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Contingency Tables

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Contingency Tables

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    1. Contingency Tables

    2. The Questions We have asked each of 150 female college students two questions: Do you smoke (yes/no)? Do you have sleep disturbances (yes/no)? Suppose that we obtain the following data (these are totally contrived, not real):

    3. Marginal Probabilities

    4. Conditional Probabilities Show Absolute Independence

    5. Multiplication Rule Given Independence Sixty of 150 have sleep disturbance and smoke, so P (Sleep ?? Smoke) = 60/150 = .40 P(A ?? B) = P(A) x P(B)

    6. Addition Rule P(A or B) = P(A ? B) = P(A) + P(B) if A and B are mutually exclusive. P(A ? B) = .2 + .3 = .5

    7. “Sleep” = Sexually Active Preacher claims those who smoke will go to Hell. And those who fornicate will go to Hell. What is the probability that a randomly selected coed from this sample will go to Hell?

    8. Addition Rule

    9. Welcome to Hell The events (sleeping and smoking) are not mutually exclusive. We have counted the overlap between sleeping and smoking (the 60 women who do both) twice. 30 + 40 + 60 = 130 of the women sleep and/or smoke. The probability we seek = 130/150 = 13/15 = .87

    10. Addition Rule For Events That Are NOT Mutually Exclusive

    11. Sleep = Sexually Active, Smoke = Use Cannabis

    12. Marginal Probabilities

    13. Conditional Probabilities Indicate Nonindependence

    14. Joint Probability What is the probability that a randomly selected coed is both sexually active and a cannabis user? There are 60 such coeds, so the probability is 60/150 = .40. Now let us see if the multiplication rule works with these data.

    15. Multiplication Rule Oops, this is wrong. The joint probability is .40. We need to use the more general form of the multiplication rule.

    16. Multiplication Rule NOT Assuming Independence Now that looks much better.

    17. Actual Data From Jury Research Castellow, Wuensch, and Moore (1990, Journal of Social Behavior and Personality, 5, 547-562 Male employer sued for sexual harassment by female employee. Experimentally manipulated physical attractiveness of both litigants

    18. Effect of Plaintiff Attractiveness P(Guilty | Attractive) = 56/73 = 77%. P(Guilty | Not Attractive) = 39/72 = 54%. Defendant found guilty more often if plaintiff was attractive.

    19. Odds and Odds Ratios Odds(Guilty | Attractive) = 56/17 Odds(Guilty | Not Attractive) = 39/33 Odds Ratio = 56/17 ? 39/33 = 2.79. Odds of guilty verdict 2.79 times higher when plaintiff is attractive.

    20. Effect of Defendant Attractiveness P(Guilty | Not Attractive) = 53/70 = 76%. P(Guilty | Attractive) = 42/75 = 56%. The defendant was more likely to be found guilty when he was unattractive.

    21. Odds and Odds Ratio Odds(Guilty | Not Attractive) = 53/17. Odds(Guilty | Attractive) = 42/33. Odds Ratio = 53/17 ? 42/33 = 2.50. Odds of guilty verdict 2.5 times higher when defendant is unattractive.

    22. Combined Effects of Plaintiff and Defendant Attractiveness Plaintiff attractive, Defendant not = 83% guilty. Defendant attractive, Plaintiff not = 41% guilty. Odds ratio = 83/17 ? 41/59 = 7.03. When attorney tells you to wear Sunday best to trial, listen.

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