# Line 1: a small demonstration of getting a
#   confidence interval for the mean 
#   of a population with unknown standard deviation.
#
#  First, we will get a population
#  In this case we will get a large population
source("../gnrnd5.R")
gnrnd5(182651734104,285002867)
#let us look at the head and tail values
head(L1)
tail(L1)
min(L1)
max(L1)
#
# now, we could find the standard deviation of the
# population, but this is supposed to be an example
# of finding confidence intervals for the mean when  
# we do not know the population standard deviation.
#
# 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 when we
# ##      do not know the population standard deviation.
# ##########################

# 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
# 
# and we will get the standard deviation of the sample
sx <- sd( L3 )
sx
#
#  Remember that the distribution of sample means
#  will be the Student's-t distribution with n-1
#  degrees of freedom, in this case 22 degrees of 
#  freedom.  And the distribution of the sample means
#  will have the same mean as the population
#  and standard deviation equal to the population 
#  standard deviation divided by the square root of
#  the sample size. However, for the Student's-t we 
#  the standard deviation of the sample divided by 
#  the square root of the size of the sample.
#
#  The long way to generate the confidence interval
#  is to find the t-value in a Student's-t 
#  distribution with 22 degrees of freedom
#  such that there is 95% of the 
#  area between -t and t.  That means that 2.5% is 
#  less than -t and 2.5% is greater that t. 
#  We can find that z value via qt.
#
t <- qt(0.025, 22, lower.tail=FALSE)
t
#
#  Then our margin of error is z*sigma/sqrt(23)
#
moe <- t*sx/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_unknown() to do this in
#   one easy step.
#
source("../ci_unknown.R")
ci_unknown( sx, 23, xbar, 0.95 )
#
#
#################################
# go back and execute lines 34-90 many more times.
#    Each time you get a different random sample.
#    Therefore, each time you get a different 
#    confidence interval. Note that the MOE changes
#    each time because the standard deviation of the
#    sample changes for each sample. By the way, the 
#    true mean of the population is about 286.62002.  
#################################