1 of 37 Key Concepts Underlying DQOs and VSP DQO Training Course Day 1 Module 4 (60 minutes) (75...
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Transcript of 1 of 37 Key Concepts Underlying DQOs and VSP DQO Training Course Day 1 Module 4 (60 minutes) (75...
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Key Concepts Underlying DQOs and VSP
DQO Training Course Day 1
Module 4
(60 minutes)(75 minute lunch break)
Presenter: Sebastian Tindall
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Key Points
Have fun while learning key statistical concepts using hands-on illustrations
This module prepares the way for a more in-depth look at the DQO Process and the use of VSP
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TheBigPicture
Decision Error
Sampling Cost
Remediation Cost
Health Risk
Waste Disposal
CostCompliance
Schedule
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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
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Our Methodology:Use Hands-On Illustrations of...
Basic statistical concepts needed for VSP and the DQO Process
Using...Visual Sample
Plan
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Our Methodology:Use Hands-On Illustrations of...
Basic statistical concepts needed for VSP and the DQO Process
Using Coin flips– Pennies
Demo #1 Demo #2
– Quarter
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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%
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Example Problem A 1-acre field was contaminated with mill
tailings in the 1960s Cleanup standard:
– “The mean 226Ra concentration in the upper 6” of soil must be less than 6.0 pCi/g.”
There is a good chance that actual mean 226Ra concentration is between 4.0 and 6.0 pCi/g
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Example Problem (cont.)
Historical data suggest a standard deviation of 1.6 pCi/g
It costs $1000 to collect, process, and analyze one sample
The maximum sampling budget is $5,000
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Chance of
Deciding Site is Dirty
1.0
0.5
0.0
6 pCi/g
Action Level
Low True Mean 226Ra Concentration High
Ideal Rule
Graph of Perfect Decision Making
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Chance of
Deciding Site is Dirty
1.0
0.5
0.0
6 pCi/g
Action Level
Low True Mean 226Ra Concentration High
Typical Curve
Graph of Typical Decision Making
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Simplified Decision Process
Take some number of samples Find the average 226Ra concentration in our
samples If we pass the appropriate QA/G-9 test, decide
the site is clean If we fail the appropriate QA/G-9 test, decide
the site is dirty
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Example of Ad Hoc Sampling Design and the Results
Suppose we choose to take 5 samples for various reasons: low cost, tradition, convenience, etc.
Need volunteer to do the sampling Need volunteer to record results We will follow QA/G-9 One-Sample t-Test
directions using an Excel spreadsheet
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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
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True Mean 226Ra 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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Key Concepts Defined
Latin Letters Concepts Greek Letters ConceptsN population size
(population unit)n number of samples
sample mean is astatistic
population mean is astatistical parameter
s sample standarddeviation is a statistic
population standarddeviation is a statisticalparameter
H0 null hypothesis(action level)
alpha error rate
beta error rate width of the gray region
x
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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
• H0 - null hypothesis
- alpha error rate - beta error rate - width of gray
region
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t-testCalculated t = (sample mean - AL)
------------------------
If calculated t is less than table value, decide site is clean
) /s( n
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Upper Confidence Limit, UCLFor 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
)]s/(*t[ df,1 nUCL X
X
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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).
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Action Level
A measurement threshold value of the Population Parameter (e.g., true mean) that provides the criterion for choosing among alternative actions.
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NTarget 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
nThe number of population units selected and measured is n
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10 x 10 FieldPopulation = All 100 Population Units
Sample = 5 Population Units
1.5
1.5
2.3
1.7
1.9
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Population Mean
The average of all N population units
i = 1
N
XiN
1
Sample Mean
The average of the n population units actually measuredX
n
1 n
i = 1
XiX
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Population Standard Deviation
The average deviation of all N population units from the population mean
N
Xi
N
i
2
1
Sample Standard Deviations
The “average” deviation of the n measured units from the sample mean
1
2
1
n
XXs
i
n
i
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The Null HypothesisH0
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)
0H : Action Level
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The Alternate HypothesisHA
The alternative hypothesis isaccepted only when there is
overwhelming proof that the Null condition is false.
H : Action LevelA
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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)
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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
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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
X