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Summarizing Risk Analysis Results To quantify the risk of an output
variable, 3 properties must be estimated: A measure of central tendency (e.g. µ
) A measure of dispersion (e.g. σ or
range) The shape of the distribution
describes which values are more likely than others to occur
Histograms and proportions
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Sources of Uncertainty Inherent risk of output variable
Measured by range and Determined by risk specified for input
variables Sampling risk
Associated with size of sample (e.g. number of iterations) and likelihood of error in parameter estimate
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Estimation Sample Statistics are used to estimate
Population Parameters is used to estimate Population Mean,
Problem: Different samples provide different estimates of the population parameter
A sampling distribution describes the likelihood of different sample estimates that can be obtained from a population
_
X
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Developing Sampling Distributions Assume there is a population … Population size N=4 Random variable, X,
is age of individuals Values of X: 18, 20,
22, 24 measured inyears A
B C
D
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1
2
1
18 20 22 2421
4
2.236
N
ii
N
ii
X
N
X
N
.3
.2
.1
0 A B C D (18) (20) (22) (24)
Uniform Distribution
P(X)
X
Developing Sampling Distributions (continued
)
Summary Measures for the Population Distribution
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1st 2nd Observation Obs 18 20 22 24
18 18,18 18,20 18,22 18,24
20 20,18 20,20 20,22 20,24
22 22,18 22,20 22,22 22,24
24 24,18 24,20 24,22 24,24
All Possible Samples of Size n=2
16 Samples Taken with Replacement
16 Sample Means1st 2nd Observation Obs 18 20 22 24
18 18 19 20 21
20 19 20 21 22
22 20 21 22 23
24 21 22 23 24
Developing Sampling Distributions
(continued)
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1st 2nd Observation Obs 18 20 22 24
18 18 19 20 21
20 19 20 21 22
22 20 21 22 23
24 21 22 23 24
Sampling Distribution of All Sample Means
18 19 20 21 22 23 240
.1
.2
.3
P(X)
X
Sample Means
Distribution
16 Sample Means
_
Developing Sampling Distributions
(continued)
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Properties of Summary Measures
i.e. is unbiased
Standard error (standard deviation) of the sampling distribution when sampling with replacement:
As n increases, decreases Sampling more decreases the uncertainty in
the estimate for
X
X
Xn
X
X
X
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Comparing the Population with its Sampling Distribution
18 19 20 21 22 23 240
.1
.2
.3 P(X)
X
Sample Means Distribution
n = 2
A B C D (18) (20) (22) (24)
0
.1
.2
.3
PopulationN = 4
P(X)
X_
21 2.236 21 1.58X X
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Central Limit TheoremAs sample size gets large enough…
the sampling distribution becomes almost normal regardless of shape of population
X
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Confidence Interval for µ One can be 95% confident that the
true mean of an output variable falls somewhere between the following limits:
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Descriptions of Shape Histograms; Percentiles Measures of symmetry
Skewness Weighted average cube of distance from
mean divided by the cube of the standard deviation
Symmetric distributions have 0 skew Positively skewed:
tail to right side is longer than that to the left Outcomes are biased towards larger values Mean > median > mode
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Population Proportions p Proportion of population having a characteristic Sample proportion provides an estimate
number of successes
sample sizeS
Xp
n
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Sampling Distribution of Sample Proportion
Mean:
Standard error:
p = population proportion
Sampling DistributionP(ps)
.3
.2
.1 0
0 . 2 .4 .6 8 1ps
Spp
1Sp
p p
n
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Confidence Interval for p One can be 95% confident that the
true proportion for an output variable that has a certain characteristic falls somewhere between the following limits: