## Confidence Interval, Difference of paired values

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 mean difference of  PAIRED data
#  when the population standard deviation is unknown.
#
#  For this example get a 95% confidence interval for
#  the mean  the difference of the pairs
#
#  We will start by generating a population, but the
#  data appears as paired values
source("../gnrnd5.R")
gnrnd5( 282091499910, 41001525010176)

tail(L1,10)
tail(L2,10)
length(L1)

#
#  Now that we have our population of pairs
#  we need to take random samples of the pairs
#  of values
#
#        ############################################
#        ##  Each time we do the following steps   ##
#        ##  we will get a different sample and as ##
#        ##  such we will get different confidence ##
#        ##  interval.                            ##
#        ############################################
n_1 <- 38  # get a sample of size 38 from L1 and L2
index_1 <- as.integer( runif( n_1, 1, 5001))
index_1
samp_1 <- L1[ index_1 ]
samp_1
samp_2 <- L2[ index_1 ]
samp_2
#
#  We really want to look at a confidence interval for L2-L1
#  so we will form a new data set based upon samp_2 - samp_1
samp_3 <- samp_2 - samp_1
samp_3
#
# then our best point estimate is the mean(samp_3)
pnt_est <- mean( samp_3 )
pnt_est
#
#  Our problem now resolves to finding the 95% confidence
#  interval for the mean on samp_3, but samp_3 is
#  disributed as a Student's-t with n_1 - 1 degrees of
#  freedom.
#
# find the t value with half of 5% to its right
t <- qt( 0.025, n_1 - 1, lower.tail=FALSE )
std_dev <- sd( samp_3 )
std_dev
# then the confidence interval will have a low value of
pnt_est - t*std_dev/sqrt( n_1 )
# and the high value of
pnt_est + t*std_dev/sqrt( n_1 )
#
# rather than take the long way for this we could just
#
source("../ci_unknown.R")
ci_unknown( sd( samp_3), length(samp_3),
mean( samp_3), cl=0.95)
#
#    ################################################
#    ##   Now, highlight and rerun, over and over, ##
#    ##   lines 31-67, to get repeated samples and ##
#    ##   thus, repeated 95% confidence intervals. ##
#    ##   While you do this be aware that the true ##
#    ##   mean of the paired values in the two     ##
#    ##   populations is 4.6501. How often do you r##
#    ##   intervals contain the true mean?         ##
#    ################################################

#  Finally, we could get a picture of the change between
# samp_1 and samp_2 via the following commands.
#  This will be for the last repetition of lines 31-67
plot(1:38,samp_1, col="darkgreen",
xlim=c(0,45), xaxp=c(0,45,9),
xlab="index values",
ylim=c(40,160), yaxp=c(40,160, 12),
ylab="item values",pch=22, las=1, cex.axis=0.7,
main="Pairs of Values from last sample"
)

points(1:38,samp_2, col="darkred", pch=20)
for(i in 1:38)
{ lines(c(i,i),c(samp_1[i],samp_2[i]))}
abline(h=seq(40,160,10),lty=3,col="darkgray")
abline(v=seq(0,40,5),lty=3,col="darkgray")
legend("topright",
legend = c("samp_1","samp_2"),
pch=c(22,20),
col=c("darkgreen","darkred"),
inset=0.03)
```