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Extra info for Statistical Analysis Techniques in Particle Physics: Fits, Density Estimation and Supervised Learning
A “good” test is one which has high power for any given significance level. A test is called Uniformly Most Powerful (UMP) if, for specified significance it is at least as powerful as any other test, for all elements of H1 . Another view of hypothesis testing is encapsulated in the p-value. The p-value for an observation is the probability, under the null hypothesis, that a fluctuation as large or larger than that observed, away from H0 , towards H1 , will occur. We will have opportunity to use both paradigms.
85) The rather low power is just telling us that θ0 and θ1 are pretty close together on the scale of the standard deviation and the value of α. 7% probability. In a sense, we were “lucky” with our actual sampling, to get such a small value for p (x is unlikely even if H1 is true). In general, we may not be able to analytically evaluate the sampling distribution for λ, under the hypothesis H0 . In this case, we resort to Monte Carlo simulation or other means to evaluate p. Care must be taken to simulate enough experiments to learn how the tails behave, since that is often where the action lies.
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