Analytical Chem istry - DePauw University

174 Analytical Chemistry 2.0Fi n d i n g t h e Sl o p e a n d y-In t e r ce p tAlthough we will not formally develop the mathematical equations for alinear regression analysis, you can find the derivations in many standardstatistical texts. 6 The resulting equation for the slope, b 1 , isb1∑n x y − x yi iiii i=22n x − ⎛ ⎞∑xii⎝⎜∑⎠⎟and the equation for the y-intercept, b 0 , isi∑∑ ∑ii5.17y −b xi 1 iiib =5.180nAlthough equation 5.17 and equation 5.18 appear formidable, it is onlynecessary to evaluate the following four summations∑ix i∑iy i∑i∑xy i i∑i2x iSee Section 5F in this chapter for detailson completing a linear regression analysisusing Excel and R.Equations 5.17 and 5.18 are written interms of the general variables x and y. Asyou work through this example, rememberthat x corresponds to C std , and that ycorresponds to S std .Many calculators, spreadsheets, and other statistical software packages arecapable of performing a linear regression analysis based on this model. Tosave time and to avoid tedious calculations, learn how to use one of thesetools. For illustrative purposes the necessary calculations are shown in detailin the following example.Example 5.9Using the data from Table 5.1, determine the relationship between S stdand C std using an unweighted linear regression.So l u t i o nWe begin by setting up a table to help us organize the calculation.x i y i x i y i2x i0.000 0.00 0.000 0.0000.100 12.36 1.236 0.0100.200 24.83 4.966 0.0400.300 35.91 10.773 0.0900.400 48.79 19.516 0.1600.500 60.42 30.210 0.250Adding the values in each column gives2∑ x i= 1.500 ∑ y i= 182.31 ∑ xy i i= 66.701 ∑ x i= 0.550iiii6 See, for example, Draper, N. R.; Smith, H. Applied Regression Analysis, 3rd ed.; Wiley: NewYork, 1998.