8.6: Exercises
- Page ID
- 290687
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)1. The following data are for a series of external standards of Cd2+ buffered to a pH of 4.6.
| [Cd2+] (nM) | 15.4 | 30.4 | 44.9 | 59.0 | 72.7 | 86.0 |
| \(S_{spike}\) (nA) | 4.8 | 11.4 | 18.2 | 26.6 | 32.3 | 37.7 |
(a) Use a linear regression analysis to determine the equation for the calibration curve and report confidence intervals for the slope and the y-intercept.
(b) Construct a plot of the residuals and comment on their significance.
At a pH of 3.7 the following data were recorded for the same set of external standards.
| [Cd2+] (nM) | 15.4 | 30.4 | 44.9 | 59.0 | 72.7 | 86.0 |
| \(S_{spike}\) (nA) | 15.0 | 42.7 | 58.5 | 77.0 | 101 | 118 |
(c) How much more or less sensitive is this method at the lower pH?
(d) A single sample is buffered to a pH of 3.7 and analyzed for cadmium, yielding a signal of 66.3 nA. Report the concentration of Cd2+ in the sample and its 95% confidence interval.
The data in this problem are from Wojciechowski, M.; Balcerzak, J. Anal. Chim. Acta 1991, 249, 433–445.
2. Consider the following three data sets, each of which gives values of y for the same values of x.
| x | y1 | y2 | y3 |
| 10.00 | 8.04 | 9.14 | 7.46 |
| 8.00 | 6.95 | 8.14 | 6.77 |
| 13.00 | 7.58 | 8.74 | 12.74 |
| 9.00 | 8.81 | 8.77 | 7.11 |
| 11.00 | 8.33 | 9.26 | 7.81 |
| 14.00 | 9.96 | 8.10 | 8.84 |
| 6.00 | 7.24 | 6.13 | 6.08 |
| 4.00 | 4.26 | 3.10 | 5.39 |
| 12.00 | 10.84 | 9.13 | 8.15 |
| 7.00 | 4.82 | 7.26 | 6.42 |
| 5.00 | 5.68 | 4.74 | 5.73 |
(a) An unweighted linear regression analysis for the three data sets gives nearly identical results. To three significant figures, each data set has a slope of 0.500 and a y-intercept of 3.00. The standard deviations in the slope and the y-intercept are 0.118 and 1.125 for each data set. All three standard deviations about the regression are 1.24. Based on these results for a linear regression analysis, comment on the similarity of the data sets.
(b) Complete a linear regression analysis for each data set and verify that the results from part (a) are correct. Construct a residual plot for each data set. Do these plots change your conclusion from part (a)? Explain.
(c) Plot each data set along with the regression line and comment on your results.
(d) Data set 3 appears to contain an outlier. Remove the apparent outlier and reanalyze the data using a linear regression. Comment on your result.
(e) Briefly comment on the importance of visually examining your data.
These three data sets are taken from Anscombe, F. J. “Graphs in Statistical Analysis,” Amer. Statis. 1973, 27, 17-21.
3. Fanke and co-workers evaluated a standard additions method for a voltammetric determination of Tl. A summary of their results is tabulated in the following table.
| ppm Tl added | Instrument Response (μμA) | ||||||
| 0.000 | 2.53 | 2.50 | 2.70 | 2.63 | 2.70 | 2.80 | 2.52 |
| 0.387 | 8.42 | 7.96 | 8.54 | 8.18 | 7.70 | 8.34 | 7.98 |
| 1.851 | 29.65 | 28.70 | 29.05 | 28.30 | 29.20 | 29.95 | 28.95 |
| 5.734 | 84.8 | 85.6 | 86.0 | 85.2 | 84.2 | 86.4 | 87.8 |
Use a weighted linear regression to determine the standardization relationship for this data. The data in this problem are from Franke, J. P.; de Zeeuw, R. A.; Hakkert, R. Anal. Chem. 1978, 50, 1374–1380.