be found on pages 15 and 16. • Total marks: 95 1 2abc 2de 2f 3 4ab 4cde 5a 5bcde 6 7 1 Continued 1. Suppose we have n = 102 pairs ( X 1 , Y 1 ) , . . . , ( X n , Y n ) and we fit the simple linear regression model Y i = β + β 1 X i + ² i . Here are some summary statistics: X = 50 Y = 100 ∑ n i =1 ( X i X ) 2 = 100 ∑ n i =1 ( Y i Y ) 2 = 200 ∑ n i =1 ( X i X )( Y i Y ) = 100 SSR = 100 (a) (5 marks) Complete the following ANOVA table: Source df SS MS F Regression 1 100 100 100 Error 100 100 1 Total 101 200 (b) (3 marks) Estimate the slope and give a 95% confidence interval for β 1 . b 1 = 100 / 100 = 1 t 100 ,. 025 . = 2 95% CI for β 1 : 1 ± 2(1) / √ 100 = (0 . 8 , 1 . 2) (c) (2 marks) Use the ANOVA table to test H : β 1 = 0 versus H 1 : β 1 6 = 0. Test statistic: F obs = 100 Approximating F 1 , 100 with F 1 , 60 distribution, p < . 001 Strong evidence that β 1 6 = 0 (d) (3 marks) Give a 90% prediction interval for a new observation at X = 50. Since X = 50 , ˆ Y at X = 50 is Y = 100 t 100 , . 05 = 1 . 71 Prediction interval: 100 ± 1 . 671(1) q 1 + 1 102 + 0 = (98 . 3 , 101 . 7) 2 Continued 2. The data in this question were collected as part of a study of the adult female Dun geness crab. While planning fishing restrictions to control crab populations, biologists want to study the growth rate of crabs. The data are measurements of the widest part of the crabs’ shells, in millimeters. Crabs molt regularly, casting off their old shells and growing new ones. Of particular interest is predicting the size of the shell before molting (variable name: presize ) having observed the size of the shell after the crab molted (variable name: postsize ). SAS output is given below for the regression of postsize on presize for 342 adult female crabs raised in a laboratory setting. The REG Procedure Number of Observations Read 342 Number of Observations Used 342 Descriptive Statistics Uncorrected Standard Variable Sum Mean SS Variance Deviation Intercept 342.00000 1.00000 342.00000 postsize 49151 143.71696 7096984 97.13490 9.85570 presize 44133 129.04357 5736616 121.84434 11.03831 The REG Procedure Model: MODEL1 Dependent Variable: presize Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 1 40192 40192 10072.0 <.0001 Error 340 1356.76275 3.99048 Corrected Total 341 41549 Root MSE 1.99762 RSquare 0.9673 Dependent Mean 129.04357 Adj RSq 0.9672 Coeff Var 1.54802 Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > t Intercept...
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 Fall '04
 Gibbs
 Regression Analysis, Variance, postsize, RSquare Adj RSq, Mean Squares Square

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