1. Chem. 31 – 9/23 Lecture Guest Lecture Dr. Roy Dixon

2. Announcements Small renewable resources company looking for interns for chemical analysis work (see bulletin board outside 446) Additional Problem with no name on it Due Today –Pipet/Buret Calibration Lab Report Today’s Lecture –Error and Uncertainty Finish up Statistical Tests –Least Squares Calibration (last part of Chapter 4)

3. Statistical Tests t Tests - Review Case 1 –used to determine if there is a significant bias by measuring a test standard and determining if there is a significant difference between the known and measured concentration Case 2 –used to determine if there is a significant differences between two methods (or samples) by measuring one sample multiple times by each method (or each sample multiple times) Case 3 –used to determine if there is a significant difference between two methods (or sample sets) by measuring multiple samples once by each method (or each sample in each set once)

4. Case 2 t test Example A winemaker found a barrel of wine that was labeled as a merlot, but was suspected of being part of a chardonnay wine batch and was obviously mis-labeled. To see if it was part of the chardonnay batch, the mis- labeled barrel wine and the chardonnay batch were analzyed for alcohol content. The results were as follows: –Mislabeled wine: n = 6, mean = 12.61%, S = 0.52% –Chardonnay wine: n = 4, mean = 12.53%, S = 0.48% Determine if there is a statistically significant difference in the ethanol content.

5. Case 3 t Test Example Case 3 t Test used when multiple samples are analyzed by two different methods (only once each method) Useful for establishing if there is a constant systematic error Example: Cl - in Ohio rainwater measured by Dixon and PNL (14 samples)

6. Case 3 t Test Example – Data Set and Calculations Conc. of Cl - in Rainwater (Units = uM) Sample #Dixon Cl - PNL Cl - 19.917.0 22.311.0 323.828.0 48.013.0 51.77.9 62.311.0 71.99.9 84.211.0 93.213.0 103.910.0 112.79.7 123.88.2 132.410.0 142.211.0 7.1 8.7 4.2 5.0 6.2 8.7 8.0 6.8 9.8 6.1 7.0 4.4 7.6 8.8 Calculations Step 1 – Calculate Difference Step 2 - Calculate mean and standard deviation in differences ave d = (7.1 + 8.7 +...)/14 ave d = 7.49 S d = 2.44 Step 3 – Calculate t value: t Calc = 11.5

7. Case 3 t Test Example – Rest of Calculations Step 4 – look up t Table –(t(95%, 13 degrees of freedom) = 2.17) Step 5 – Compare t Calc with t Table, draw conclusion –t Calc >> t Table so difference is significant

8. t- Tests Note: These (case 2 and 3) can be applied to two different senarios: –samples (e.g. sample A and sample B, do they have the same % Ca?) –methods (analysis method A vs. analysis method B)

9. F - Test Similar methodology as t tests but to compare standard deviations between two methods to determine if there is a statistical difference in precision between the two methods (or variability between two sample sets) As with t tests, if F Calc > F Table, difference is statistically significant S 1 > S 2

10. Grubbs Test Example Purpose: To determine if an “outlier” data point can be removed from a data set Data points can be removed if observations suggest systematic errors Example: Cl lab – 4 trials with values of 30.98%, 30.87%, 31.05%, and 31.00%. Student would like less variability (to get full points for precision) Data point farthest from others is most suspicious (so 30.87%) Demonstrate calculations

11. Dealing with Poor Quality Data If Grubbs test fails, what can be done to improve precision? –design study to reduce standard deviations (e.g. use more precise tools) –make more measurements (this may make an outlier more extreme and should decrease confidence interval)

12. Statistical Test Questions 1.A chemist has developed a new test to measure gamma hydroxybutyrate that is expected to be faster and more precise than a standard method. What test should be used to test for improved precision? Are multiple samples needed or multiple analyses of a single sample? 2.The chemist now wants to compare the accuracy for measuring gamma hydroxybutyrate in alcoholic beverages. Describe a test to determine if the method is accurate.

13. Calibration For many classical methods direct measurements are used (mass or volume delivered) Balances and Burets need calibration, but then reading is correct (or corrected) For many instruments, signal is only empirically related to concentration Example Atomic Absorption Spectroscopy –Measure is light absorbed by “free” metal atoms in flame –Conc. of atoms depends on flame conditions, nebulization rate, many parameters –It is not possible to measure light absorbance and directly determine conc. of metal in solution –Instead, standards (known conc.) are used and response is measured Light beam To light Detector

14. Method of Least Squares Purpose of least squares method: –determine the best fit curve through the data –for linear model, y = mx + b, least squares determines best m and b values to fit the x, y data set –note: y = measurement or response, x = concentration, mass or moles How method works: –the principle is to select m and b values that minimize the sum of the square of the deviations from the line (minimize Σ[y i – (mx i + b)] 2 ) –in lab we will use Excel to perform linear least squares method

15. Example of Calibration Plot Best Fit Line Equation Best Fit Line Deviations from line

16. Assumptions for Linear Least Squares Analysis to Work Well Actual relationship is linear All uncertainty is associated with the y- axis The uncertainty in the y-axis is constant

17. Calibration and Least Squares - number of calibration standards (N) NConditions 1Must assume 0 response for 0 conc.; standard must be perfect; linearity must be perfect 2Gives m and b but no information on uncertainty from calibration Methods 1 and 2 result in lower accuracy, undefined precision 3Minimum number of standards to get information on validity of line fit 4Good number of standards for linear equation (if standards made o.k.) More standards may be needed for non-linear curves, or samples with large ranges of concentrations

18. Use of Calibration Curve Mg Example: An unknown solution gives an absorbance of 0.621 Use equation to predict unknown conc. y = mx + b x = (y – b)/m x = (0.621 + 0.0131)/2.03 x = 0.312 ppm Can check value graphically Calibration “Curve”

19. Use of Calibration Curve - Uncertainty in Unknown Concentration Uncertainty given by S x (see below): Notes on equation: m = slope, S y = standard error in y n = #calibration stds k = # analyses of unknown, x i = indiv std conc., y i = unknown response The biggest factors are S y and m Two other parameters that often indicate calibration quality are R 2 and b. R 2 should be close to 1 (good is generally >0.999); b should be small relative to y of lowest standard.

20. Use of Calibration Curve - Quality of Results Quality of Results Depends on: –Calibration Results R 2 value (measure of variability of response due to conc.) Reasonable fit –Range of Unknown Concentrations next slide Better fit by curve

21. Use of Calibration Curve - Quality of Results Quality of Results Depends on: –Calibration Results on last slide –Range of Unknown Concentrations Extrapolation outside of range of standards should be avoided Best concentration range Range of Standards (0.02 to 0.4 ppm) Absolute Uncertainty Relative Uncertainty Best Range: upper 2/3rds of standard range

22. Calibration Question A student is measuring the concentrations of caffeine in drinks using an instrument. She calibrates the instruments using standards ranging from 25 to 500 mg/L. The calibration line is: Response = 7.21*(Conc.) – 47 The response for caffeine in tea and in espresso are 1288 and 9841, respectively. What are the caffeine concentrations? Are these values reliable? If not reliable, how could the measurement be improved?