1. 1 of 37 Key Concepts Underlying DQOs and VSP DQO Training Course Day 1 Module 4 (60 minutes) (75 minute lunch break) Presenter: Sebastian Tindall

2. 2 of 37 Key Points n Have fun while learning key statistical concepts using hands-on illustrations n This module prepares the way for a more in- depth look at the DQO Process and the use of VSP

3. 3 of 37 The Big Picture Decision Error Sampling Cost Remediation Cost Health Risk Waste Disposal Cost Compliance Schedule

4. 4 of 37 Our Focus Sampling Cost $$

5. 5 of 37 Balance in Sampling Design The statistician’s aim in designing surveys and experiments is to meet a desired degree of reliability at the lowest possible cost under the existing budgetary, administrative, and physical limitations within which the work must be conducted. In other words, the aim is efficiency-- the most information (smallest error) for the money. Some Theory of Sampling, Deming, W.E., 1950

6. 6 of 37 Our Methodology: Use Hands-On Illustrations of... n Basic statistical concepts needed for VSP and the DQO Process n Using... Visual Sample Plan

7. 7 of 37 Our Methodology: Use Hands-On Illustrations of... n Basic statistical concepts needed for VSP and the DQO Process n Using Coin flips – Pennies n Demo #1 n Demo #2 – Quarter

8. 8 of 37 How Many Samples Should We Take? 5? 50?

9. 9 of 37 How Many Times Should I Flip a Coin Before I Decide it is Contaminated (Biased Tails)? One tail, 50%Six tails, 1.6% Two tails, 25%Seven tails, 0.8% Three tails, 12.5%Eight tails, 0.4% Four tails, 6%Nine tails, 0.2% Five tails, 3%Ten tails, 0.1%

10. 10 of 37 Football Field One-Acre Football Field 30'0"

11. 11 of 37 Example Problem n A 1-acre field was contaminated with mill tailings in the 1960s n Cleanup standard: –“The mean 226 Ra concentration in the upper 6” of soil must be less than 6.0 pCi/g.” n There is a good chance that actual mean 226 Ra concentration is between 4.0 and 6.0 pCi/g

12. 12 of 37 Example Problem (cont.) n Historical data suggest a standard deviation of 1.6 pCi/g n It costs $1000 to collect, process, and analyze one sample n The maximum sampling budget is $5,000

13. 13 of 37 Chance of Deciding Site is Dirty 1.0 0.5 0.0 6 pCi/g Action Level Low True Mean 226 Ra Concentration High Ideal Rule Graph of Perfect Decision Making

14. 14 of 37 Chance of Deciding Site is Dirty 1.0 0.5 0.0 6 pCi/g Action Level Low True Mean 226 Ra Concentration High Typical Curve Graph of Typical Decision Making

15. 15 of 37 Marbles 9Black 8Blue 7Dark Yellow 6Red 5Green 4White 3Clear Ra-226, pCi/gColor

16. 16 of 37 Simplified Decision Process n Take some number of samples n Find the average 226 Ra concentration in our samples n If we pass the appropriate QA/G-9 test, decide the site is clean n If we fail the appropriate QA/G-9 test, decide the site is dirty

17. 17 of 37 Example of Ad Hoc Sampling Design and the Results n Suppose we choose to take 5 samples for various reasons: low cost, tradition, convenience, etc. n Need volunteer to do the sampling n Need volunteer to record results n We will follow QA/G-9 One-Sample t-Test directions using an Excel spreadsheet

18. 18 of 37 One-Sample t-Test Equation from EPA’s Practical Methods for Data Analysis, QA/G-9 Calculated t = (sample mean - AL) ------------------------ std. dev/sqrt(n) If calculated t is less than table value, decide site is clean

19. 19 of 37 True Mean 226 Ra Concentration Action Level X 2 3 4 5 6 7 8 X X X 4 - 6 = -2 5 - 6 = -1 Comparing UCL to Action Level is Like Student’s t-Test 7 - 6 = 1 8 - 6 = 2 UCL = 4 UCL = 5 UCL = 7 UCL = 8

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21. 21 of 37 Key Concepts Defined

22. 22 of 37 Learn the Jargon t-test UCL - upper confidence limit AL - action level N - target population n - population units sampled  - population mean x - sample mean  - population standard deviation s - sample standard deviation H 0 - null hypothesis  - alpha error rate  - beta error rate  - width of gray region

23. 23 of 37 t-test Calculated t = (sample mean - AL) ------------------------ If calculated t is less than table value, decide site is clean

24. 24 of 37 Upper Confidence Limit, UCL For a 95% UCL and assuming sufficient n: If you repeatedly calculate 95% UCLs for many independent random sampling events, in the long run, you would be correct 95% of the time in claiming that the true mean is less than or equal to your UCLs. Note: Different s will produce different UCLs

25. 25 of 37 Upper Confidence Limit, UCL More commonly, but some experts dislike: For a single UCL, you are 95% confident that the true mean is less than or equal to your calculated UCL. (See Hahn and Meeker in Statistical Intervals A Guide for Practitioners, p. 31).

26. 26 of 37 Action Level A measurement threshold value of the Population Parameter (e.g., true mean) that provides the criterion for choosing among alternative actions.

27. 27 of 37 N Target Population: The set of N population units about which inferences will be made Population Units: The N objects (environmental units) that make up the target or sampled population n The number of population units selected and measured is n

28. 28 of 37 10 x 10 Field Population = All 100 Population Units

29. 29 of 37 10 x 10 Field Population = All 100 Population Units Sample = 5 Population Units 1.5 2.3 1.7 1.9

30. 30 of 37 Population Mean  The average of all N population units i = 1 N XiXi Sample Mean The average of the n population units actually measured n i = 1 XiXi

31. 31 of 37 Population Standard Deviation  The average deviation of all N population units from the population mean Sample Standard Deviation s The “average” deviation of the n measured units from the sample mean

32. 32 of 37 The Null Hypothesis H 0 The initial assumption about how the true mean relates to the action level Example: The site is dirty. (We’ll assume this for the rest of this discussion)

33. 33 of 37 The Alternate Hypothesis H A The alternative hypothesis is accepted only when there is overwhelming proof that the Null condition is false.

34. 34 of 37 The Alpha Error Rate (Type 1, False +)  The chance of deciding that a dirty site is clean when the true mean is equal to the action level The Beta Error Rate (Type 2, False -)  The chance of deciding a clean site is dirty when the true mean is equal to the lower bound of the gray region (LBGR) (Null Hypothesis = Site is Dirty)

35. 35 of 37 The Width of Gray Region AL -   =  Gray Region = AL - LBGR The lower bound of the gray region (   ) is defined as the hypothetical true mean concentration where the site should be declared clean with a reasonably high probability

36. 36 of 37 Decisions about population parameters, such as the true mean, , and the true standard deviation, , are based on statistics such as the sample mean,, and the sample standard deviation, s. Since these decisions are based on incomplete information, they can be in error. Summary

37. 37 of 37 End of Module 4 Thank you Questions? We will now take a 75 minute lunch break. Please be back at 1:00 pm.