ch8sol - Chapter 8 Hypothesis Testing 8.1 Let X = # of...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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( ), and we observed X = 10. To assess if the accident rate has dropped, we could calculate P ( X 10 | = 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 ( y ) = sup L ( | y 1 ,...,y m ) sup L ( | y 1 ,...,y m ) . Let y = m i =1 y i , and note that the MLE in the numerator is min { y/m, } (see Exercise 7.12) while the denominator has y/m as the MLE (see Example 7.2.7). Thus ( y ) = 1 if y/m ( ) y (1- ) m- y ( y/m ) y (1- y/m ) m- y if y/m > , and we reject H if ( ) y (1- ) 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 ( y ) is decreasing in y so that ( y ) < c occurs for y > b > m . It is easier to work with log ( y ), and we have log ( y ) = y log + ( m- y ) log (1- )- y log y m- ( m- y ) log m- y m , 8-2 Solutions Manual for Statistical Inference and d dy log ( y ) = log - log(1- )- log y m- y 1 y + log m- y m + ( m- y ) 1 m- y = log y/m ( m- y m ) 1- ! . For y/m > , 1- y/m = ( m- y ) /m < 1- , so each fraction above is less than 1, and the log is less than 0. Thus d dy log < 0 which shows that is decreasing in y and ( y ) < c if and only if y > b . 8.4 For discrete random variables, L ( | x ) = f ( x | ) = P ( X = x | ). So the numerator and denomi- nator of ( x ) are the supremum of this probability over the indicated sets. 8.5 a. The log-likelihood is log L ( , | x ) = n log + n log - ( + 1)log Y i x i ! , x (1) , where x (1) = min i x i . For any value of , this is an increasing function of for x (1) . So both the restricted and unrestricted MLEs of are = x (1) . To find the MLE of , set log L ( ,x (1) | x ) = n + n log x (1)- log Y i x i !...
View Full Document

Page1 / 21

ch8sol - Chapter 8 Hypothesis Testing 8.1 Let X = # of...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online