### Chapter 18: Sampling Distribution Models

```Chapter 18: Sampling
Distribution Models
AP Statistics
Overview of Chapter
• We have already discussed samples and
descriptive statistics, like sample proportions and
sample means.
• We know that if we take a large enough sample,
our results should be close to what we would get
if we asked the entire population (as long as
sample is random, etc)
• In this chapter, we look at many samples of to
help us do many things—maybe most important
of those things is to determine what is
statistically significant
Modeling the Distribution of Sample
Proportions
Suppose that a poll was conducted in
September in which 1000 people were asked
if they supported sending more troops to
Afghanistan and 45% said yes. A few days
later, a different polling organization asked the
same question to 1000 people and instead
found that 42% said yes. Which one is
correct? Should we be surprised with these
different results? Why or why not?
Modeling the Distribution of Sample
Proportions
What would have to do to answer those
questions, is to assume that one of those
proportions is “correct” and then imagine
what would happen if I looked at the results of
many, many different samples of 1000 people.
How much would those samples differ? What
would the distribution of those who said yes
look like?
Modeling the Distribution of Sample
Proportions
What we would find out is that the distribution
of those many, many samples would be
symmetric and unimodal—centering on the
true population proportion (or what you are
calling the true proportion). From this
symmetric and unimodal distribution, we can
then model the sample proportions as a
normal model—AS LONG AS CERTAIN
ASSUMPTIONS AND CONDITIONS ARE
SATISFIED!!!!
Modeling the Distribution of Sample
Proportions
Once we can establish the use of the normal
model, we are then able to find the standard
deviation of the distribution and therefore,
our model has the parameters

N  p,


pq 

n 
Modeling the Distribution of Sample
Proportions
Visual of How A Model of a Sampling
Distribution of Proportions is Formed
Summary of Modeling the Distribution
of Sample Proportions
Summary of Modeling the Distribution
of Sample Proportions
Normal Model for the Distribution of
the Percent of American Who Believe
we Should Send More Troops to
Afghanistan
Assumptions and Conditions
We can only use the Normal Model for the
Distribution of Sample Proportions IF two
assumptions are met:
1. The sampled values must be independent of
each other.
2. The sample size, n, must be large enough
Assumptions and Conditions
It is difficult (if not sometimes impossible) to
check or satisfy those assumptions.
Therefore, we can verify certain conditions
assumptions. Those conditions are
1. Randomization Condition
2. 10% Condition
3. Success/Failure Condition
The Three Conditions for using a
Normal Model for Sampling
Distribution of Proportions
Randomization Condition: The sample should
be an SRS (or at least very confident it is not
biased)
10% Condition: If the sample has not been
made with replacement, the sample size must
be no larger than 10% of the population.
Success/Failure Condition: The sample size has
to be big enough so that both np and nq are at
least 10.
Models
• No longer is a proportion something we just
compute, we now see it as a random quantity
that has a distribution.
• These models now can tell us the amount of
variation to expect if we sample (and what we
shouldn’t expect)
• Sampling Distributions act as a bridge between
the real world of data and an imaginary model.
This bridge and the model that results has huge
implications in statistics
Example #1
Assume that 30% of all students at a university
wear contact lenses. We randomly pick 100
student and want to know the approximate
probability that more than one-third of those
students wear contacts.
(In the process of answering this question,
specify the appropriate model, the mean and
the standard deviation. Be sure the verify that
the conditions are met.)
Modeling Distributions of Sample
Means
Below is the distribution of the numbers on the
face of a die if 10,000 dice were rolled.
Modeling Distributions of Sample
Means
Below are the distributions of rolling 2, 3, 5 and
20 dice and taking the mean of the rolls.
What do you
notice?
Modeling Distributions of Sample
Means
The Distribution of Sample Means (like Sample
Proportions) will produce a symmetric and
unimodal distribution. As long as a few
assumptions/conditions are met, then that
distribution can be modeled using the Normal
Model. This concept (along with a few other
important points) is called the Central Limit
Theorem (CLT). Sometimes, because of its
importance, it is called the Fundamental
Theorem of Statistics.
The Central Limit Theorem
Very simply, the Central Limit Theorem states:
The mean of a random sample has a sampling
distribution whose shape can be
approximated by a Normal Model. The
larger the sample size, the better the
approximation will be.
The Central Limit Theorem
• The sampling distribution of any mean becomes
more nearly normal as the sample size grows.
• The distribution of the population does NOT
matter—the distribution of sample means will
always approximate the Normal Curve.
• Need to verify two assumptions: the
observations are independent and collected with
randomization. We use conditions to help us
satisfy those important assumptions.
Conditions for Central Limit Theorem
In order to justify those assumptions, you can check these
three conditions:
1. Randomization Condition: Data must be sampled
randomly.
2. 10% Condition: If the sample has not been made
with replacement, the sample size must be no larger
than 10% of the population. This satisfies the
Independence Assumption.
3. Large Enough Sample Size: This gets discussed more
in chapter 24, but for now just think about how your
sample size relates to the population size.
Limit Theorem
The CLT does NOT talk about the distribution of
the data from the sample. It talks about the
sample means and sample proportions of
many different randeomsamples drawn from
the same population
Normal Model for the Distribution of
Sample Means
A few things to remember:
* Will be centered at the population mean.
• Means have smaller standard deviations than
individuals.
• The standard deviation of the sample mean
falls as the sample size grows. The
relationship between the standard deviation
of the mean and the sample size can be
shown by the formula:   y   SD  y   
n
Normal Model for the Distribution of
Sample Means
The Normal Model for the Distribution of
Sample Means has the parameters
 

N ,

n

Normal Model for the Distribution of
Sample Means
Example #2
Assume that SAT scores are normally distributed
with a mean of 500 and a standard deviation
of 10. Describe the distribution of sample
means if we randomly pick 50 students. Verify
that conditions are met.
Do you think it would be unreasonable to have a
randomly selected group of 50 students who
had mean of 550? Justify, using statistics.
Standard Error
The Standard Error is what we call our
estimation of the standard deviation of a
sampling distribution when we don’t know
the population proportion or the standard
deviation.
For sampling distribution of sample proportions:
SE  pˆ  
pˆ qˆ
n
For sampling distribution of sample means:

SE y 
s
n
Sampling Distribution Models (Visual
of Logic)
Problems to Look Out For
• Don’t confuse the sampling distribution with
the distribution of the sample.
• Beware of observations that are not
independent.
• Watch out for small samples from skewed
populations. --Will take large sample sizes to
“undo” the skewness and create symmetric
sampling distributions.
```