## Confidence Interval, Two Pop., Diff means, sigmas unknown

On your USB drive, create a new directory, copy model.R to that directory, rename the file in the new directory, double click on the file to open Rstudio. Then copy all of the text below the line and paste it into your Rstudio editor pane.
```
# line 1
#  Look at finding a confidence interval for
#  the difference of two means when the population
#  standard deviations are unknown.
#
#  For this example get a 96% confidence interval for
#  the difference  mu_1 - mu_2.
#
#  We will start by generating two populations
source("../gnrnd5.R")
gnrnd5( 275552499901, 10025010176)
L2 <- L1
tail(L2,10)
length(L2)
#
gnrnd5(277661599901, 10413010732)
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 <- 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
# and we need to get the standard deviation of each sample
sx1 <- sd( samp_1 )
sx1
sx2 <- sd( samp_2 )
sx2
#
# So our best point estimate is
pnt_est <- xbar_1 - xbar_2
pnt_est
#
#  The distribution of the difference of the means will
#  be Student's-t with standard deviation equal to
#  sqrt( sx1^2/n_1 + sx2^2/n_2 )
sd_difference <- sqrt( sx1^2/n_1 + sx2^2/n_2)
sd_difference
#
#   The, for a 96% confidence interval we need to
#   have the 4% we are missing split evenly on both
#   tails.  But, since this is a Student's-t distribution
#   the two t-values will just be opposites.  We will
#   just find the upper value.

#  But for the Student's-t we also need the degrees of
#      freedom.
#  This simple way is to use one less than the smaller
#      sample size.
df <- n_1 - 1
if( n_2 < n_1 )
{ df <- n_2- 1 }
df
#
t <- qt( 0.04/2, df, lower.tail=FALSE )
t
#  then our simple confidence interval has a low end of
pnt_est - t*sd_difference
#  and a high end of
pnt_est + t*sd_difference
#
#  Alternatively, we have a much more complex way to
#  compute the degrees of freed and this more complex
#  approach gives us a higher degree of freedom and
#  therefore a lower t score so we get a smaller
#  confidence interval.
#
# compute the complex degrees of freedom
d1 <- sx1^2 / n_1
d2 <- sx2^2 / n_2
d_f_complex <- ( d1 + d2)^2 /
(d1^2/(n_1 - 1) + d2^2/(n_2 - 1))
d_f_complex
#
#
t <- qt( 0.04/2, d_f_complex, lower.tail=FALSE )
t
#  then our complex confidence interval has a low end of
pnt_est - t*sd_difference
#  and a high end of
pnt_est + t*sd_difference
#
#
#  We could have taken the shorcut and used the
#  function that is provided.
#
source("../ci_2unknown.R")
ci_2unknown( sx1, n_1, xbar_1,
sx2, n_2, xbar_2, 0.96 )
#
#    ################################################
#    ##   Now, highlight and rerun, over and over, ##
#    ##   lines 32-114, 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 8.09059.  How often do your           ##
#    ##   intervals contain the true mean?         ##
#    ################################################
```