1. David Housman for Math 323 Probability and Statistics Class 05 Ion Sensitive Electrodes

2. David Housman for Math 323 Probability and Statistics Review of Regression Derivation Population Data Regression equation and error estimates

3. David Housman for Math 323 Probability and Statistics Ion Sensitive Electrodes Chemists often use Ion Sensitive Electrodes to measure the ion concentration of aqueous solutions. These devices measure the migration of the charge of these ions and give a reading in millivolts (mV). A standard curve is produced by measuring known concentrations (in ppm) and fitting a line to the millivolt data. Here is data for calcium ISE. Use File > Open worksheet to obtain the data. Obtained from Exercise 7.8-4 of "Probability and Statistical Inference, 6th edition" by Robert V. Hogg & Elliot A. Tanis.

4. David Housman for Math 323 Probability and Statistics Ion Sensitive Electrodes Use Stat > Basic Statistics > Display Descriptive Statistics Variables ppm mV Variable N N* Mean SE Mean StDev ppm 18 0 95.8 15.8 67.1 mV 18 0 3.368 0.414 1.754 Variable Minimum Q1 Median Q3 Maximum ppm 0.000 50.0 87.5 150.0 200.0 mV 1.680 2.093 2.645 4.480 6.670

5. David Housman for Math 323 Probability and Statistics Use Stat > Regression > Regression, response variable ‘mV’ and predictor ‘ppm’. The regression equation is mV = 0.981 + 0.0249 ppm Predictor Coef SE Coef T P Constant 0.9810 0.2302 4.26 0.001 ppm 0.024906 0.001986 12.54 0.000 S = 0.549564 R-Sq = 90.8% R-Sq(adj) = 90.2% Analysis of Variance Source DF SS MS F P Regression 1 47.491 47.491 157.24 0.000 Residual Error 16 4.832 0.302 Total 17 52.323 Ion Sensitive Electrodes

6. David Housman for Math 323 Probability and Statistics Exercise 7.8-4 Use Graph > Scatterplot > With Regression, y-variable ‘mV’ and x-variable ‘ppm’.

7. David Housman for Math 323 Probability and Statistics Ion Sensitive Electrodes Two reasonable alternatives for describing a point on the regression line: 1.The average calibration test was with an ion concentration of 96 ppm generating a potential of 3.4 mV. 2.A solution with no ion concentration generates a potential of 0.98 mV. Two reasonable alternatives for describing the regression line slope: 1.An increase of 1 ppm in the ion concentration typically increases the potential 0.025 mV. 2.An increase of 50 ppm in the ion concentration typically increases the potential 1.25 mV. mV = 0.981 + 0.0249 ppm Variable Mean ppm 95.8 mV 3.368

8. David Housman for Math 323 Probability and Statistics Ion Sensitive Electrodes Four reasonable alternatives for describing the quality of the regression line: 1.This linear relationship accounts for 91% of the variation in the observed potentials. 2.Only 9% of the variation in the observed potentials is due to factors other than the ion concentration. 3.The deviation of the predicted potential from the actual potential will be 0.5 mV on average. 4.The predicted potential should be within 1.1 mV of the actual value about 95% of the time (at least for ion concentrations no greater than 200 ppm). S = 0.549564 R-Sq = 90.8%

9. David Housman for Math 323 Probability and Statistics Ion Sensitive Electrodes But notice that there seems to be a pattern to the errors: the observed potentials fall above the predictions for low and high ion concentrations and the observed potentials fall below the predictions for non-extreme ion concentrations. This indicates a quadratic relationship may be a better fit.

10. David Housman for Math 323 Probability and Statistics Ion Sensitive Electrodes Use Graph > Fitted Line Plot Response ‘mV’, Predictor ‘ppm’., and Type of Regression Model quadratic.