# this is a script to accompany the video for
# Topic 17
#
# We have a large population.  H0 is that for some 
# characteristic the proportion of items in the population
# with that characteristic is 0.23,        H0: p = 0.23
# and the alternative is that the proportion of items with
# that characteristic is less than 0.23,   H1: p < 0.23
# We want to run the test at the 0.02 level of significance.

# We will take a sample of size 147.
#   That is less than 5% of the large population.
#   We need to check n*p and n*(1-p) be be sure both are
#   greater than 10.
147*0.23
147*(1-0.23)
#   Both work so we can use the normal approximation 
#   for the distribution of sample proportions.  If H0
#   is true then that distribution will be 
#   N( 0.23, sqrt(0.23*(1-0.23)/147) )

######################################
###  The critical value method
######################################
#   compute sqrt(0.23*(1-0.23)/147)
sqrt(0.23*(1-0.23)/147)
#  Then the critical low value will be
qnorm( 0.02, mean=0.23, sd=0.0347 )
#
#  We take our sample and we find that there are 22
#  items in the sample with the desired characteristic.
#  So, our sample proportion is
p_hat <- 22/147
p_hat
#  That is less than our critical low value.  Therefore,
#  we reject  H0 in favor of H1 at the 0.02 level
#  of significance.


######################################
###  The attained significance method
######################################

#  We took our sample and found 22 items with the 
#  characteristic. So, p_hat is
p_hat  <- 22/147
p_hat
#  Assuming H0 is true, then how "strange" is it to 
#  get a sample proportion that low or lower?
pnorm( 22/147, mean=0.23, sd=sqrt( 0.23*(1-0.23)/147 ) ) 
#  That attained significance is less than our 0.02
#  stated level of significance so we reject H0 in
#  favor of H1.

##############################
#  Or we could just use our hypoth_test_prop() function
#    to compute both approaches
source("../hypo_prop.R")
#      we will use -3 to indicate that H1 is "<"
hypoth_test_prop( 0.23, 22, 147, -3, 0.02)