# Line 1: a small demonstration of getting a 
#   confidence interval for the mean 
#   of a population with known standard deviation.
#
#  First, we will get a population
#  In this case we will get a large population
source("../gnrnd5.R")
gnrnd5(170972734104,235001365)
#let us look at the head and tail values
head(L1)
tail(L1)
min(L1)
max(L1)
source("../pop_sd.R")
#  Now we will find the population standard deviation
sigma <- pop_sd( L1 )
sigma
# just a quick look at L1
hist(L1)
boxplot(L1, horizontal=TRUE)
source("../assess_normality.R")
assess_normality( L1 )
#
#  L1 sure looks like a Normal distribution.
#
# ##########################
# ##  Problem: find the 95% confidence interval
# ##      for the mean of the population
# ##########################

# take a simple random sample of size 23
#
#  Be careful:  Every time we do this we get 
#               a different random sample
#
L2 <- as.integer( runif(23, 1, 7343) )
# L2  holds the index values of our simple random sample
L2
L3 <- L1[ L2 ]   # L3 holds the simple random sample
L3
# we will get the mean of L3
xbar <- mean(L3)
xbar
#
#  Remember that the distribution of sample means
#  will be normal with the same mean as the population
#  and standard deviation equal to the population 
#  standard deviation divided by the square root of
#  the sample size.
#
#  The long way to generate the confidence interval
#  is to find the z-value in a standard normal
#  distribution such that there is 95% of the 
#  area between -z and z.  That means that 2.5% is 
#  less than -z and 2.5% is greater that z. We can find
#  that z value via qnorm.
#
z <- qnorm(0.025, lower.tail=FALSE)
z
#
#  Then our margin of error is z*sigma/sqrt(23)
#
moe <- z*sigma/sqrt(23)
moe
#
#  and our confidence interval is between
#  xbar-moe and xbar+moe
#
xbar - moe
xbar + moe
#
#   Of course, we could use the function
#   ci_known() to do this in
#   one easy step.
#
source("../ci_known.R")
ci_known( sigma, 23, xbar, 0.95 )
#
#
#################################
# go back and execute lines 36-77 many more times.
#    Each time you get a different random sample.
#    Therefore, each time you get a different 
#    confidence interval.  Note that the margin
#    of error stays the same because we know the
#    population standard deviation. By the way, the 
#    true mean of the population is 136.4781.  
#################################