We see the power of a test depends on its critical

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We see the power of a test depends on its critical region (rule). Denote the power function by r C ( θ ) = P θ { ( X 1 , · · · , X n ) C c } , θ Θ . Given two critical regions, C 1 and C 2 , which are both of size of α , we claim C 1 is better than C 2 if r C 1 ( θ ) r C 2 ( θ ) , θ Θ 1 . Jimin Ding, Math WUSTL Math 494 Spring 2018 37 / 44
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Example 1: Binomial Model Jimin Ding, Math WUSTL Math 494 Spring 2018 38 / 44
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Jimin Ding, Math WUSTL Math 494 Spring 2018 39 / 44
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Example 2: Normal Model Jimin Ding, Math WUSTL Math 494 Spring 2018 40 / 44
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P-value So far, we have derived decision rules and rejection regions using α . We see they depend on the choice of significance level α . But no data information, except sample size, was used in decision rules. Jimin Ding, Math WUSTL Math 494 Spring 2018 41 / 44
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P-value So far, we have derived decision rules and rejection regions using α . We see they depend on the choice of significance level α . But no data information, except sample size, was used in decision rules. In practice, however, the data are often already collected, and one may not know a good choice of α in advance , or want to know all decision rules for a set values of α . For example, the observed sample mean is given ¯ x = 5 , should we reject H 0 for α = 0 . 01 , or α = 0 . 05 , or α = 0 . 1 ... Jimin Ding, Math WUSTL Math 494 Spring 2018 41 / 44
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P-value So far, we have derived decision rules and rejection regions using α . We see they depend on the choice of significance level α . But no data information, except sample size, was used in decision rules. In practice, however, the data are often already collected, and one may not know a good choice of α in advance , or want to know all decision rules for a set values of α . For example, the observed sample mean is given ¯ x = 5 , should we reject H 0 for α = 0 . 01 , or α = 0 . 05 , or α = 0 . 1 ... In this case, a p-value instead of a rejection region is often used to provide more information . Precisely, p-value is, given that the H 0 is true, the conditional probability of observing more extreme data in the direction of H 1 . The p-value can be viewed as an observed significance level. Jimin Ding, Math WUSTL Math 494 Spring 2018 41 / 44
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Example: Zea mays Growth (Example 4.5.1,4.5.5) In 1878, Darwin recorded the heights of Zea mays plants to see the difference between cross- and self- fertilization. On each of the 15 plots, one cross-fertilized plant and one self-fertilized plant were planted to grow and then measured. The height difference of the cross-fertilized plant and the self-fertilized plant in the same plot is recorded. For 15 plots, the mean is ¯ x = 2 . 62 and standard deviation is s = 4 . 72 . Assume the difference in height is independent, and normally distributed. Is cross-fertilized plant taller than self-fertilized plant? I Population I Sample I Probability model Jimin Ding, Math WUSTL Math 494 Spring 2018 42 / 44
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Example: Zea mays Growth (Example 4.5.1,4.5.5) In 1878, Darwin recorded the heights of Zea mays plants to see the difference between cross- and self- fertilization. On each of the 15 plots, one cross-fertilized plant and one self-fertilized plant were planted to grow and then measured. The height difference of the cross-fertilized plant and the self-fertilized plant in the same plot is recorded. For 15 plots, the mean is ¯ x = 2 . 62 and standard deviation is s = 4 . 72 . Assume the difference in height is independent, and normally distributed. Is cross-fertilized plant
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