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solmidterms06 - PSTAT 120C Midterm Solutions May 9 2006 1(a...

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PSTAT 120C: Midterm Solutions May 9, 2006 1. (a) The dependent variable is the height from the floor and the independent variable is the mass on the spring. (b) In our typical regression equation, height = β 0 + β 1 mass where β 0 is the height when there is no mass on the spring and - β 1 = k . ˆ β 1 = r s y s x = - . 9867 14 . 523 64 . 55 = - 0 . 222 . The value for k = 0 . 222 The 95% confidence interval depends on an estimate of the residual variance s 2 e = n - 1 n - 2 ( 1 - r 2 ) s 2 y = 3 2 ( 1 - . 9867 2 ) 14 . 523 2 = 8 . 3596 . The standard error of ˆ β 1 is s e ( n - 1) s 2 x = 8 . 3596 3(64 . 55) 2 = 0 . 02586 Leading to the confidence interval, ( t critical value with 2 degrees of freedom is 4.303), 0 . 222 - 4 . 303(0 . 02586) = 0 . 111 0 . 222 + 4 . 303(0 . 02586) = 0 . 333 [0 . 111 , 0 . 333] (c) The estimate for ˆ β 0 is ˆ β 0 = ¯ y - ˆ β 1 ¯ x = 81 . 25 + 0 . 222(125) = 109 . The prediction is thus ˆ β 0 + 300 ˆ β 1 = 109 - 300(0 . 222) = 42 . 4. The prediction interval is 42 . 4 ± t 2 ,. 005 s e 1 + 1 n + ( x - ¯ x ) 2 ( n - 1) s 2 x = = 42 . 4 ± 9 . 925 × 2 . 891 1 + 1 4 + (300 - 125) 2 3(64 . 55) 2 = = 42 . 4 ± 55 . 19 [ - 12 . 79 , 97 . 59] (d) This interval is bit long in part because there are only 4 observations, and thus the estimate of the variance may not be too good. However, the real practical concern here
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