# line 1
#  Look at the null hypothesis that the 
#  difference of two proportions in two 
#  populations is zero against the alternative
#  hypothesis that the _1 - p_2 > 0.  
#
#  For this example we will use a 3% level of
#  significance and we will look at the proportion
#  of 3's in the two populations where p_1 is the proportion
#  of 3's in population one and p_2 is the proportion
#  of 3's in population two.
#
#  We will start by generating two populations
source("../gnrnd5.R")
gnrnd5( 94472799907, 54367637)
L2 <- L1
head(L2,10)
tail(L2,10)
length(L2)
#
gnrnd5(45861699907, 25457427)
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 <- 95  # get a sample of size 95 from L1
index_1 <- as.integer( runif( n_1, 1, 7001))
index_1
samp_1 <- L1[ index_1 ]
samp_1
#
n_2 <- 89  # get a sample of size 89 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
#  3 in each of the two samples.
table_one <- table( samp_1 )
table_one
x_1 <- table_one[3]
x_1

table_two <- table( samp_2 )
table_two
x_2 <- table_two[3]
x_2
# and we need to get the proportion of 3's in each sample
prop_1 <- x_1/n_1
prop_1
prop_2 <- x_2/n_2
prop_2
#
# Then our test statistic is p_1 - p_2 is
#
test_stat <- prop_1 - prop_2
test_stat
#
# the standard deviation of the test statistic under 
# the null hypothesis is given by first finding phat
# as the pooled proportion
phat <- (x_1+x_2) /(n_1 + n_2)
# then the standard deviation is  
# sqrt( prop_1(1-prop_1)/n_1 + prop_2(1-prop_2)/n_2)
stdev <- sqrt(  phat*(1-phat)/n_1 +
                phat*(1-phat)/n_2 )
stdev
#
# To find the critical value for this one-sided test
# we need the z value that has 3% of the are to its right.
z <- qnorm(0.03, lower.tail=FALSE)
z
#  Then our critical high value will be z*stdev
z*stdev
# reject H0 if test statistic is greater than the 
# critical value
#
# For the attained significance approach we will
# need to find out how strange it is to get our
# test statistic.
#
pnorm( test_stat, mean=0, sd=stdev, lower.tail=FALSE )
#
# Reject H0 if the attained significance is less than
# our alpha level.

#  We could have taken the shortcut and used the 
#  function that is provided.
#
source("../hypo_2popproport.R")
hypoth_2test_prop( x_1, n_1,
                   x_2, n_2, 33, 0.03 )
#
#    ################################################
#    ##   Now, highlight and rerun, over and over, ##
#    ##   lines 36-103, to get repeated samples and##
#    ##   thus, repeated hypothesis test.          ##
#    ##   While you do this be aware that the true ##
#    ##   difference between the population        ##
#    ##   proportions is 0.03667857.               ##
#    ################################################