CLASS LOG FOR NOVEMBER 13, 2017 IN-CLASS DEMONSTRATION ON BOOTSTRAP CONFIDENCE INTRVALS
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> library(MASS)
> set.seed(2222)
> xvec = rpois(80, 2.3)
  lam.hat = mean(xvec)
> CI0 = lam.hat + 1.96*c(-1,1)*sqrt(lam.hat/80)
  CI0
[1] 2.095376 2.779624
> lam.hat
[1] 2.4375

> NPBarr = array( xvec[ sample(1:80, 80*1000, replace=T) ], c(1000,80))
  Batch.meanNPB = apply(NPBarr,1, mean)
> BootQ1 = quantile(Batch.meanNPB - lam.hat, prob=c(.025, .975))
  BootQ1
      2.5%      97.5% 
-0.3125000  0.3378125 
> CI1 = lam.hat - BootQ1[c(2,1)]
  CI1
   97.5%     2.5% 
2.099688 2.750000 

> BootQ2 = quantile((Batch.meanNPB - lam.hat)/sqrt(Batch.meanNPB/80), prob=c(.025, .975))
  BootQ2
     2.5%     97.5% 
-1.917412  1.813694 
> CI2 = lam.hat - sqrt(lam.hat/80)*BootQ2[c(2,1)]
  CI2
   97.5%     2.5% 
2.120914 2.772190 


> PBarr = array( rpois(80*1000, lam.hat), c(1000,80))
  Batch.meanPB = apply(PBarr,1,mean)
> BootQ4 = quantile(Batch.meanPB-lam.hat, prob=c(.025,.975))
  CI3 = lam.hat-BootQ4[c(2,1)]
  CI3
   97.5%     2.5% 
2.074688 2.750000 

### ======================================

> hist(galaxies, nclass=15, prob=T)
> lines(density(galaxies), lty=3, col="red")