Fundamentals of Probability and Statistics for Engineers

Linear Models and Linear Regression 341Table 11. 1 Percentage yield, y i , with process temperature, x i , for Example 11.1Ex ample 11. 1. Problem: it is expected that the average percentage yield, Y ,from a chemical process is linearly related to the process temperature, x, inC.Determine the least-square regression line for E fY g on the basis of 10 observationsgiven in Table 11.1.Answer: in view of Equations (11.7) and (11.8), we need the followingquantities:The substitution of these values into Equations (11.7) and (11.8) givesThe estimated regression line together with observed sample values is shownin Figure 11.2.It is noteworthy that regression relationships are valid only for the rangeof x values represented by the data. Thus, the estimated regression line inExample 11.1 holds only for temperatures between 45Cand90C.Extrapolationof the result beyond this range can be misleading and is not valid in general.Another word of caution has to do with the basic linear assumption betweenE fY g and x. Linear regression analysis such as the one performed in Example11.1 is based on the assumption that the true relationship between E fY g andx is linear. Indeed, if the underlying relationship is nonlinear or nonexistent,i1 2 3 4 5 6 7 8 9 10x (C) 45 50 55 60 65 70 75 80 85 90y 43 45 48 51 55 57 59 63 66 68X ṋi 1x ˆ 1 X nx i ˆ 1 …45 ‡ 50 ‡‡90† ˆ67:5;n 10iˆ1y ˆ 1 X ny i ˆ 1 …43 ‡ 45 ‡‡68† ˆ55:5;n 10iˆ1X niˆ1…x i x† 2 ˆ 2062:5;… x i x†…y i y† ˆ1182:5:^ ˆ 1182:52062:5 ˆ 0:57;^ ˆ 55:5 0:57…67:5† ˆ17:03:TLFeBOOK