Announcements Turn in Pipet/Buret Calibration Report Wednesday –AP1.2 due + quiz Today’s Lecture –Chapter 4 Material Statistical Tests Calibration and Least Square’s Analysis
Chapter 4 – Gaussian Distributions Now for a “real” limit problem example: A man wants to get life insurance. If his measured cholesterol level is over 240 mg/dL (2,400 mg/L), his premium will be 25% higher. His level is measured and found to be 249 mg/dL. His uncle, a biochemist who developed the test, tells him that a typical standard deviation on the measurement is 25 mg/dL. What is the chance that a second measurement (with no crash diet or extra exercise) will result in a value under 240 mg/dL (e.g. beat the test)?
Statistical Tests t Tests Case 1 –used to determine if there is a significant bias by measuring a test standard (concentration known) 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 two samples) by measuring one sample multiple time 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 sample once by each method (or each sample in each set once)
Case 1 t test Methylmannopyranoside (MMP) example Added as an internal standard at 5 ppm Analysis will tell if sample causes a bias compared to standard
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.
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)
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
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
t- Tests Note: These (case 2 and 3) can be applied to two different senarios: –samples (e.g. do fish caught in a lake near a power plant and far from the plant have the Hg concentration) –methods (analysis method A vs. analysis method B)
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
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
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)
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.
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
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: –not required to know math to determine m and b –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
Example of Calibration Plot Best Fit Line Equation Best Fit Line Deviations from line
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