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The Language of Statistical Decision Making

The Language of Statistical Decision Making. Lecture 3 Section 1.3 Mon, Jan 21, 2008. Errors. Recall our conclusion that the distribution of M&M colors agreed with what the company said. Could our conclusion have been wrong? What would be the cause of our error?. Errors.

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The Language of Statistical Decision Making

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  1. The Language of Statistical Decision Making Lecture 3 Section 1.3 Mon, Jan 21, 2008

  2. Errors • Recall our conclusion that the distribution of M&M colors agreed with what the company said. • Could our conclusion have been wrong? • What would be the cause of our error? The Language of Statistical Decision Making - Part 2

  3. Errors • Had we concluded that the distribution was not what the company said it was, could we have been wrong? • What would be the cause of our error? The Language of Statistical Decision Making - Part 2

  4. Possible Errors • We might reject H0 when it is true. • This is a Type I error. • We might accept H0 when it is false. • This is a Type II error. • See Making Intelligent Errors, by Walter Williams. The Language of Statistical Decision Making - Part 2

  5. Decisions and Errors State of Nature H0 true H0 false Correct Type II Error Accept H0 Decision Type I Error Correct Reject H0 The Language of Statistical Decision Making - Part 2

  6. Decisions and Errors True distribution It is what company says It is not what company says Correct Type II Error It is what company says Decision Type I Error Correct It is not what company says The Language of Statistical Decision Making - Part 2

  7. Decisions and Errors State of Nature H0 true H0 false Correct Type II Error Accept H0 Decision Type I Error Correct Reject H0 The Language of Statistical Decision Making - Part 2

  8. Decisions and Errors State of Nature H0 true H0 false Correct Type II Error Accept H0 Decision Type I Error Correct Reject H0 The Language of Statistical Decision Making - Part 2

  9. Case Study 2 • Hair May Help Reveal Eating Disorders The Language of Statistical Decision Making - Part 2

  10. Example • Consider a study to determine the effectiveness of a new drug. • What are the two possible conclusions (hypotheses)? • Which should get the benefit of the doubt? • What are the two possible errors? • Which is more serious? The Language of Statistical Decision Making - Part 2

  11. Example • Now consider a study to determine the safety of a new drug. • What are the two possible conclusions (hypotheses)? • Which should get the benefit of the doubt? • What are the two possible errors? • Which is more serious? The Language of Statistical Decision Making - Part 2

  12. Type I Error vs. Type II Error • See Making Intelligent Errors, by Walter Williams. The Language of Statistical Decision Making - Part 2

  13. Significance Level • Significance Level – The likelihood of rejecting H0 when it is true, i.e., the likelihood of committing a Type I error. •  – The likelihood of a Type I error. •  – The likelihood of a Type II error. • That is,  is the significance level. The Language of Statistical Decision Making - Part 2

  14. Two Unusual Dice • Suppose that we have two unusual dice. • Die A rolls a 1 80% of the time and a 6 only 20% of the time. (It never lands 2, 3, 4, or 5.) • Die B rolls a 1 only 10% of the time and a 6 90% of the time. (It never lands 2, 3, 4, or 5.) • Visually, the two dice are indistinguishable. The Language of Statistical Decision Making - Part 2

  15. Which Die Did We Pick? • We pick up one of the dice. • Suppose the null hypothesis is that we picked up die A and the alternative hypothesis is that we picked up die B. • We will roll the die one time and, based on the outcome, decide which die we think it is. The Language of Statistical Decision Making - Part 2

  16. The Decision • What should be our criterion (decision rule) for choosing between the two hypotheses? • That is, if the die turns up 1, which hypothesis do we choose? What if it turns up 6? • Describe a Type I error. • Describe a Type II error. The Language of Statistical Decision Making - Part 2

  17. The Significance Level • What is the value of ? • What is the value of ? The Language of Statistical Decision Making - Part 2

  18. Are Two Rolls Better Than One? • Suppose now that we roll the chosen die twice and average the two rolls. • We must get either • A pair of 1s, with an average of 1. • A 1 and a 6, with an average of 3.5. • A pair of 6s, with an average of 6. The Language of Statistical Decision Making - Part 2

  19. Are Two Rolls Better Than One? • What would be a good criterion for deciding which die it is? • Based on this criterion, • What is ? • What is ? The Language of Statistical Decision Making - Part 2

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