Chapter 6.3 - Math3200 Intermediate Probability and Statistics Prof Nan Lin Department of Mathematics Washington University Outline Hypothesis testing

Chapter 6.3 - Math3200 Intermediate Probability and...

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Math3200 Intermediate Probability and Statistics Prof. Nan Lin Department of Mathematics Washington University
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Outline Hypothesis testing Null and alternative hypotheses Type I and type II errors Significance level Power Pvalue One-side or two-sided Math3200 Nan Lin, Washington Unviersity 2
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Hypothesis testing Model: 𝑋 1 , … , 𝑋 𝑛 ∼ 𝑓 𝜃 𝑥 i.i.d. Question: e.g. is 𝜃 = 𝜃 0 ? Let ? = ?(𝑋 1 , … , 𝑋 𝑛 ) be a test statistic Answer: If |? 𝑥 1 , … , 𝑥 𝑛 − 𝜃 0 | > 𝑐 , we think 𝜃 ≠ 𝜃 0 . Math3200 Nan Lin, Washington Unviersity 3
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Elements of hypothesis testing Null hypothesis - Statement regarding the value(s) of unknown parameter(s). (Often contain an equality) Alternative hypothesis - Statement contradictory to the null hypothesis (often contain an inequality) Test statistic - Quantity based on sample data and null hypothesis used to test between the null and alternative hypotheses Rejection region - Values of the test statistic for which we reject the null in favor of the alternative hypothesis Math3200 Nan Lin, Washington Unviersity 4
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Choice of null and alternative hypotheses The null hypothesis is a hypothesis which the researcher tries to disprove, reject or nullify. The 'null' often refers to the common view of something, while the alternative hypothesis is what the researcher really thinks is the cause of a phenomenon. 𝐻 0 : Tomato plants do not exhibit a higher rate of growth when planted in compost rather than soil. 𝐻 1 : Tomato plants exhibit a higher rate of growth when planted in compost rather than in soil. Practically, many researchers see accepting the null as a failure of the experiment. Math3200 Nan Lin, Washington Unviersity 5
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A criminal trial is an example of hypothesis testing without the statistics. In a trial a jury must decide between two hypotheses. The null hypothesis is H 0 : The defendant is innocent The alternative hypothesis or research hypothesis is H 1 : The defendant is guilty The jury does not know which hypothesis is true. They must make a decision on the basis of evidence presented. Math3200 Nan Lin, Washington Unviersity pronounced H “nought” 6
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“Convicting the defendant” is called rejecting the null hypothesis in favor of the alternative hypothesis . That is, the jury is saying that there is enough evidence to conclude that the defendant is guilty (i.e., there is enough evidence to support the alternative hypothesis). If the jury “acquits”, it is stating that there is not enough evidence to support the alternative hypothesis. Notice that the jury is not saying that the defendant is innocent, only that there is not enough evidence to support the alternative hypothesis. That is why we never say that we accept the null hypothesis, although most people in industry will say “ We accept the null hypothesis You may say “I retain the null hypothesis” or “I fail to reject the null hypothesis”.
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  • Spring '08
  • SAWYER
  • Statistics, Probability, Null hypothesis, Statistical hypothesis testing, Statistical significance, Nan Lin, Prof. Nan Lin

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