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# This lab might seem similar to the previous lab, and it is!
# But there are some subtle differences. The main difference is
# the emphasis this lab places on the concept of a confidence interval,
# and what it is expected/designed to do.
# The first task of this lab is to show that the formula for the confidence
# interval for the population mean actually does what it is designed to do.
# Recall that the formula is
#
sample.mean + z_star * sample.sd/root(n)
# and it is designed to cover the population mean in 95% of samples taken
# from the population. The nontrivial part of this formula is the
# sample.sd/root(n), also called the standard error (std err) of the sample mean.
# The second task is to show that we can actually get similar answers,
# WITHOUT using the formula for the std err of the mean. This is important
# when simple formulas for the std err do not exist, e.g., for sample median.
# The main idea is called the bootstrap, and it basically treats the single
# sample that you have in a realistic situation as if it were the population!
# So, instead of sampling from the population (i.e., what we did last time),
# bootstrap resamples from the sample.
# It's like magic, but you'll see how it works below.
# The third task is to actually use the bootstrap idea to build a CI for
# the population median, i.e. a situation where simple formulas for the
# std. err. of the median do not exist.
# Type in a script window, first, because you'll be using similar
# lines of code for different purposes.
# The final task is to learn about a function in R  called t.test() 
# which computes CIs under special conditions (normality, etc.). We'll
# learn more about the t.test() in the next lab.
##########################################################################
#
Cut/paste the following block from
#
http://www.stat.washington.edu/marzban/390/lab7_supp.txt
rm(list=ls(all=TRUE))
N = 100000
# Here is our population.
set.seed(1)
pop=rgamma(N,2,3)
pop.mean=mean(pop)
pop.sd=sd(pop)
pop.median=median(pop)
c(pop.mean,pop.sd,pop.median)
hist(pop,breaks=400)
# Population is not Normal. Check it out.
############################################################################
## 1) Build CIs for population mean, using the formula for the std err of mean.
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## Draw n.trial=100 samples of size sample.size=90 from the population made above.
## We want to confirm that the CI (as computed with our formula) covers the
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 Spring '08
 Normal Distribution, #, Std Err, *each sample*

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