# R-program for Supplementary Exercises 8.84 and 8.85 of IPS7e

# 8.84: confidence intervals
# note: built-in function prop.test uses different method
prop.test(68, 100, conf.level=0.95)

# function to compute classical CI for one proportion
zprop.ci = function(x, n, conf.level=0.95){
   p <- x/n
   lower <- p - qnorm(1-(1-conf.level)/2)*sqrt(p*(1-p)/n)
   upper <- p + qnorm(1-(1-conf.level)/2)*sqrt(p*(1-p)/n)
   return(as.vector(c(p,lower,upper)))
}
# classical (z) CI
zprop.ci(68, 100)
* plus four CI, manually constructed
zprop.ci(70, 104)
# "exact" CI based on the binomial distribution
binom.test(68, 100, conf.level=0.95)

# 8.85: tests
# note: built-in function prop.test computes chi-square test (=z^2)
prop.test(68, 100, p=0.75, correct=F)

# function to compute classical z-test for one proportion
zprop.test = function(x, n, p=0.5, alternative = c("two.sided")){
   z <- (x/n-p)/sqrt(p*(1-p)/n)
   if (alternative == "two.sided")
     pval <- 2*pnorm(-abs(z))
   if (alternative == "less")
     pval <- pnorm(z)
   if (alternative == "greater")
     pval <- pnorm(z, lower.tail=F)
   return(as.vector(c(z,pval)))
}
# z-test based on normal approximation
zprop.test(68, 100, p=0.75)
# exact test based on the binomial distribution - preferable!
binom.test(68, 100, p=0.75)