# lec3 - Statistical Inference for FE Professor S Kou...

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Statistical Inference for FE Professor S. Kou, Department of IEOR, Columbia University Lecture 3. Hypothesis Testing 1 Introduction to Hypothesis Testing 1.1 Con f dence Intervals and Hypothesis Testing How to test the hypothesis H 0 : μ =3 versus the alternative hypothesis H a : μ 6 =3 . Assume we have large sample size. One possible solution is simply by using the idea of con f dence interval. More precisely, f rst we construct a ( 1 α ) c.i. for μ : ¯ X ± z α / 2 S n . Then we reject H 0 if 3 is not within the ( 1 α ) c.i; otherwise, we cannot reject it as a plausible value for μ . In other words, we reject H 0 if 3 < ¯ X z α / 2 S n or 3 > ¯ X + z α / 2 S n . The above formula can be rewritten as ¯ X 3 S/ n < z α / 2 or ¯ X 3 S/ n >z α / 2 . By using such a test, we may make a wrong decision with probability α , if we reject the null hypothesis when it is indeed true. This is called type I error. There is another possible error, which is the error we fail to reject the null hypothesis when it is false. This is called type II error. 1.2 Connection between Hypothesis Testing and Jury Trials The jury trail of an accused murderer is very similar to the statistical hy- pothesis testing. 1. H 0 In a jury trial, the null hypothesis is that the accused is not guilty. The status-quo hypothesis in the American system of justice is innocence, which is assumed to be true until proven otherwise. 2. H a . The alternative hypothesis is guilt, which is accepted only when su cient evidence exists to establish its truth. 3. Test Statistics. The test statistics in a trial is the f na lvo teo fthe jury, i.e., the number of the jury members who vote “guilty.” 4. Rejection Region. In a murder trial the jury vote must be unanimous in favor of guilt before the null hypothesis of innocence is rejected in favor of the alternative hypothesis of guilt. Thus, for a 12-member jury trial, the rejection region is x

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5. Assumption. The primary assumption made in jury trail concerns the method of selecting the jury. The jury is assumed to represent a random sample of citizens who have no prejudice concerning the case. 6. Sample and calculate the test statistics. Sample data are generated through the trial jury’s deliberations. The f nal vote of the jury is analogous to the calculation of the test statistics. 7.Conclusion. If the vote of the jury is unanimous in favor of guilt, the null hypothesis of innocence is rejected and the court concludes that the accused murderer is guilty. 8. Although the court does not, in general, know the probability, the type one error, that the conclusion is in error, the system relies on the belief that the value is made very small by requiring a unanimous vote before guilt is concluded. 9. Any vote other than a unanimous one for guilt results in the court
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lec3 - Statistical Inference for FE Professor S Kou...

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