HW11Solutions - Homework 11 1 a = 1 β expected change in flow rate(y associated with a one inch increase in pressure drop(x =.095 b We expect flow

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Unformatted text preview: Homework 11 1. a. = 1 β expected change in flow rate (y) associated with a one inch increase in pressure drop (x) = .095. b. We expect flow rate to decrease by 475 . 5 1 = β . c. ( ) , 83 . 10 095 . 12 . 10 = + − = ⋅ Y μ and ( ) 305 . 1 15 095 . 12 . 15 = + − = ⋅ Y μ . d. ( ) ( ) 4207 . 20 . 025 . 830 . 835 . 835 . = > = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − > = > Z P Z P Y P ( ) ( ) 3446 . 40 . 025 . 830 . 840 . 840 . = > = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − > = > Z P Z P Y P e. Let Y 1 and Y 2 denote pressure drops for flow rates of 10 and 11, respectively. Then , 925 . 11 = ⋅ Y μ so Y 1 ‐ Y 2 has expected value .830 ‐ .925 = ‐ .095, and s.d. ( ) ( ) 035355 . 025 . 025 . 2 2 = + . Thus ( ) 0036 . 69 . 2 035355 . 095 . ) ( ) ( 2 1 2 1 = > = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + > = > − = > Z P z P Y Y P Y Y P 2. a. 100 50 100 90 80 70 60 50 40 30 20 10 x: y: Rainfall volume (x) vs Runoff volume (y) Yes, the scatterplot shows a strong linear relationship between rainfall volume and runoff volume, thus it supports the use of the simple linear regression model. b. 200 . 53 = x , 867 . 42 = y , ( ) 4 . 586 , 20 15 798 63040 2 = − = xx S , ( ) 7 . 435 , 14 15 643 999 , 41 2 = − = yy S , and ( ) ( ) 4 . 024 , 17 15 643 798 232 , 51 = − = xy S . 82697 . 4 . 586 , 20 4 . 024 , 17 ˆ 1 = = = xx xy S S β and . ( ) 1278 . 1 2 . 53 82697 . 867 . 42 ˆ − = − = β c. ( ) 2207 . 40 50 82697 . 1278 . 1 50 = + − = ⋅ y μ . d. ( )( ) 07 . 357 4 . 324 , 17 82697 . 7 . 435 , 14 ˆ 1 = − = − = xy yy S S SSE β . 24 . 5 13 07 . 357 2 ˆ = = − = = n SSE s σ . e. 9753 . 7 . 435 , 14 07 . 357 1 1 2 = − = − = SST SSE r . So 97.53% of the observed variation in runoff volume can be attributed to the simple linear regression relationship between runoff and rainfall. 3. Let 1 β denote the true average change in runoff for each 1 m 3 increase in rainfall. To test the hypotheses : 1 = β o H vs. : 1 ≠ β a H , the calculated t statistic is 64 . 22 03652 . 82697 . ˆ 1 ˆ 1 = = = β β s t which (from the printout) has an associated p ‐ value of P = 0.000. Therefore, since the p ‐ value is so small, H o is rejected and we conclude that there is a useful linear relationship between runoff and rainfall. A confidence interval for 1 β is based on n – 2 = 15 – 2 = 13 degrees of freedom. , so the interval estimate is 160 . 2 13 , 025 ....
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This note was uploaded on 09/06/2009 for the course ENGRD 2700 taught by Professor Staff during the Spring '05 term at Cornell University (Engineering School).

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HW11Solutions - Homework 11 1 a = 1 β expected change in flow rate(y associated with a one inch increase in pressure drop(x =.095 b We expect flow

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