## Hypothesis Test, Two Pop., Diff means, sigmas known

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 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
tail(L2,10)
source("../pop_sd.R")
sigma_2 <- pop_sd( L2 )
sigma_2
length(L2)
#
gnrnd5(245402599901, 10413010732)
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
tail(L2,10)
#
sigma_2 <- pop_sd( L2 )
sigma_2
length(L2)
#
gnrnd5(279402599901, 13413010732)
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?               ##
#    #################################################

```