Transcript 23 October 2001

20 October 2003
5.2 The Sampling Distribution of a Sample Mean
The expected mean of x-bar
The expected variance of x-bar
The expected histogram of x-bar
Sampling from a known population
 Suppose we have a population whose mean and
variance (or standard deviation) is known.
 Suppose we draw a simple random sample
(SRS) of n people from that population, and
calculate the mean.
 Suppose we draw another SRS of n people from
that population, and calculate its mean.






Suppose we draw another SRS of n people from that population, and calculate its mean.
Suppose we draw another SRS of n people from that population, and calculate its mean.
Suppose we draw another SRS of n people from that population, and calculate its mean.
Suppose we draw another SRS of n people from that population, and calculate its mean.
Suppose we draw another SRS of n people from that population, and calculate its mean.
Suppose we draw another SRS of n people from that population, and calculate its mean.
Sampling from a known population
After drawing lots and lots of simple random
samples, we will have a list of lots and lots of
sample means.
(The mean will vary from sample to sample,
depending on just who gets selected.)
What should we expect these sample means to
look like?
The Central Limit Theorem
If you draw simple random samples of size n
from a population with mean m and variance s2
then
 the expected mean of x-bar is m
 the expected variance of x-bar is
s2 / n
 the expected histogram of x-bar is
approximately normal
If you draw simple random samples of size n
from a population with mean m and variance s2

mx  m
If you draw simple random samples of size n
from a population with mean m and variance s2

s
2
X

s
n
2
If you draw simple random samples of size n
from a population with mean m and variance s2

s
X

s
n
If you draw simple random samples of size n
from a population with mean m and variance s2
 The expected histogram of the mean
will be approximately normal if n is large,
or if the histogram of the population is
approximately normal.
If you draw small samples from a very non-normal
population, then you should not expect your means
to have an approximately normal histogram.
The expected histogram of the mean
will be approximately normal if
large
sample
small
sample
approx
normal
population


non-normal
population


The expected histogram of the mean
will be approximately normal if
large
sample
small
sample
approx
normal
population


non-normal
population


The expected histogram of the mean
will be approximately normal if
large
sample
small
sample
approx
normal
population


non-normal
population


The expected histogram of the mean
will be approximately normal if
large
sample
small
sample
approx
normal
population


non-normal
population


The expected histogram of the mean
will be approximately normal if
large
sample
small
sample
approx
normal
population


non-normal
population


A Central-Limit-Theorem Example


Heights of American adult women are approximately
normally distributed, with a mean of 64.5 inches and
a standard deviation of 2.5 inches.
If we draw samples of 100 women from the
population over and over again, then we would
expect that the means of these samples would have
an approximately normal distribution, with a mean of
64.5 inches and a standard deviation of 0.25 inches.

According to Table A, about 68% of our samples should
have means between 64.25 and 64.75 inches.

According to Table A, about 95% of our samples should
have means between 64 and 65 inches.
Homework Problem Sets
Homework 7 27 Oct
5.2 28, 31, 36, 46, 49
Homework 8 10 Nov 6.1 3, 4, 7, 10, 16, 17, 19