【正文】 4, is large。 2, the sampling fraction n/N is large。 5, the correlation R is close to 1. 222V ( ) ( , )?( ) 1 x x yUUB S R S Sn B x C o v x yE B BN n x x? ???? ? ? ??????BUxxSThe proof for it is: ()? UUx x xy y B x y B xB B Bx x x x????? ? ? ? ? ?? ? () UUUy B x x xy B xx x x??????? ? ? ? ()? UUy B x x xE B B Exx????? ? ??? ???? ?21 () UUE y B x x xx? ? ? ?????? ?S i n c e : 0UUE y B x y B x? ? ? ? ? ?21 ()U U UUE y y y B x x xx? ? ? ? ? ?????? ? ? ?21 ( ) ( )U U U UUE y y x x y B x x xx? ? ? ? ? ? ?????? ? ? ?222211 1y x x x x yUUnR S S B S B S R S Sx N n x??? ? ? ? ? ?????? ? ? ?21 ( ) ( )U U U UUE y y x x B x B x x xx? ? ? ? ? ? ?????? ? () 1 amp。 2, the sampling fraction n/N is large。 Average yield per acre Per capita ine The ratio of liabilities to assets The ratio of the fish caught to the hours spent fishing ?Sometimes we want to estimate a population total, but the population size N is unknown。 whereas “central limit theorem ” says that the distribution of any sample mean converges to normal distribution if n is sufficiently large, with the same or different distribution. 212 2... 1l i m P ( ) ( ) d x2tynnX X X n y y en????????? ? ? ? ? ? ? ? ?l i m P( ) 1npn? ???? ? ?Bernoulli’s law of large numbers is: Linderberg and levy’s central limit theorem is: ? Sample Size Estimation ?An investigator often measures several variables and has a number of goals for a survey. Anyone designing an SRS must decide what amount of sampling error in the estimates is tolerable and must balance the precision of the estimates with the cost of the survey. ? Follow these steps to estimate the sample size: ? Ask questions as: A: What is expected of the sample? B: How much precision do I need? C: What are the consequences of the sample results? D: How much error is tolerable? ?Find an equation relating the sample size n and our expectations of the sample ?Estimate any unknown quantities and solve for n. ?If the sample size you calculated in last step is much larger than you can afford. Go back and adjust some of the expectations for the survey and try again. ?Specify the tolerable error P ( ) 1Uy y e ?? ? ? ?o r P ( ) 1UUyy ey ?? ? ? ??Find an equation ? Solving for n, we have: In relative precision case, we have 2 1nSezN n???222 022022 1zS nnnzSe NN??????? ?2 2 2 2222 2 2 22 22 2C V ( )C V ( )Uz S z ynz S z ye y eNN????????? Randomization Theory Results for Simple Random Sampling* ? To verify and Define , then we have [] UE y y?2[ ] 1 nSVyNn????????1 i f u n i t i s i n t h e s a m p l0 o t h e r w i s eiieZ?? ??1Niiii S iyyyZnn?????? ? ?P ( 1 ) s e l e c t u n i t i n s a m p l eiZ P i??num be r of sa m pl e s i nc l uding uni t num be r of a l l possi bl e sa m pl e si?11Nn nN Nn???????????????? As a consequence of equation () in order to calculate the variance of ,note that: ? ? ? ?? ?222 1i i in n n nV Z E Z E ZN N N N? ? ? ???? ? ? ? ? ?? ? ? ??? ? ? ? ?? ?1 1 1N N Ni i iiUi i iy y ynE y E Z yn N n N? ? ???? ? ? ?????? ? ?? ? 2ii nE Z E Z N??????y ? ?,i j i j i jC o v Z Z E Z Z E Z E Z? ? ? ? ? ???? ? ? ? ? ?( 1 a n d 1 )i j i jE Z Z P Z Z?? ? ? ???fo r ij?1( 1 | 1 ) ( 1 ) *1j i innP Z Z P ZNN?? ? ? ? ??211*1n n n n nN N N N N N? ? ? ? ?? ? ? ? ?? ? ? ??? ? ? ? ? 221 1 1N N N Ni i i ji i j j iy y y y? ? ? ??? ??????? ? ? ?From:221 1 1N N N Ni j i ij j i i iy y y y? ? ? ?????????? ? ? ?, w e c a n g e t :And:? ? 22 2 21111S y yN 1 N 1NNi U i Uiiy y N????? ? ? ???????22111N ( N 1 )NNiiiiN y y???? ???? ?? ???????? 221 1 111( ) ,N N Ni i i i j ji i jV y V Z y C o v Z y Z ynn ? ? ??????? ?????? ??? ? ?22111 ( ) ( , )N N Ni i j j i ji j j iy V Z y y C o v Z Zn ? ? ?????????? ? ?221111111N N Ni j ji j j in n n ny y yn N N N N N? ? ???? ? ? ?? ? ? ???? ? ? ??? ? ? ??? ? ? ? 221111111N N Ni i ji j j in n n ny y yn N N N N N? ? ???? ? ? ?? ? ? ???? ? ? ??? ? ? ??? ? ? ?22111111N N Ni i ji j j inn y y yn N N N? ? ?????? ? ???????? ??? ? ?2221 1 1111 ( 1 )( 1 )N N Ni i ii i inN y y yn N N N ? ? ??? ????? ? ? ? ????? ????? ????? ? ?2 22111111( 1 )NNiiiin n SN y yn N N N N n???? ??? ? ? ?? ? ? ? ???? ? ? ????? ? ? ??????? Now let us prove: Proof: ? ? ? ?22111E11Ni i Ui S iy y y ynN???? ? ? ??????? ??22E o rsS?? ???? ? ? ? 22E E ) (i i U Ui S i Sy y y y y y??? ? ? ?? ? ? ? ?????? ? ? ?? ? ? ??? ? ? ? ?22E i U UiSy y n y y???? ? ? ??????? ? 21E ( )Ni i UiZ y y n V y???? ? ??????? ? 2 211NiUinny y SNN???? ? ? ??????2 2 2( 1 ) ( 1 )n N N nS S n SNN??? ? ? ?? CHAPTER 3