DEMONSTRATION LOG OF PROBABILITY SAMPLING  using R functions.
==================================================== 9/7/05

NOTE: lines of R coding are given following the
  standard prompt > , while comments and summaries 
  are given following #. 

> yvals <- c(1,2,4,4,7,7,7,8)

# These are the attribute values y_i given in 
Example 2.1, p. 26.

# Consider all possible samples of size 4
# They are placed below into a matrix with 70 rows.
# NB: these rows are all distinct, but not all of
the corresponding samples of yvals are !!

> sampmat <- NULL
  for(i in 1:5) for (j in (i+1):6) for (k in (j+1):7) 
      for(m in (k+1):8) sampmat <- rbind(sampmat, c(i,j,k,m))
> dim(sampmat)
[1] 70  4

> tvec <- apply(sampmat,1, function(irow) 2*sum(yvals[irow]))
## This compact line of R coding applies the same rule ---
##   twice the sum of the yvals values with 4 specified 
##   sampled indices --- for each possible sample of indices.
## It produces a vector of length 70: all possible t-hat 
##   estimators of the y-attribute total based on the 70 
##   equally likely samples that could be drawn.

> table(tvec)
tvec
22 28 30 32 34 36 38 40 42 44 46 48 50 52 58 
 1  6  2  3  7  4  6 12  6  4  7  3  2  6  1 

## Each of these values divided by 70 gives probability 
## of resulting estimate of total of y attributes.

> mean(tvec)
[1] 40
> var(tvec)*69/70  ## correction because this is a theoretical
[1] 54.85714       ## not a 'sample' variance !

## Formulas in Ch.2 are: mean = sum(yvals), and
##  var = (8^2/4)*(1-4/8)*var(yvals)
> sum(yvals)
[1] 40
> (8^2/4)*(1-4/8)*var(yvals)
[1] 54.85714

====================================================
NOTE for IN-CLASS comparison with the WITH-REPLACEMENT 
SAMPLING DESIGN:

> mean(yvals)         ## Mean
[1] 5
> (7/8)*var(yvals)    ## and Theoretical Variance 
[1] 6                 ## of single equiprobably sampled
		      ## value from   yvals