10.4 Inference as Decision. The 1995 O.J. Simpson trial: the situation
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1 10.4 Inference as Decision The 1995 O.J. Simpson trial: the situation Nicole Brown Simpson and Ronald Goldman were brutally murdered sometime after 10:00 pm on June 12, Nicole was the wife of O.J. Simpson, former NFL player (winner of 1968 Heisman Trophy, Hall of Famer). O.J., like any American accused of a crime, is assumed innocent. Evidence collected: bloody size 12 Bruno Magli shoeprint, bloody glove, blood spots on white Ford Bronco, neighbor heard a thump on the condominium wall, Criminal trial: O.J. is accused of the murders Civil trial: O.J. is accused of wrongfully causing the deaths 1
2 O.J. Simpson trial: jury decisions Criminal trial: did not reject assumption of innocence. Behave as if O.J. is innocent. Civil trial: reject assumption of innocence. Behave as if O.J. is guilty. Was an error made in either trial? 2
3 Errors in Trials Truth (what is actually the case) Innocent Guilty Jury decision Not Guilty Guilty ERROR ERROR O.J. innocent error made in civil trial. O.J. guilty error made in criminal trial. 3
4 Errors in Hypothesis Testing Decision Do not reject Ho Reject Ho H o really true TYPE I ERROR H o really false TYPE II? β ERROR α power Type I false alarm. Type II a miss. 4
5 Example: Grade inflation? Has the average college gpa increased beyond 2.7? Ho: μ = 2.7 Ha: μ > 2.7 Random sample of students Decision Rule We'll choose α = Statistics n = 36 σ = 0.6 and is also found 5
6 If = (Distribution if Null is true) Each time, p value 0.05, so we reject Ho. If =2.95 (Distribution if Null is true).0062 If =3.0 (Distribution if Null is true)
7 Alternative (older) method: Find critical value z* that gives p value = 0.05 & reject H o if your z score lands in the critical (rejection) region. critical (rejection) region.05 (Distribution if Null is true) critical value z*=
8 Significance level, α = P(Type I error). Common values: α = 0.01, 0.05, or Choose value based on seriousness of Type I error. In criminal trials: Type I error: innocent sentenced to imprisonment or death. "Beyond a reasonable doubt." α really small, such as In civil trials: Type I error: innocent fined. "Preponderance of evidence." α not so small, such as
9 Example: Serious Type I Error A new drug is supposed to reduce blood pressure, but can have serious side effects. Null Hypothesis: new drug has no impact on blood pressure Decision Do not reject Ho Reject Ho H o is true Approve drug, though not effective. H o is false Don't approve drug, which is effective.? β α power Result: Expose patients to serious side effects without real benefit. Therefore, we make α small, perhaps 0.01 or less. 9
10 Example: Serious Type II Error Snow is forecast. H o : No big changes in road conditions due to snow Null: snow won't be bad enough to cancel school. (no effect or no difference from a normal day) Decision Do not reject Ho Reject Ho H o is true have school, roads good no snow & no school? β α power H o is false Result: People in danger trying to get to school. bad roads, & in school school off, bad roads 10
11 Example: Not so serious Type I Error A university suspects grade inflation is occurring. Truth (No grade inflation Avg. GPA still the same.) Decision Do not reject Ho Reject Ho H o is true Mistakenly conclude avg. GPA has increased.? β α power H o is false Decide GPAs are inflated, raise expectations Result: Instructors grade harder, students become unhappy, but may be better prepared for careers. Therefore, we make α bigger, perhaps up to
12 Type II Error: Fail to reject false Ho. If Ho is false, (Ha is true) P(correctly reject Ho Ho is false) + P(incorrectly retain Ho Ho is false) =1 Power + P(Type II error) = 1 So, Power = 1 P(Type II error) The Power of a test is the probability of correctly rejecting Ho when it's false. 12
13 Power is probability of correctly rejecting false Ho. 0 is bad! 1 is good! Calculating is not on the AP Exam. Knowing about is on the AP Exam. Change: Move Ha distribution, α, σ, or n. 13
14 These affect power... Difference between value under Ho and the actual value P(Type I error) = α Standard deviation Sample size Power is higher when... the true mean is far from the mean specified in H o. α is higher (though this increases the chance of a Type I error). σ is smaller. n is larger. 14
15 H o really true H o really false Decision Do not reject Ho Reject Ho 15
16 H o really true H o really false Decision Do not reject Ho Reject Ho 16
17 H o really true H o really false Decision Do not reject Ho Reject Ho 17
18 18
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