Analytical Chem istry - DePauw University

190 Analytical Chemistry 2.0y1086420r = 0.9930 2 4 6 8 10xFigure 5.17 Example of fitting astraight-line to curvilinear data.See Section 4F.2 and Section 4F.3 for areview of the F-test.See Section 4F.1 for a review of the t-test.is a reason for this. For most straight-line calibration curves the correlationcoefficient will be very close to +1, typically 0.99 or better. There isa tendency, however, to put too much faith in the correlation coefficient’ssignificance, and to assume that an r greater than 0.99 means the linearregression model is appropriate. Figure 5.17 provides a counterexample.Although the regression line has a correlation coefficient of 0.993, the dataclearly shows evidence of being curvilinear. The take-home lesson here is:don’t fall in love with the correlation coefficient!The second table in Figure 5.16 is entitled ANOVA, which stands foranalysis of variance. We will take a closer look at ANOVA in Chapter 14.For now, it is sufficient to understand that this part of Excel’s summaryprovides information on whether the linear regression model explains asignificant portion of the variation in the values of y. The value for F is theresult of an F-test of the following null and alternative hypotheses.H 0 : regression model does not explain the variation in yH A : regression model does explain the variation in yThe value in the column for Significance F is the probability for retainingthe null hypothesis. In this example, the probability is 2.5×10 –6 %, suggestingthat there is strong evidence for accepting the regression model. Asis the case with the correlation coefficient, a small value for the probabilityis a likely outcome for any calibration curve, even when the model is inappropriate.The probability for retaining the null hypothesis for the data inFigure 5.17, for example, is 9.0×10 –7 %.The third table in Figure 5.16 provides a summary of the model itself.The values for the model’s coefficients—the slope, b 1 , and the y-intercept,b 0 —are identified as intercept and with your label for the x-axis data, whichin this example is Cstd. The standard deviations for the coefficients, s b 0 ands b , are in the column labeled Standard error. The column t Stat and the1column P-value are for the following t-tests.slope H 0 : b 1 = 0, H A : b 1 ≠ 0y-intercept H 0 : b 0 = 0, H A : b 0 ≠ 0The results of these t-tests provide convincing evidence that the slope isnot zero, but no evidence that the y-intercept significantly differs fromzero. Also shown are the 95% confidence intervals for the slope and they-intercept (lower 95% and upper 95%).Pr o g r a m t h e Fo r m u l a s Yo u r s e l fA third approach to completing a regression analysis is to program a spreadsheetusing Excel’s built-in formula for a summation=sum(first cell:last cell)and its ability to parse mathematical equations. The resulting spreadsheetis shown in Figure 5.18.