# line 1
#  Look at finding a confidence interval for
#  the difference of two means when the population
#  standard deviations are known.
#
#  For this example get a 93% confidence interval for
#  the difference  mu_1 - mu_2.
#
#  We will start by generating two populations
source("../gnrnd5.R")
gnrnd5( 276932499901, 10025010176)
L2 <- L1
head(L2,10)
tail(L2,10)
source("../pop_sd.R")
sigma_2 <- pop_sd( L2 )
sigma_2
length(L2)
#
gnrnd5(245961599901, 10413010732)
head(L1,14)
tail(L1,14)
sigma_1 <- pop_sd(L1)
sigma_1
length(L1)
#
#  Now that we have our two populations 
#  and now that we know the standard deviation of each
#  population, 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 <- 43  # get a sample of size 43 from L1
index_1 <- as.integer( runif( n_1, 1, 6001))
index_1
samp_1 <- L1[ index_1 ]
samp_1
#
n_2 <- 56  # get a sample of size 56 from L2
index_2 <- as.integer( runif( n_2, 1, 5001))
index_2
samp_2 <- L2[ index_2 ]
samp_2
#
#  Then we need to find the mean of each sample
xbar_1 = mean( samp_1 )
xbar_1
xbar_2 = mean( samp_2 )
xbar_2
#
# So our best point estimate is 
pnt_est <- xbar_1 - xbar_2
pnt_est
#
#  The distribution of the difference of the means will
#  be normal with standard deviation equal to 
#  sqrt( sigma_1^2/n_1 + sigma_2^2/n_2 )
sd_difference <- sqrt( sigma_1^2/n_1 + sigma_2^2/n_2)
sd_difference
#
#   The, for a 93% confidence interval we need to 
#   have the 7% we are missing split evenly on both
#   tails.  But, since this is a normal distribution
#   the two z-values will just be opposites.  We will
#   just find the upper value.
#
z <- qnorm( 0.07/2, lower.tail=FALSE )
z
#  then our confidence interval has a low end of 
pnt_est - z*sd_difference
#  and a high end of
pnt_est + z*sd_difference
#
#  We could have taken the shorcut and used the 
#  function that is provided.
#
source("../ci_2known.R")
ci_2known( sigma_1, n_1, xbar_1,
           sigma_2, n_2, xbar_2, 0.93 )
#
#    ################################################
#    ##   Now, highlight and rerun, over and over, ##
#    ##   lines 38-84, to get repeated samples and ##
#    ##   thus, repeated 93% confidence intervals. ##
#    ##   While you do this be aware that the true ##
#    ##   difference between the population means  ##
#    ##   is 7.811552.  How often do your          ##
#    ##   intervals contain the true mean?         ##
#    ################################################