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midterm1solu

# midterm1solu - Midterm I Solution Problem 1(1a 1 = y1 y =...

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Midterm I Solution Problem 1 (1a) ˆ τ 1 = ¯ y 1 · - ¯ y ·· = 17 . 6 - 19 . 933 = - 2 . 333 ˆ τ 2 = ¯ y 2 · - ¯ y ·· = 18 . 4 - 19 . 933 = - 1 . 533 ˆ τ 3 = ¯ y 3 · - ¯ y ·· = 23 . 8 - 19 . 933 = 3 . 867 ˆ σ 2 = MSE = SSE N - a = 3 i =1 ( n i - 1) S 2 i N - a = (5 - 1) × ( 2 . 881 2 + 3 . 050 2 + 4 . 438 2 ) 15 - 3 = 12 . 4328 (1b) H 0 : τ 1 = τ 2 = τ 3 = 0 v.s. H a : not all τ i are 0 SSE = (5 - 1)(2 . 881 2 + 3 . 050 2 + 4 . 438 2 ) = 149 . 194 . F 0 = MS treatment MSE = (SST - SSE) / ( a - 1) SSE / ( N - a ) = (262 . 933 - 149 . 194) / (3 - 1) 149 . 194 / (15 - 3) = 4 . 574 F 0 > F 0 . 05 (2 , 12) = 3 . 89, reject H 0 , there exists difference in fuel dfficiency between the three SUV brands. (1c) (i) Let L 1 = μ 3 - μ 1 , H 0 : L 1 = 2 v.s. H a : L 1 6 = 2. ˆ L 1 = ¯ y 3 · - ¯ y 1 · = 23 . 8 - 17 . 6 = 6 . 2 S.E. ˆ L 1 · = q MSE X c 2 i /n i = r 12 . 433 × 2 5 = 2 . 23 test statistic t 0 = ˆ L 1 - 2 S.E. ˆ L 1 · = 1 . 883 < t 0 . 025 (12) = 2 . 179 Fail to reject H 0 , that is, fail to reject manufacturer C’s claim that brand C outperforms brand A by a margin of 2 MPG. (ii) Let L 2 = μ 3 - μ 2 , H 0 : L 2 = 2 v.s. H a : L 2 6 = 2 ˆ L 2 = ¯ y 3 · - ¯ y 2 · = 23 . 8 - 18 . 4 = 5 . 4 S.E. ˆ L 2 · = q MSE X c 2 i /n i = r 12 . 433 × 2 5 = 2 . 23 test statistic t 0 = ˆ L 1 - 2 S.E.

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midterm1solu - Midterm I Solution Problem 1(1a 1 = y1 y =...

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