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Statistical Decision Making

Statistical Decision Making. Analysts must often make decisions about some condition in the real world. Assume that you have finished your MA in policy studies and have been hired as the environmental affairs officer for the city of Morgantown. You must make the following assessment:

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Statistical Decision Making

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  1. Statistical Decision Making • Analysts must often make decisions about some condition in the real world. • Assume that you have finished your MA in policy studies and have been hired as the environmental affairs officer for the city of Morgantown. • You must make the following assessment: • Does the water supply in Morgantown comply with safe drinking water standards?

  2. Hypothesis Testing We can decide the facility is In Compliance. or We can decide the facility is in Violation. (not in compliance)

  3. If we decide: The facility is in compliance – and it is. or We decide the facility is in violation – and it is. Then we are fine. We have made a correct decision But sometimes… No, make that often… we’re not correct. We make mistakes. Statistics gives us some rules to reduce the likelihood of making these mistakes.

  4. Example of a statistical test • In order to test the facilities water we will collect several bottles of water at different times. • We take this sample… • (Do not be confused by the chemistry. Each bottle of water is a sample to the chemist, while all of the separate bottles of water are the sample to the statistician.) • And we calculate the average amount of the pollutant – e.g. lead.

  5. The average of the samples is thus Our hypothesis is thus ppb (the standard for lead) Because we will say that the standard actually refers to a population distribution with a mean equal to the standard, we can restate this is conventional statistical terms For info on EPA standards

  6. Hypotheses • The hypothesis is: • The facility is in violation. • Sometimes referred to as “The alternate hypothesis.” • The null hypothesis is: • The facility is in compliance. • In this instance, you want to be able to reject the alternate hypothesis, or more properly, fail to reject the null hypothesis.

  7. Decision

  8. Decision In Compliance

  9. Decision In Compliance In Violation

  10. Decision In Compliance In Violation Real World

  11. Decision In Compliance In Violation In Compliance Real World

  12. Decision In Compliance In Violation In Compliance Real World In Violation

  13. Decision In Compliance In Violation In Compliance Real World In Violation

  14. Decision In Compliance Do not reject H0 In Violation Reject H0 In Compliance Real World In Violation

  15. Decision In Compliance Do not reject H0 In Violation Reject H0 In Compliance Correct Real World In Violation

  16. Decision In Compliance Do not reject H0 In Violation Reject H0 In Compliance Correct Real World Correct In Violation

  17. Decision In Compliance Do not reject H0 In Violation Reject H0 Incorrect In Compliance Correct Real World Correct In Violation

  18. Decision In Compliance Do not reject H0 In Violation Reject H0 Incorrect In Compliance Correct Real World Incorrect Correct In Violation

  19. Decision In Compliance Do not reject H0 In Violation Reject H0 Incorrect Prob. =  In Compliance Correct Real World Incorrect Correct In Violation

  20. Decision In Compliance Do not reject H0 In Violation Reject H0 Incorrect Prob. =  In Compliance Correct Prob. = 1- Real World Incorrect Correct In Violation

  21. Decision In Compliance Do not reject H0 In Violation Reject H0 Incorrect Prob. =  In Compliance Correct Prob. = 1- Real World Incorrect Prob. =  Correct In Violation

  22. Decision In Compliance Do not reject H0 In Violation Reject H0 Incorrect Prob. =  (Type I error) In Compliance Correct Prob. = 1- Real World Incorrect Prob. =  Correct In Violation

  23. Decision In Compliance Do not reject H0 In Violation Reject H0 Incorrect Prob. =  (Type I error) In Compliance Correct Prob. = 1- Real World Incorrect Prob. =  (Type II error) Correct In Violation

  24. Decision In Compliance Do not reject H0 In Violation Reject H0 Incorrect Prob. =  (Type I error) In Compliance Correct Prob. = 1- Real World Incorrect Prob. =  (Type II error) Correct Prob. = 1- In Violation

  25. Decision In Compliance Do not reject H0 In Violation Reject H0 Incorrect Prob. =  (Type I error) In Compliance Correct Prob. = 1- Real World Incorrect Prob. =  (Type II error) Correct Prob. = 1- (Power of the test) In Violation

  26. Alpha (α) • Note that alpha (α) is: • The probability of rejecting the null hypothesis when the null is in fact true. • It is this the probability of making a Type I error • By convention, we usually set α=.05 (1 time out of 20 by chance alone) • A good working rule is to always use α=.05 until you know when not to….

  27. Beta () • Beta () is: • The probability of failing to reject the null hypothesis when the alternate is in fact true. • It is thus the probability of making a Type II error. • We can never really know , because we never know the “true” situation. • α and  are inversely related.

  28. Environmental vs. Industrial Protection •  provides us a measure of environmental protection. • αprovides us a measure of industrial protection.

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