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Summarizing Risk Analysis Results

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

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Summarizing Risk Analysis Results

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  1. 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

  2. Estimation _ • Sample Statistics are used to estimate Population Parameters • X = 50 is used to estimate Population Mean,  • Problem: Different samplesprovidedifferent estimates of the Population Parameter • A sampling distribution describes the likelihood of different sample estimates that can be obtained from a population

  3. 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 C B D A

  4. Developing Sampling Distributions (continued) Summary Measures for the Population Distribution P(X) .3 .2 .1 0 X A B C D (18) (20) (22) (24) Uniform Distribution

  5. Developing Sampling Distributions All Possible Samples of Size n=2 (continued) 16 Sample Means 16 Samples Taken with Replacement

  6. Developing Sampling Distributions Sampling Distribution of All Sample Means (continued) Sample Means Distribution 16 Sample Means P(X) .3 .2 .1 _ 0 X 18 19 20 21 22 23 24

  7. Comparing the Population with its Sampling Distribution Population N = 4 Sample Means Distribution n = 2 P(X) P(X) .3 .3 .2 .2 .1 .1 _ 0 0 X AB C D (18)(20)(22)(24) 18 19 20 21 22 23 24 X

  8. Properties of Summary Measures • i.e. is unbiased • Standard error (standard deviation) of the sampling distribution is less than the standard error of other unbiased estimators • For sampling with replacement: • As n increases, decreases

  9. Unbiasedness P(X) Unbiased Biased

  10. Effect of Large Sample Larger sample size P(X) Smaller sample size

  11. Sample Size • Determining adequate sample size n: Large samples give better estimates. Large samples are more costly. • How much dispersion is therein the sample mean? • Standard Error = Assuming sampling with replacement or sampling without replacement from a large population

  12. Central Limit Theorem the sampling distribution becomes almost normal regardless of shape of population As sample size gets large enough…

  13. Confidence Interval for µ • One can be 95% confident that the true mean of an output variable falls somewhere between the following limits:

  14. Population Proportions • Proportion of population having a characteristic • Sample proportion provides an estimate

  15. Sampling Distribution of Sample Proportion • Mean: • Standard error: Sampling Distribution P(ps) .3 .2 .1 0 ps 0 . 2 .4 .6 8 1 p = population proportion

  16. 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:

  17. Other Descriptions of Shape • 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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