# line 1
#  Look at finding a confidence interval for
#  the difference of two proportions in two 
#  populations.
#
#  For this example get a 96% confidence interval for
#  the difference  p_1 - p_2 where p_1 is the proportion
#  of 4's in population one and p_2 is the proportion
#  of 4's in population two.
#
#  We will start by generating two populations
source("../gnrnd5.R")
gnrnd5( 29472799907, 54363637)
L2 <- L1
head(L2,10)
tail(L2,10)
length(L2)
#
gnrnd5(40561699907, 35476337)
head(L1,14)
tail(L1,14)
length(L1)
#
#  Now that we have our two populations 
#  we need to take random samples of the 
#  two populations
#
#        ############################################
#        ##  Each time we do the following steps   ##
#        ##  we will get different samples and as  ##
#        ##  such we will get different confidence ##
#        ##  intervals.                            ##
#        ############################################
n_1 <- 93  # get a sample of size 93 from L1
index_1 <- as.integer( runif( n_1, 1, 7001))
index_1
samp_1 <- L1[ index_1 ]
samp_1
#
n_2 <- 104  # get a sample of size 104 from L2
index_2 <- as.integer( runif( n_2, 1, 8001))
index_2
samp_2 <- L2[ index_2 ]
samp_2
#
#  Then we need to find the frequency of the value
#  4 in each of the two samples.
table_one <- table( samp_1 )
table_one
x_1 <- table_one[4]
x_1

table_two <- table( samp_2 )
table_two
x_2 <- table_two[4]
x_2
# and we need to get the proportion of 4's in each sample
prop_1 <- x_1/n_1
prop_1
prop_2 <- x_2/n_2
prop_2
#
# Then our best point estimate for p_1 - p_2 is
#  prop_1 - prop_2
pe <- prop_1 - prop_2
#
# the standard deviation of the test statistic
# is given by 
# sqrt( prop_1(1-prop_1)/n_1 + prop_2(1-prop_2)/n_2)
stdev <- sqrt(  prop_1*(1-prop_1)/n_1 +
                prop_2*(1-prop_2)/n_2 )
stdev
#
# since we want a 96% confidence interval we want to 
# find the z value tht has 2% above it
z <- qnorm(0.02, lower.tail=FALSE)
z
# then the low end of the confidence interval is
pe - z*stdev
# and the high end of the confidence interval is 
pe + z*stdev
#
#
#  We could have taken the shortcut and used the 
#  function that is provided.
#
source("../ci_2popproport.R")
ci_2popproportion( n_1, x_1,
                   n_2, x_2, 0.96 )
#
#    ################################################
#    ##   Now, highlight and rerun, over and over, ##
#    ##   lines 34-89, to get repeated samples and ##
#    ##   thus, repeated 96% confidence intervals. ##
#    ##   While you do this be aware that the true ##
#    ##   difference between the population        ##
#    ##   proportions is 0.02448214.  How often do ##
#    ##   your intervals contain the true          ##
#    ##                                            ##
#    ################################################