#This is an example from Topic 16
#  The problem is:
#   1) we have a population with an unknown mean
#      and unknown standard deviation.
#   2) we have a null hypothesis that the mean of the 
#      population is 18.6
#   3) we have an alternative hypothesis that the mean
#      of the population is less than 18.6
#   4) we want to test the null hypothesis against the 
#      alternative hypothesis at the 0.025 level of
#      significance.
#   5) to do this we take a random sample of size 37
#   6) compute the sample mean and standard deviation;
#      sample mean = 17.21   sample standard deviation=4.8
#   -----------------------
#   7) at this point we could use the critical value
#      approach and find the critical low value for the
#      sample mean, the value that has probability that a 
#      sample mean will be less than that value = 0.025
#   8) if the sample mean is less than the critical low
#      value then we reject H0 in favor of H1, if not then 
#      we do not have enough evidence to reject H0 in favor 
#      of H1.
#   ------- alternatively  -----
#   9) we could use the attained significance approach and
#      determine the probability of getting a sample mean 
#      as low or lower than we jsut got.
#  10) if that probability is less than the level of 
#      significance then we reject H0 in favor of H1, 
#      if not the we do not have enough evidence to reject
#      H0 in favor of H1
################################################

#  for the critical value approach first
#  find the t value with 0.025 to its left
#      for 36 degrees of freedom
low_t <- qt( 0.025,36 )
low_t
#
#   our critical low will be 
18.6 + low_t*4.8/sqrt( 37 )  # remember that low_t is negative
#  Because our sample mean, 17.21 is not less than the
#  critical low value we do not reject H0 in favor of H1

#  or use the attained significance approach
#  First normalize the sample mean sor we can use 
#  the pt() function.
#
standard_t <- (17.21-18.6)/(4.8/sqrt(37))
standard_t  # this becomes our test statistic
# find the probability of getting that value or lower
pt( standard_t, 36 )
#   that answer, 0.0433, is not less than our level of 
#   significance, 0.025 so we do not reject H0 
#   in favor of H1

#
#   Or we could have just used our 
#       hypoth_test_unknown function to compute both
#       approaches
source("../hypo_unknown.R")
#              note the    -1   to indicate that H1: mean < 18.6
hypoth_test_unknown( 18.6, -1, 0.025, 37, 17.21, 4.8)