184 the type ii error rate β is 1 1600 p 1600 r r

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18.4. The type II error rate β is 1 - (1600) = P 1600 { R < r } . This is the distribution function of a hypergeometric random variable and thus these probabilities can be computed using the phyper command For varying significance, we have the R commands: > t<-200;N<-1600 > k<-400 > alpha<-c(0.05,0.02,0.01) > ralpha<-c(49,51,53) > beta<-1-phyper(ralpha-1,t,N-t,k) > data.frame(alpha,beta) alpha beta 1 0.05 0.5993010 2 0.02 0.4609237 3 0.01 0.3281095 Notice that the type II error probability is high for = 0 . 05 and increases as decreases. For varying recapture size, we continue with the R commands: 287
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Introduction to the Science of Statistics Composite Hypotheses > k<-c(400,600,800) > ralpha<-c(49,70,91) > beta<-1-phyper(ralpha-1,t,N-t,k) > data.frame(k,beta) k beta 1 400 0.5993010 2 600 0.8043988 3 800 0.9246057 Notice that increasing recapture size has a significant impact on type II error probabilities. 18.6. The i -th observation satisfies P { X i x } = Z x 0 1 d ˜ x = x Now, X ( n ) x occurs precisely when all of the n -independent observations X i satisfy X i x . Because these random variables are independent, F X ( n ) ( x ) = P { X ( n ) x } = P { X 1 x, X 1 x, . . . , X n x } = P { X 1 x } P { X 1 x } , · · · P { X n x } = x ⌘ ⇣ x · · · x = x n 18.10. Yes, the p -value is below 0.05. No, the p -value is above 0.01. 288
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  • Spring '17
  • KASMIS MISGANEW
  • Null hypothesis, Statistical hypothesis testing, Type I and type II errors, power function, Science of Statistics

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