AP STATS: Warm-Up
Complete the “quiz” 9.1/9.2 with a partner. Don’t use
notes or your textbook.
If you want to have your quiz graded, hand it in, and it
will count as a 25 point quiz. If not, you can choose to
just keep it.
“I shall persevere until I find something that is certain - or, at least until I find
for certain that nothing is certain.”
Rene Descartes
(1596 - 1650)
Agenda
Thursday: Sample Means
 Friday: Review
 Monday: Unit Test – Chapter 9

OVERVIEW:
This section contains one of the most important
of all statistical theorems, the Central Limit
Theorem of Statistics.
 It also emphasizes that the Greek letters m and s
are conventionally used for the population
parameters mean and standard deviation, and
 that x(bar) and s conventionally represent the
mean and standard deviation for samples.

Sample Means
Today we will talk about sample means.
Give me an example of a sample mean we
might be interested in finding.
Give me an example of a sample
proportion we might be interested in
finding.
Sample Means: Stock Diversification
Normal Distribution
Recall that the sampling distribution of p-hat is
approximately normal under certain conditions.
The same is true for x-bar. When the population is
normal, the sampling distribution is also normal.
However, the central limit theorem tells us even
more… namely that even when the population
distribution is NOT normal, the sampling distribution
will become normal when n is sufficiently large.
This is true of any size n. i.e. the mean of x-bar ALWAYS equals μ.
Use the standard deviation formula when the population is at least ten times the size of the sample.
Example
What is the probability that a randomly selected woman is 66.5 inches or taller?
What is the probability that the mean height of an
SRS of 10 randomly selected women is 66.5 inches or taller?
The Central Limit Theorem

Consider an SRS of size n from any population with
mean μ and standard deviation σ. When n is large, the
sampling distribution of x-bar has the following
properties:
◦ It is approximately normal.
◦ The mean of the distribution is x-bar ( = μ).
 s 
◦ The standard deviation of the distribution is s.  

n

A Few Notes on the CLT
1.) When the population
has a normal distribution –
shape of the sampling
distribution is normal,
regardless of the sample
size.
2.) Any population shape,
for small n – the shape of
the sampling distribution is
similar to the shape of the
population distribution.
3.) Any population
shape, large n – shape
of the sampling
distribution is close to
normal (CLT).
**This is the CLT!!
Here is an example illustrating (b) and (c) of
the Central Limit Theorem.

Consider the population P = {2,4,6}.

For this population, m = 4 and

s = sqrt[((2-4)2 + (4-4)2 + (6-4)2)/3]
= 1.632993162
Now, consider all possible samples of size 2
(with replacement). There would be 32 = 9 such
samples.
Sample
2,2
2,4
2,6
4,2
4,4
4,6
6,2
6,4
6,6
Sample
Mean
2
3
4
3
4
5
4
5
6
The collection of sample means
is M = {2,3,3,4,4,4,5,5,6}.
 The mean of M = 4 = m,
illustrating (b) of the Central
Limit Theorem.
 The standard deviation of M =
1.154700538 = s/sqrt(2) =
1.632993162/Sqrt(2), illustrating
(c) of the Central Limit Theorem.




Example:
A tire manufacturer advertised that a new brand of tires
had a mean life of 40,000 miles with a standard
deviation of 2,000 miles. A research team examined a
random sample of 100 of these tires and determined that
the tires in the sample had a mean life of 39,000 miles.
If the mean life is indeed 40,000 miles, how likely is it
that a random sample of 100 would have a mean life of
39,000 miles?
Considering the set W of all sample means of size 100,
the mean of W is 40,000, and the standard deviation of
W is 2000/sqrt(100) = 200. The probability of getting a
sample with a mean of 39,000 or less is
normalcdf(-1E99, 39000, 40000, 200) = .000000287.
In other words, it would be very unlikely to get a
sample of 100 with a mean of 39,000 miles if the
manufacturers claim is true.
Summarizing what we learned
HW #4
Read section 9.3
 Complete exercises 9.41, 9.42, 9.45, 9.46
