# 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)