STAT 515 - Chapter 6 Supplement
Brian Habing - University of South Carolina
Last Updated: January 11, 2002
S6 - More on Sampling Distributions: The t,χ2, and FDistributions
As we saw in Section 6.3, the normal distribution plays a pivotal roll in describing
how the sample mean ¯xwill behave when you have a random sample x1, x2,...xn.
Unfortunately the central limit theorem only applies when the sample size is large.
Additionally, it only tells us about the sampling distribution of the sample mean, and
not about the sampling distribution of the sample variance s2. These limitations can
be overcome if we can believe that the sample was taken from a population that was
normal to begin with. That is, if we apply the methods in Section 5.4 and verify the
data is normal, we can get the sampling distribution for ¯xwhen nis small, and can
also get the sampling distribution for s2.
S6.1 - ¯xand the Normal Distribution
A fact that is proved in STAT 512 is: if the random sample is drawn from a
population that follows a normal distribution, then Z=¯x−µ
σ/√nis exactly standard
normal. In other words, if the base population is already normal, the central limit
theorem result applies even when n= 1! The only difficulty in this is that we rarely,
if ever, know the value of the parameter σ. Because of this we can’t use this fact
directly.
S6.2 - s2and the χ2(chi-squared) Distribution
The χ2distribution can be defined as follows. If z1, z2,...z(n−1) are independent
and each follows the standard normal distribution , then
X2=z2
1+z2
2+···z2
(n−1)
