# line 1
#  Look at testing the hypothesis that there is no
#  difference between two means when the population
#  standard deviations are known.
#
#  For this example we will run the test at the 0.02
#  level of significance and the alternative hypotheis
#  is that the difference  mu_1 - mu_2 != 0.
#
#  We will start by generating two populations
source("../gnrnd5.R")
gnrnd5( 256472499901, 11225010176)
L2 <- L1
head(L2,10)
tail(L2,10)
source("../pop_sd.R")
sigma_2 <- pop_sd( L2 )
sigma_2
length(L2)
#
gnrnd5(245402599901, 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 a different          ##
#        ##  test of the null hypothesis.          ##
#        ############################################
n_1 <- 37  # get a sample of size 37 from L1
index_1 <- as.integer( runif( n_1, 1, 6001))
index_1
samp_1 <- L1[ index_1 ]
samp_1
#
n_2 <- 49  # get a sample of size 49 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 difference of the sample means is
samp_diff <- xbar_1 - xbar_2
samp_diff
#
#  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
#
#  First the critical value approach:
#
#   To run the test at the 0.02 level of significance 
#   have the 2% 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( 0.02/2, lower.tail=FALSE )
z
#  then our low critical value will be 
-z*sd_difference
#  and our high critical value will be
z*sd_difference
#
#  REJECT H0 if samp_diff is less than the low
#  critical value or greater than the high value
#
#  Second, the attained significance approach
#
#  Because this is a two-tailed test we need to 
#  find how strange it is to get the difference of
#  the means to be this strange, the value of samp_diff,
#  or stranger.
#
#  Because this is a dynamic section of the script 
#  samp_diff could be positive or negative.  Here
#  we will use the fact that the normal distribution
#  is symmetric so we can just use the absolute 
#  value of the difference of the sample means.
#
abs_diff <- abs( samp_diff)
#
#  The the question is how strange would it be to 
#  get that value of higher from a population that
#  is Normal with mean=0 and standard deviation 
#  equal to sd_difference
#
attained <- pnorm( abs_diff, mean=0, sd=sd_difference,
       lower.tail=FALSE)
attained
#
#  Then, double that value, because this is a two-tailed
#  test, and if the doubled value is less than our
#  level of significance we reject H0.
attained*2


#  We could have taken the shorcut and used the 
#  function that is provided.
#
source("../hypo_2known.R")
hypoth_2test_known( sigma_1, n_1, xbar_1,
           sigma_2, n_2, xbar_2, 
           0, 0.02 )
#
#    #################################################
#    ##   Now, highlight and rerun, over and over,  ##
#    ##   lines 39-121, to get repeated samples and ##
#    ##   thus, repeated 2% tests of the null       ##
#    ##   hyothesis that the means are the same.    ##
#    ##   While you do this be aware that the true  ##
#    ##   difference between the population means   ##
#    ##   is 1.163402.  How often do your tests     ##
#    ##   reject the null hypothesis?               ##
#    #################################################

# -----------------------------------------------------
#
#   We can do this all over again, but this time we will
#   create two populations that have means that differ by
#   a significant amount
#
#  We will start by generating two populations
#
gnrnd5( 213472499901, 11225010176)
L2 <- L1
head(L2,10)
tail(L2,10)
#
sigma_2 <- pop_sd( L2 )
sigma_2
length(L2)
#
gnrnd5(279402599901, 13413010732)
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 a different          ##
#        ##  test of the null hypothesis.          ##
#        ############################################
n_1 <- 37  # get a sample of size 43 from 37
index_1 <- as.integer( runif( n_1, 1, 6001))
index_1
samp_1 <- L1[ index_1 ]
samp_1
#
n_2 <- 49  # get a sample of size 49 from L1
index_2 <- as.integer( runif( n_1, 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 difference of the sample means is
samp_diff <- xbar_1 - xbar_2
samp_diff
#
#  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
#
#  First the critical value approach:
#
#   To run the test at the 0.02 level of significance 
#   have the 2% 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( 0.02/2, lower.tail=FALSE )
z
#  then our low critical value will be 
-z*sd_difference
#  and our high critical value will be
z*sd_difference
#
#  REJECT H0 if samp_diff is less than the low
#  critical value or greater than the high value
#
#  Second, the attained significance approach
#
#  Because this is a two-tailed test we need to 
#  find how strange it is to get the difference of
#  the means to be this strange, the value of samp_diff,
#  or stranger.
#
#  Because this is a dynamic section of the script 
#  samp_diff could be positive or negative.  Here
#  we will use the fact that the normal distribution
#  is symmetric so we can just use the absolute 
#  value of the difference of the sample means.
#
abs_diff <- abs( samp_diff)
#
#  The the question is how strange would it be to 
#  get that value of higher from a population that
#  is Normal with mean=0 and standard deviation 
#  equal to sd_difference
#
attained <- pnorm( abs_diff, mean=0, sd=sd_difference,
                   lower.tail=FALSE)
attained
#
#  Then, double that value, because this is a two-tailed
#  test, and if the doubled value is less than our
#  level of significance we reject H0.
attained*2


#  We could have taken the shorcut and used the 
#  function that is provided.
#
source("../hypo_2known.R")
hypoth_2test_known( sigma_1, n_1, xbar_1,
                    sigma_2, n_2, xbar_2, 
                    0, 0.02 )
#
#    #################################################
#    ##   Now, highlight and rerun, over and over,  ##
#    ##   lines 169-250, to get repeated samples and##
#    ##   thus, repeated 2% tests of the null       ##
#    ##   hyothesis that the means are the same.    ##
#    ##   While you do this be aware that the true  ##
#    ##   difference between the population means   ##
#    ##   is 15.74752.  How often do your tests     ##
#    ##   reject the null hypothesis?               ##
#    #################################################