## Hypothesis test, 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 testing the null hypothesis that
#  the mean difference of  PAIRED data is zero
#  against the alternative that the mean difference in
#  the pairs is greater than 0,
#  when the population standard deviation is unknown.
#  Perform the test at the 0.025 level of significance.
#
#  We will start by generating a population, but the
#  data appears as paired values
source("../gnrnd5.R")
gnrnd5( 282553499910, 23001625005076)

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 geta different sample and as  ##
#        ##  such we will perform a different      ##
#        ##  test of the null hypothesis.          ##
#        ############################################
n_1 <- 42  # get a sample of size 42 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 test the hypothesis that
#  mean difference in the pairs, computed as
#  the second value - the first value is 0
#  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 the hypothesis that
#  the population mean = 0 by looking at the mean of samp_3.
#  But the means of samples of size n_1 are
#  disributed as a Student's-t with n_1 - 1 degrees of
#  freedom.
#
# find the t value with 2.5%  of the area to its right
t <- qt( 0.025, n_1 - 1, lower.tail=FALSE )
std_dev <- sd( samp_3 )
std_dev
# then the critical value will be
0+t*std_dev/sqrt( n_1 )
#
# reject H0 if the sample mean is greater than
# that critical value
#
# or to use the attained significance approach,
# we find the probability that we get a sample
# mean as strange or stranger than what we found
#
pt( pnt_est/(sd(samp_3)/sqrt( n_1)), n_1 - 1, lower.tail=FALSE)
#  Reject H0 if that value is less than the level
#  of significance stated, namely, 0.025
#
# rather than take the long way for this we could just
#
source("../hypo_unknown.R")
hypoth_test_unknown( 0, 1, 0.025,
length(samp_3),
mean( samp_3), sd(samp_3))
#
#    ################################################
#    ##   Now, highlight and rerun, over and over, ##
#    ##   lines 31-81, to get repeated samples and ##
#    ##   thus, repeated the test of the null      ##
#    ##   hypothesis against the alternative one.  ##
#    ##   While you do this be aware that the true ##
#    ##   mean of the differences in the paired    ##
#    ##   values in the overall population of the  ##
#    ##   pairs is 1.626204. How often do you      ##
#    ##   reject the null hypothesis?              ##
#    ################################################
```