# This lab looks long, but it's not!
# Most of the material is on the web, where you can cut/paste from.
# The majority of this document is explanations - lots of it. So,
# read carefuly.
# 1) Last time, we computed CIs in three ways: a) Using the formula for CI,
# b) Using the function t.test(), and c) using bootstrap, i.e., literally
# doing the resampling and building the empirical sampling distribution
# for the parameter of interest. The 1st method is based on the normal
# distribution. Now, we will do the same thing, but with the t-distribution.
# Three different ways of computing CIs based on t-distribution.
# Copy/paste the next block from the web:
# http://www.stat.washington.edu/marzban/390/lab_CI_supp.txt .
rm(list=ls(all=TRUE))
set.seed(123)
samp.size = 30
# Take a SMALL sample of size 30,
x = rnorm(samp.size,0,1)
# from a normal population with mean=0, sd=1.
# Note: last time, we made a non-normal population and then took samples
# from it. This time, though, we are taking a sample from a Normal
# population, because that's the only population for which the t.test()
# applies.
###################################################################
## First way: using the formulas that apply to a single sample mean only.
## Note: last week, we computed the z-interval, and so we needed quantiles
## of the normal distribution, i.e. qnorm(). This time, we are building
## t-intervals, and so we use qt(), which gives quantiles of the t-distribution.
lower = mean(x) + qt(0.05/2,samp.size-1)*sd(x)/sqrt(samp.size)
upper = mean(x) - qt(0.05/2,samp.size-1)*sd(x)/sqrt(samp.size)
c(lower,upper)
###################################################################
## Second way: t.test itself, which does exactly what the first way does.
## Obviously, this is the easiest way, but make sure you know what
## assumptions underlie it.
t.test(x)
#
Here is the portion of the output which we need for todays lab:
#
TAs explain these quantities.
#
t = -0.263, df = 29,
# The observed t-value, and the df.
#
95 percent confidence interval: