Week 11 Wed Nov 10 - Posted before class WEEK 11 Wed, Nov...

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1 Posted before class WEEK 11 – Wed, Nov 10 HYPOTHESIS TESTING – The p-value Approach (Background material is in Chapter 7) Conventions in Hypothesis Testing: 1. One usually selects as the null hypothesis, the hypothesis that we will retain (live with) unless there is sufficiently strong evidence against it. 2. One usually selects as the alternative hypothesis, the hypothesis that one hopes to “prove” to be correct; in academic circles is sometimes called the research hypothesis. Decision (Reject H 0 in favor of H 1 when p-value is small) Data Example. Innocent or Nor Innocent) . A person is taken to court charged with a crime. One hypothesis is that the person is innocent of the crime; the negation is that he/she is guilty of the crime. In the country’s legal system, innocence is assumed and rejected only when there is sufficient evidence against innocence. Thus, in terms of our conventions for the null hypothesis and alternative hypothesis we would specify H 0 : Person is Innocent and H 1 : Person is Guilty The court and/or jury should find the person Guilty if and only if there is sufficiently strong evidence against the null hypothesis H 0 . Decision (Reject innocence if and only if evidence is strong against it) H 0 : Person is Innocent H 1 : Person is Guilty Jury Evidence Null Hypothesis H 0 is True Alternative Hypothesis H 1 is True Null Hypothesis H 0 is True Alternative Hypothesis H 1 is True
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2 Example 1. Is the Die Balanced ? In an experiment earlier in the semester, we used an 8-sided die to make selections from {1, 2, 3, 4, 5, 6, 7, 8}. In the past I noticed the face labeled with a 4 seemed to come-up more frequently than expected. Let us now experiment with the die and conduct a formal test as to whether the die is imbalanced in favor of Face 4 or not. As with the ESP example covered last time, we use this example to illustrate many of the ideas from Chapter 7 in regard to the p-value approach to testing hypotheses. We will discuss setting up two hypotheses concerning the probability p that Face 4 turns-up when the die is given a vigorous roll. Consider my research hypothesis that the die is imbalanced with p > 1/8. I will label this as the hypothesis H 1 : p > 1/8. This is called the Alternative Hypothesis . Its negation is called the Null Hypothesis . Here the Null Hypothesis is H 0 : p ≤ 1/8. It is conceptually simpler to think of testing H 0 : p =1/8 versus H 1 : p > 1/8. (You may prefer to use the decimal representation 0.125 for the fraction 1/8.) From a visual inspection of the die, we see no evidence that the die is imbalanced in favor of the outcome Face 4. We decide that the evidence must be strong against the null hypothesis H 0 : p =1/8 and in favor of the alternative hypothesis H 1 : p > 1/8 before we reject the null hypothesis 1 . To develop objective evidence, we decide to roll the die
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Week 11 Wed Nov 10 - Posted before class WEEK 11 Wed, Nov...

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