Analytical Chem istry - DePauw University

Chapter 5 Standardizing Analytical Methods20111. In Chapter 4 we used a paired t-test to compare two analytical methodsused to independently analyze a series of samples of variable composition.An alternative approach is to plot the results for one method versusthe results for the other method. If the two methods yield identicalresults, then the plot should have an expected slope, b 1 , of 1.00 andan expected y-intercept, b 0 , of 0.0. We can use a t-test to compare theslope and the y-intercept from a linear regression to the expected values.The appropriate test statistic for the y-intercept is found by rearrangingequation 5.23.bt = − b=exps sβ 0 0 0b0 b0Rearranging equation 5.22 gives the test statistic for the slope.texpβ b . b= − 1 1100−1=s sb1 b1Reevaluate the data in problem 25 from Chapter 4 using the samesignificance level as in the original problem.Although this is a common approach forcomparing two analytical methods, itdoes violate one of the requirements foran unweighted linear regression—that indeterminateerrors affect y only. Becauseindeterminate errors affect both analyticalmethods, the result of unweighted linearregression is biased. More specifically, theregression underestimates the slope, b 1 ,and overestimates the y-intercept, b 0 . Wecan minimize the effect of this bias byplacing the more precise analytical methodon the x-axis, by using more samplesto increase the degrees of freedom, andby using samples that uniformly cover therange of concentrations.For more information, see Miller, J. C.;Miller, J. N. Statistics for Analytical Chemistry,3rd ed. Ellis Horwood PTR Prentice-Hall:New York, 1993. Alternativeapproaches are found in Hartman, C.;Smeyers-Verbeke, J.; Penninckx, W.; Massart,D. L. Anal. Chim. Acta 1997, 338,19–40, and Zwanziger, H. W.; Sârbu, C.Anal. Chem. 1998, 70, 1277–1280.12. Consider the following three data sets, each containing value of y forthe same values of x.Data Set 1 Data Set 2 Data Set 3x y 1 y 2 y 310.00 8.04 9.14 7.468.00 6.95 8.14 6.7713.00 7.58 8.74 12.749.00 8.81 8.77 7.1111.00 8.33 9.26 7.8114.00 9.96 8.10 8.846.00 7.24 6.13 6.084.00 4.26 3.10 5.3912.00 10.84 9.13 8.157.00 4.82 7.26 6.425.00 5.68 4.74 5.73(a) An unweighted linear regression analysis for the three data sets givesnearly identical results. To three significant figures, each data sethas a slope of 0.500 and a y-intercept of 3.00. The standard deviationsin the slope and the y-intercept are 0.118 and 1.125 for each