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Unformatted text preview: Chapter 8 Hypothesis Testing 8.1 Let X = # of heads out of 1000. If the coin is fair, then X ∼ binomial(1000 , 1 / 2). So P ( X ≥ 560) = 1000 X x =560 1000 x 1 2 x 1 2 n x ≈ . 0000825 , where a computer was used to do the calculation. For this binomial, E X = 1000 p = 500 and Var X = 1000 p (1 p ) = 250. A normal approximation is also very good for this calculation. P { X ≥ 560 } = P X 500 √ 250 ≥ 559 . 5 500 √ 250 ≈ P { Z ≥ 3 . 763 } ≈ . 0000839 . Thus, if the coin is fair, the probability of observing 560 or more heads out of 1000 is very small. We might tend to believe that the coin is not fair, and p > 1 / 2. 8.2 Let X ∼ Poisson( λ ), and we observed X = 10. To assess if the accident rate has dropped, we could calculate P ( X ≤ 10  λ = 15) = 10 X i =0 e 15 15 i i ! = e 15 1+15+ 15 2 2! + ··· + 15 10 10! ≈ . 11846 . This is a fairly large value, not overwhelming evidence that the accident rate has dropped. (A normal approximation with continuity correction gives a value of .12264.) 8.3 The LRT statistic is λ ( y ) = sup θ ≤ θ L ( θ  y 1 ,...,y m ) sup Θ L ( θ  y 1 ,...,y m ) . Let y = ∑ m i =1 y i , and note that the MLE in the numerator is min { y/m,θ } (see Exercise 7.12) while the denominator has y/m as the MLE (see Example 7.2.7). Thus λ ( y ) = 1 if y/m ≤ θ ( θ ) y (1 θ ) m y ( y/m ) y (1 y/m ) m y if y/m > θ , and we reject H if ( θ ) y (1 θ ) m y ( y/m ) y (1 y/m ) m y < c. To show that this is equivalent to rejecting if y > b , we could show λ ( y ) is decreasing in y so that λ ( y ) < c occurs for y > b > mθ . It is easier to work with log λ ( y ), and we have log λ ( y ) = y log θ + ( m y ) log (1 θ ) y log y m ( m y ) log m y m , 82 Solutions Manual for Statistical Inference and d dy log λ ( y ) = log θ log(1 θ ) log y m y 1 y + log m y m + ( m y ) 1 m y = log θ y/m ( m y m ) 1 θ ! . For y/m > θ , 1 y/m = ( m y ) /m < 1 θ , so each fraction above is less than 1, and the log is less than 0. Thus d dy log λ < 0 which shows that λ is decreasing in y and λ ( y ) < c if and only if y > b . 8.4 For discrete random variables, L ( θ  x ) = f ( x  θ ) = P ( X = x  θ ). So the numerator and denomi nator of λ ( x ) are the supremum of this probability over the indicated sets. 8.5 a. The loglikelihood is log L ( θ,ν  x ) = n log θ + nθ log ν ( θ + 1)log Y i x i ! , ν ≤ x (1) , where x (1) = min i x i . For any value of θ , this is an increasing function of ν for ν ≤ x (1) . So both the restricted and unrestricted MLEs of ν are ˆ ν = x (1) . To find the MLE of θ , set ∂ ∂θ log L ( θ,x (1)  x ) = n θ + n log x (1) log Y i x i !...
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 Spring '08
 Peruggia,M
 Binomial, Order theory, Prediction interval, Monotonic function, Student's tdistribution

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