# 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)

head(L1,10)
head(L2,10)
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 
# jump to doing hypoth_test_unknown().
# 
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?              ##
#    ################################################