Some solutions to Sheet 1

Problem 1
Here is an Splus function that reads in the population data for the 29 towns (assuming it is stored in a text file called town_pops.txt), then calculates the variances for the three sampling estimators.

```> get.towns
function()
{
n <- 1000       #sample size
# first read in the date, and discard unneeded fields
tp <- read.table("town_pops.txt", header = T, sep = ",")
tp <- tp[, 4:10]
names(tp) <- c("pop", "white", "black", "amer", "asian","other","hisp")
Ssize <- tp\$pop # strata sizes
hisp.p <- tp\$hisp/tp\$pop        #proportion hisp in each stratum
var.within <- hisp.p * (1 - hisp.p)
Hisp <- sum(tp\$hisp)/sum(tp\$pop)        # prop hisp in whole pop
weights <- Ssize/sum(Ssize)
var.srs <- Hisp * (1 - Hisp)
var.prop <- sum(weights * var.within)
var.opt <- (sum(weights * sqrt(var.within)))^2
cat("1000 times variances:\n", "srs= ", round(var.srs, 4),
"\nprop= ", round(var.prop, 4), "\noptimal= ", round(var.opt, 4),"\n")
}
```
Here is the output from the function:

```> get.towns()
1000 times variances:
srs=  0.075
prop=  0.0632
optimal=  0.0462
```
Notice that the sample size (1000) has the same effect on each estimator.

Problem 3
Here is a not very elegant Splus function to generate the histograms. First it reads the data (assumed to be in a file breast.cancer.data) to a data.frame called bc, with columns labelled mort and pop. The random sampling is carried out by calls to the sample() functions. The plots themselves are drawn by the hist() function. The cat() function is used to print out the values of several means, variances, and standard deviations.

I found myself getting confused about the differences between the "population of mortalities" (the 301 values) and the population (sizes).

I used the cut() function to form strata. It puts about 75 counties in each stratum. Perhaps it would be better to place the larger counties in a stratum consisting of a smaller number of counties. For this problem it did not seem worth the trouble, but perhaps I am wrong.

If you like playing with Splus, try to rewrite my function by replacing it by another for-loop.

```function()
{
names(bc) <- c("mort", "pop")   #
###part (i)
popmean <- mean(bc\$pop)
NN <- length(bc\$pop)
popvar <- sum((bc\$pop - popmean)^2)/NN
)
cat("Pop mean= ", round(mean(bc\$pop), 1), "  total mort= ",sum(bc\$mort ), "\n")
cat("Pop variance= ", round(popvar, 1), " pop std= ",round(sqrt(popvar), 1), "\n")    #
###part (ii)
srs100 <- 1:100 # dummy values
for(i in 1:100) {
srs100[i] <- (NN/25) * sum(sample(bc\$mort, 25))
}
cat("srs mean = ", mean(srs100), " srs sd = ",sqrt(var(srs100)), "\n")
#
###part (iii)
prop100 <- 1:100        # dummy values
strata <- cut(bc\$pop, c(0, quantile(bc\$pop, (1:4/4))))  #form strata
Strata.sizes <- as.vector(table(strata))
samp.sizes <- c(6, 6, 6, 7)
ssam <- 1:4     #dummy values
for(i in 1:100) {
ssam[1] <- sum(sample(bc\$mort[strata == 1], 6))
ssam[2] <- sum(sample(bc\$mort[strata == 2], 6))
ssam[3] <- sum(sample(bc\$mort[strata == 3], 6))
ssam[4] <- sum(sample(bc\$mort[strata == 4], 7))
prop100[i] <- sum(Strata.sizes * ssam/samp.sizes)
}
cat("prop mean = ", mean(prop100), " prop sd=",sqrt(var(prop100)),"\n")

# Now draw the histograms.
par(mfrow = c(3, 1))
hist(bc\$mort, nclass = 16)
hist(srs100)
hist(prop100)
}

```