#  Examples Test of Binomial proportion
#  Edps 587
#  C.J.Anderson
#
#  Exact tests and confidence intervals for 
#    US presidents
#    GSS regarding abortion
#    2000 election: Bush vs Gore
#
library(binom)   # For asympotic CI and alternative method

#######################################################
# Example 1: US presidents                            #
#######################################################
# US presidents:  Taller candidate won (as in the lecture notes)
N<- 15
y<- 13

#  Exact tests:
# --- one sided
(binom.test(y,N,alternative=c("less"),conf.level=.95))

# --- two sided
(binom.test(y,N,alternative=c("two.sided"),conf.level=.95))

# Asymptotic test statistic:
# --- no correction for continuity
(prop.test(y,N,p=.5, alternative="two.sided",conf.level=0.95,correct=FALSE))

# --- Note on asympototic:  "X-squared is z^2".  
sqrt(8.0667)                             # compare this with.... 

# To check the asympotitc w/o continunity
p <- y/N
z <- (p-.5)/sqrt(.5*(1-.5)/N)

# --- with Yate's correction for continuity
(prop.test(y,N,p=.5, alternative="two.sided",conf.level=0.95,correct=TRUE))


# More on confidence intervals
# - here are 8 different methods
binom.confint(y,N,conf.level=.95, methods="all")
# Among these are asymptotic, exact and "wilson", which is the 
#   one based on finding pi such that would retain null.



################################################################
#   Example 2: GSS                                             #
################################################################
# GSS:  ok for pregnant woman to have aborition if can't afford it ?
yes <- 971
no <- 954 
N <- yes+no

# Exact: 
(binom.test(yes,N,alternative=c("two.sided"),conf.level=.95))

# Asymptotic test:
(prop.test(yes,N,p=.5, alternative="two.sided",conf.level=0.95,correct=FALSE))
binom.confint(yes,N,conf.level=.95, methods="all")

##############################################################
# Example 3: 2000 election: Bush vs Gore in Florida          #
##############################################################
 
bush <- 2912790
gore <- 2912253
n <- bush + gore
(binom.test(bush,n,alternative=c("greater"),conf.level=.999))

# An setting alpha=.001 and doing 2-sided
(binom.test(bush,n,alternative=c("two.sided"),conf.level=.999))

# This should be *very* close to asymptotic result
(prop.test(bush,n,alternative="two.sided",conf.level=0.999,correct=TRUE))
binom.confint(y,N,conf.level=.99, methods="all")