# Example 1: ratio estimation of number and proportion of females
# p. 38 of notes
x <- c(10,18,10,12,25,15,10)
y <- c(7,12,4,6,19,7,5)
n <- 7
N <- 20
plot(x,y,pch=16,xlab="Number of animals",ylab="Number of females")

# Estimation of proportion of females
r <- mean(y)/mean(x)
r
sr2 <- (1/(n-1))*sum((y-r*x)^2) 
sr2
SE.r <- sqrt((1-n/N)*sr2/(mean(x)^2*n))  # assumes mu_x not known
SE.r
c(r - qt(.975,n-1)*SE.r,r + qt(.975,n-1)*SE.r)

# Estimation of total number of females
tau.x <- 350  # total number of animals on all 20 plots (given)
tau.hat <- r*tau.x
SE.tau <- sqrt(N*(N-n)*sr2/n)
c(tau.hat,SE.tau)
c(tau.hat - qt(.975,n-1)*SE.tau,tau.hat + qt(.975,n-1)*SE.tau)