HW11

ISyE 3030 Homework 11 Hannah McClure 11/27/2021 Exercise 11.2.3 Compressive strength x , intrinsic permeability y . n = 14 , ∑ y i = 572 , ∑ y 2 i = 23530 , ∑ x i = 43 , ∑ x 2 i = 157 . 42 , ∑ x i y i = 1697 . 80 a) ˆ β 1 = S xy S xx = ( ∑ i x i y i ) - n ¯ x ¯ y ( ∑ i x 2 i ) - n ¯ x 2 ¯ x = ∑ x i n = 43 14 = 3 . 0714 , ¯ y = ∑ y i n = 572 14 = 40 . 8571 ˆ β 1 = 1697 . 80 - 14 × 3 . 07 × 40 . 8571 157 . 42 - 14 × 3 . 0714 2 = - 59 . 05727 25 . 34853 = - 2 . 32981 ˆ β 0 = ¯ y - ˆ β 1 ¯ x = 40 . 8571 - ( - 2 . 32981 × 3 . 0714) = 48 . 01292 ˆ σ 2 = MSE = SSE n - 2 = SSTO - ˆ β 1 S xy n - 2 SSTO = i y 2 i - n ¯ y 2 = 23530 - 14 × 40 . 8571 2 = 159 . 7633 ˆ σ 2 = SSTO - ˆ β 1 S xy n - 2 = 159 . 7633 - ( - 2 . 32981 × (1697 . 80 - 14 × 3 . 0714 × 40 . 8571)) 14 - 2 = 1 . 847591 plot( 0 , 48.01292 , type = "n" , xlim = c( 0 , 10 ), xlab = "compressive strength (x)" , ylab = "intrinsic permeability (y)" ) abline( 48.01292 , - 2.32981 , col = "green" , lwd = 3 ) 1

0 2 4 6 8 10 30 40 50 60 compressive strength (x) intrinsic permeability (y) b) Compressive strength x = 4 . 3 . ˆ y i = ˆ β 0 + ˆ β 1 x i = 48 . 01292 - 2 . 32981 × 4 . 3 = 37 . 99474 c) Compressive strength x = 3 . 7 . μ y i | x i = E ( y i | x i ) = ˆ β 0 + ˆ β 1 x i = 48 . 01292 - 2 . 32981 × 3 . 7 = 39 . 39262 d) Observed value of permeability at x = 3 . 7 is y = 46 . 1 . e i = y i - ˆ y i ˆ y i = ˆ β 0 + ˆ β 1 x i = 48 . 01292 - 2 . 32981 × 3 . 7 = 39 . 39262 e i = 46 . 1 - 39 . 39262 = 6 . 70738 Exercise 11.2.4 Sound pressure level in decibels x , blood pressure rise in millimeters of mercury y . a) Yes, a simple linear regression model seems reasonable in this situation because it follows a linear pattern. input = read.table( "11-10.txt" , header = TRUE) attach(input) plot(x,y, main = "Relationship Between Noise Exposure and Hypertension" , xlab = "sound pressure level in decibels (x)" , ylab = "blood pressure rise in millimeters of mercury (y)" ) 2