Comments on Assignment 8 

1.(a) Strictly speaking, a bivariate normal distribution makes no sense 
      for a finite population (N = 82 here).  This distribution has mass 
      on at most 82 points in 2-dimensional space, so very discrete!  

  (b) trivial. 

  (c) Usual approach yields r +/- 1.28*SE(r), as the approximate CI for 
      rho.  To implement, you need to replace SE(r)=(1-rho^2)/sqrt(n) 
      by estimated SE (replacing rho by r).

  (d) Same approach as in (c) yields z_r +/- 1.28*SE(z_r), where z_r is 
      Fisher's transform and SE(z_r)=1/sqrt(n-3), as the approximate 
      CI for Fisher's transform of rho.  Then you need to transform to 
      the rho scale to get the corresponding approximate CI for rho.  
      If (L,U) is the CI for Fisher's transform of rho, the CI for
      rho is [(exp(2L)-1)/(exp(2L)+1),(exp(2U)-1)/(exp(2U)+1)].  

  (e) Once you have written a function to do the JK calculations,
      all is straightforward.

  (f) straightforward

  (g) You need to save the bootstrap estimates for each bootstrap sample.  
      When you've finished the BT sampling, use the quantile function to 
      pick off the desired quantiles, or sort the vector of boot estimates 
      and then pick off the two desired quantiles.

  (h) Your observations will depend on the particular simple random sample
      you drew.  A couple of things can be said in general:
      - when using Fisher's transform, the CIs for rho will always be a 
   	subset of the possible range (-1,+1)
      - the use of bootstrap distribution in (g) yields same CIs for rho 
        whether based on r or Fisher's transform of r (as the latter is a 
        monotone function of r)