#examples with sampling distributions

#go over the example in class
#my example had the following units:
ids=c(66,86,87,3,34,51,15,48,97,97)
#the responses for those ids were:
resp=c(10,5,8,1,12,16,9,6,4,4)
#the mean was:
ybar=mean(resp)
#the standard deviation was:
s=sd(resp)
#the standard error for Ybar was:
se=sd(resp)/sqrt(10)

#let us compute a histogram for all your values
your.ybars=c()
hist(your.ybars)
#where is the population mean? 

#In this example, I know the data for the entire population, 
#so I can actually go and repeat this exercise by drawing more samples. 

#let's read in the data first.
areas.pop=scan("100Areas.txt")
#compute the mean for the sample from class
mean(areas.pop[ids])

#draw another sample, size 20 this time
resp=sample(areas.pop,20)
#compute mean and std.err
mean(resp)
sd(resp)/sqrt(20)

#draw many samples and compute the average each time. Then draw a histogram.We need to use a ``function'' for this
#the function is written in a separate file which we will
#read into R by using the command ``source''.
#The file name is Lab2Sampling.r
source("Lab2Sampling.r")
#the name of the function we need to use is repeat.samples
#and its arguments are (in this order):``population''(all responses),``sample.size''(denoted by n in class) and ''M'' (the number of samples to be drawn from the population)
	
#to run this function you will need to specify three things: the population to sample from, the sample size and how many sample you would like to draw. For example, type:
repeat.samples(areas.pop,20,100)
#this reported 100 sample means of samples of size 20

#to draw histograms of the result you will need to save these results
mult.ybars=repeat.samples(areas.pop,20,100)
#then draw a histogram
hist(mult.ybars)

#do this 1000 times
mult.ybars=repeat.samples(areas.pop,20,1000)
#then draw a histogram
hist(mult.ybars)

#increase the sample size to 30
mult.ybars=repeat.samples(areas.pop,30,1000)
#then draw a histogram
hist(mult.ybars)

#draw 10000 random samples of size 5
mult.ybars.5=repeat.samples(areas.pop,5,10000)
hist(mult.ybars.5)
#draw 10000 random samples of size 10
mult.ybars.10=repeat.samples(areas.pop,10,10000)
hist(mult.ybars.10)
#draw 10000 random samples of size 50
mult.ybars.50=repeat.samples(areas.pop,50,10000)
hist(mult.ybars.50)

#add a red line on the histogram indicating the true mean
abline(v=mean(areas.pop),col=2)

#draw a graph containing the following 4 figures: a histogram of the population, 10000 sample means for sample size 5, 10 and 50, indicating the true average (mu) on all these graphs

par(mfrow=c(2,2))
hist(areas.pop)
abline(v=mean(areas.pop),col=2)
hist(mult.ybars.5)
abline(v=mean(areas.pop),col=2)
hist(mult.ybars.10)
abline(v=mean(areas.pop),col=2)
hist(mult.ybars.50)
abline(v=mean(areas.pop),col=2)