Hypothesis Test, Two Pop., Diff means, sigmas unknown

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 unknown.
#
#  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( 213572499901, 11225010176)
L2 <- L1
tail(L2,10)
#
length(L2)
#
gnrnd5(258202599901, 10413010732)
tail(L1,14)
#
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
#
#  And we need to find the standard deviation of
# each sample
sx1 <- sd( samp_1 )
sx1
sx2 <- sd( samp_2 )
sx2
#
# 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 Student's-t with standard deviation equal to
#  sqrt( sx1^2/n_1 + sx2^2/n_2 )
sd_difference <- sqrt( sx1^2/n_1 + sx2^2/n_2)
sd_difference
#
#  But we still needd to determine the degrees of freedom
#  to use.  One way will be to use one less than the smaller
#  of n_1 and n_2.  We could improve our test by taking
#  the much more complex computed number of degrees of
#  freedom
d_f_simple <- n_1 - 1
if( n_2 < n_1 )
{ d_f_simple <- n_2 - 1 }
d_f_simple
#
#  or we do the computation for the complex version
d1 <- sx1^2 / n_1
d2 <- sx2^2 / n_2
d_f_complex <- ( d1 + d2)^2 / (d1^2/(n_1 - 1) + d2^2/(n_2 - 1))
d_f_complex
#
#  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 Student's-t distribution
#   the two t-values will just be opposites.  We will
#   just find the upper value.
#
t_simple <- qt( 0.02/2, d_f_simple, lower.tail=FALSE )
t_simple
#  then our low critical value will be
-t_simple*sd_difference
#  and our high critical value will be
t_simple*sd_difference
#
#   or to use the complex degrees of freedom
t_complex <- qt( 0.02/2, d_f_simple, lower.tail=FALSE )
t_complex
#  then our low critical value will be
-t_complex*sd_difference
#  and our high critical value will be
t_complex*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 Student's 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 Student's-t with mean=0 and standard deviation
#  equal to sd_difference and for our choice of
#  simple or complex degrees of freedom
#
#   First, using the simple degrees of freedom
attained <- pt( abs_diff/sd_difference,
d_f_simple, 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

# Then using the complex degrees of freedom
attained <- pt( abs_diff/sd_difference,
d_f_complex, 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_2unknown.R")
hypoth_2test_unknown( sx1, n_1, xbar_1,
sx2, n_2, xbar_2,
0, 0.02 )
#
#    #################################################
#    ##   Now, highlight and rerun, over and over,  ##
#    ##   lines 36-160, 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 1.30584.  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( 273502499901, 11225010176)
L2 <- L1
tail(L2,10)
#
sigma_2 <- pop_sd( L2 )
sigma_2
length(L2)
#
gnrnd5(214825599901, 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
#
#  And we need to find the standard deviation of
# each sample
sx1 <- sd( samp_1 )
sx1
sx2 <- sd( samp_2 )
sx2
#
# 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 Student's-t with standard deviation equal to
#  sqrt( sx1^2/n_1 + sx2^2/n_2 )
sd_difference <- sqrt( sx1^2/n_1 + sx2^2/n_2)
sd_difference
#
#  But we still needd to determine the degrees of freedom
#  to use.  One way will be to use one less than the smaller
#  of n_1 and n_2.  We could improve our test by taking
#  the much more complex computed number of degrees of
#  freedom
d_f_simple <- n_1 - 1
if( n_2 < n_1 )
{ d_f_simple <- n_2 - 1 }
d_f_simple
#
#  or we do the computation for the complex version
d1 <- sx1^2 / n_1
d2 <- sx2^2 / n_2
d_f_complex <- ( d1 + d2)^2 / (d1^2/(n_1 - 1) + d2^2/(n_2 - 1))
d_f_complex
#
#  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 Student's-t distribution
#   the two t-values will just be opposites.  We will
#   just find the upper value.
#
t_simple <- qt( 0.02/2, d_f_simple, lower.tail=FALSE )
t_simple
#  then our low critical value will be
-t_simple*sd_difference
#  and our high critical value will be
t_simple*sd_difference
#
#   or to use the complex degrees of freedom
t_complex <- qt( 0.02/2, d_f_simple, lower.tail=FALSE )
t_complex
#  then our low critical value will be
-t_complex*sd_difference
#  and our high critical value will be
t_complex*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 Student's 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 Student's-t with mean=0 and standard deviation
#  equal to sd_difference and for our choice of
#  simple or complex degrees of freedom
#
#   First, using the simple degrees of freedom
attained <- pt( abs_diff/sd_difference,
d_f_simple, 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

# Then using the complex degrees of freedom
attained <- pt( abs_diff/sd_difference,
d_f_complex, 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_2unknown.R")
hypoth_2test_unknown( sx1, n_1, xbar_1,
sx2, n_2, xbar_2,
0, 0.02 )
#
#    #################################################
#    ##   Now, highlight and rerun, over and over,  ##
#    ##   lines 208-332, 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 16.57508.  How often do your tests     ##
#    ##   reject the null hypothesis?               ##
#    #################################################

```