> # read in data, extract rows for douglas fir
> trees <- read.csv('http://www.ms.unimelb.edu.au/%7Es620374/sampling/ufc.csv')
> summary(trees)
      Plot             Tree           Species         Dbh        
 Min.   :  1.00   Min.   : 1.000   WC     :251   Min.   : 100.0  
 1st Qu.: 30.00   1st Qu.: 2.000   GF     :185   1st Qu.: 222.5  
 Median : 58.00   Median : 3.000   DF     : 77   Median : 329.0  
 Mean   : 63.23   Mean   : 3.465   WP     : 44   Mean   : 356.6  
 3rd Qu.: 97.00   3rd Qu.: 5.000   WL     : 34   3rd Qu.: 457.0  
 Max.   :144.00   Max.   :13.000   SF     : 14   Max.   :1120.0  
                                   (Other): 32   NA's   :  10.0  
     Height     
 Min.   :  0.0  
 1st Qu.:190.0  
 Median :240.0  
 Mean   :240.7  
 3rd Qu.:290.0  
 Max.   :480.0  
 NA's   :248.0  
> trees.df <- trees[trees$Species=="DF",]
> 
> # plot Dbh against Height for all trees and douglas fir
> par(mfrow=c(1,2))
> max(trees$Height, na.rm=T)
[1] 480
> max(trees$Dbh, na.rm=T)
[1] 1120
> plot(c(0, 1120), c(0, 480), type='n', xlab='Dbh', ylab='Height',  main='All trees')
> points(trees$Dbh, trees$Height)
> max(trees.df$Height, na.rm=T)
[1] 420
> max(trees.df$Dbh, na.rm=T)
[1] 998
> plot(c(0, 998), c(0, 420), type='n', xlab='Dbh', ylab='Height', main='Douglas Firs')
> points(trees.df$Dbh, trees.df$Height)
> 
> # extract sample observations
> trees.sample <- trees.df[!is.na(trees.df$Height),]
> N <- length(trees.df[[1]])
> n <- length(trees.sample[[1]])
> 
> # SRS estimate of average Height
> mu.srs <- mean(trees.sample$Height)
> var.srs <- var(trees.sample$Height)*(1-n/N)/n
> cat('SRS: mean', mu.srs, 'std dev', sqrt(var.srs), '\n')
SRS: mean 254.3036 std dev 4.619371 
> 
> # Ratio estimate
> r <- mean(trees.sample$Height)/mean(trees.sample$Dbh)
> mu.ratio <- r*mean(trees.df$Dbh)
> var.ratio <- (var(trees.sample$Height) - 
+ 2*r*cov(trees.sample$Height, trees.sample$Dbh) + r^2*var(trees.sample$Dbh))*(1-n/N)/n
> cat('Ratio: mean', mu.ratio, 'std dev', sqrt(var.ratio), '\n')
Ratio: mean 246.203 std dev 5.000876 
> 
> # Linear Regression estimate
> b <- cov(trees.sample$Height, trees.sample$Dbh)/var(trees.sample$Dbh)
> mu.lr <- mean(trees.sample$Height) + b*(mean(trees.df$Dbh) - mean(trees.sample$Dbh))
> var.lr <- (var(trees.sample$Height) - cov(trees.sample$Height, trees.sample$Dbh)^2/
+ var(trees.sample$Dbh))*(n-1)/(n-2)*(1-n/N)/n
> cat('LR: mean', mu.lr, 'std dev', sqrt(var.lr), '\n')
LR: mean 250.5279 std dev 3.123521 
>