spr03_1 - STA 6208 Spring 2003 Exam 1 Print Name: SSN: All...

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STA 6208 – Spring 2003 – Exam 1 Print Name: SSN: All questions are based on the following two regression models, where SIMPLE REGRESSION refers to the case where p =1 , and X is of full column rank (no linear dependencies among the predictor variables) Model 1: Y i = β 0 + β 1 X i 1 + ··· + β p X ip + ε i i ,...,n ε i NID (0 2 ) Model 2: Y = X β + ε X n × p ± β p ± × 1 ε N ( 0 2 I ) 1) For Model 1 , derive the normal equations. 2) A researcher is interested in the relationship between the education level and salaries in rural counties in the U.S. He obtains the percentage of adults over 25 with a college education in each county ( X ) and the per capita income of the county ( Y , in $1000s). He obtains the following summary statistics, based on a sample of n = 30 counties: (12 points) ± ( X - X ) 2 = 2207 . 45 ± ( X - X )( Y - Y )=658 . 37 ± ( Y - Y ) 2 = 654 . 86 X =41 . 92 Y =35 . 83 a) Give least squares estimates of the parameters of Model 1 : β 0 , β 1 . b) It can be shown that: ± ( Y - ˆ Y ) 2 = ± ( Y - Y ) 2 - ( ( X - X )( Y - Y )) 2 ( X - X ) 2 Give an unbiased estimate of σ 2 .
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spr03_1 - STA 6208 Spring 2003 Exam 1 Print Name: SSN: All...

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