Watters et al. have discussed uncertainties associated with calibration curves

Chapter 10, Problem 10-14

(choose chapter or problem)

Watters et al. have discussed uncertainties associated with calibration curves forfCP-OES and procedures for optimizing the results of least-squares analysis ofsuch data.'1(a) The central issue addressed by these workers is whether a weighted or anunweighted least-squares procedure is appropriate for fCP calibration. Whatis the primary criterion for deciding which procedure to use"(b) What do the terms homoscedasticity and heteroscedasticitv mean')(c) The model used by these workers for thc lCP curves is rc'presented bv thefollowing equation: .Y,j = a + bXj + errorllDefine each variable in the equation. and describe the significance of therelationship embodied in it.(d) Walters et al. chose to model the error in [CP working curves in terms ofconcentration rather than intensity. What is their rationale for this choice')(e) One of the suggested models for the error in the calibration curves is(T(X) = c + dx + ex'Describe the significance of each variable. and characterize the nature of themodel.(f) How is (T(X) determined in practice')(g) Shot noise and source flicker noise correspond to which variables in theexpression shown in (e)"(h) What experimental noise sources are constant" Which variable in theexpression shown in (e) corresponds to these sources (or source)~(i) What is the relationship of the following alternative model to the modelin (e)"(T'(X) = g + hx + h'(j) What is the purpose of the models in (e) and (i)') .(k) What is the significance of each of the variables a and b')(I) The authors conclude that "if heteroscedasticity is ignored, confidence intervalswill be too narrow at the high end and too wide at the low end ofthe fCP calibration curve. The magnitude of these effects will depend onthe particular dilution scheme used to make the calibration standard solutions."How does the dilution scheme for standards affect the results ofleast-squares analysis on fCP calibration curves')(m) Create an Excel spreadsheet similar to the one shown below containingdata from Table f of the paper by Watters et aI., and perform both unweightedand weighted least-squares analysis of the data. First use L1NESTto perform the unweighted analysis, and then carry out an unweightedanalysis using Solver to minimize cell C25 by varying B14 and C14.Compare the results obtained by both methods. Compare and contrastadvantages of each method." Repeat the analysis for the ten-replicate data using the formulas providedin the spreadsheet documentation. Formulas entered in row 16 must becopied into rows 17-24. Minimize cell F25 by varying cells E14 and F14 usingSolver to obtain estimates of the slope and intercept. Repeat the analysisfor the four-replicate data using Sol\'er to minimize cell 125 by varyingcells H14 and Il4. Compare your procedures and results to those of Watterset al. and comment on any differences. What advantage does weightedleast-squares analysis have over unweighted analysis"(n) Add a section to your Excel spreadsheet to compute the mean and standarddeviation of the concentration of an analyte given a number of measurementsof sample IC? emission intensity."(0) There are a number of commercial and noncommercial sources on theInternet for Excel add-ins, which are programs or function packs thatsupplement the built-in functions of Excel. For example, Solver is actuallyan add-in that is produced by an independent contractor and that is availablefrom the vendor in an enhanced version. Use a search engine such asGoogle to locate the websites of programmers and vendors that offer addinsto perform weighted least-squares analysis. One such vendor is XLSTAT(www.xlstat.com). Download the demonstration version of XLSTAT, installit on your computer, and use it to perform weighted and unweighted leastsquaresanalysis on the data of (n). Note that you must calculate the weightingfactors from the standard deviations given in the paper and that youmay need to scale them. You should calculate columns in your spreadsheetcontaining lIs2, and then divide each cell by the largest value in the column.Weighting factors need be only proportional to the variance, so you canscale them in any convenient way. Compare the results from XLSTAT withthe results from your spreadsheet, and comment on the ease of use andfunctionality of XLSTAT. XLSTAT contains many other useful statisticsand numerical analysis functions and is available at modest cost to studentswho desire permanent use of the add-in.