Chapter 18: Sampling Distribution Models
How can surveys conducted at essentially the same time by organizations asking
the same questions get different results? You probably easily accept that
observations vary but how much variability among polls should we expect to see?
The proportions vary from sample to sample because the samples are composed
of different people.
The Central Limit Theorem for Sample Proportions
We want to imagine the results from all the random samples of the same size
that we didn’t take. What would the histogram of all the sample proportions
look like? The center of the histogram will be at the true proportion in the
population, and we call that p.
How about the shape of the histogram? To figure this out, we could actually
simulate a bunch of random samples that we didn’t really draw. It should be no
surprise that we don’t get the same proportion for each sample we draw. Each
Ù
p comes from a different simulated sample. If you create a histogram of all the
proportions from all the possible samples it would be called the sampling
distribution of the proportions.
It ends up that the sampling distribution of the proportions is centered at the
true proportion in the population (p) and is unimodal and symmetric in shape. It
looks like we can use the Normal model for the histogram of the sample
proportions.
Sampling proportions vary from sample to sample and that is one of the very
powerful ideas of this course. A sampling distribution model for how a sample
proportion varies from sample to sample allows us to quantify that variation and
to talk about how likely it is that we’d observe a sample proportion in any
particular interval. (for example, if you would take several different 10 person
random samples asking whether or not people believed in ghosts, how likely is it
that one group will get 10% while another will get 90%??? Is there a reason why
these results occurred?? Was the survey biased??? What’s going on!?)
To use the Normal model we need two parameters – mean and standard
deviation. We will put the mean, m, at the center of the histogram, p.
Once we know the mean p, we can find the standard deviation with the
formula: s ( pˆ ) = SD( pˆ ) =
pq
.
n
As long as the number of samples is reasonably large, we can model the
distribution of sample proportions with a probability model that is N(p,
pq
).
n
And don’t forget that once you have a Normal model, you can use the 68-95-99.7
rule or look up other probabilities using a table or calculator.
This is what we mean by sampling error – it’s not really an error at all! Its just
variability you’d expect to see from one sample to another. A better term
would be sampling variability.
The Normal model becomes a better and better representation of the
distribution of the sample proportions as the sample size gets bigger. (formally,
we say the claim is true in the limit as n grows)
Assumptions & Conditions
To use a model, we usually must make some assumptions. To use the sampling
distribution model for sample proportions, we need two assumptions:
1. The Independence Assumption – the sampled values must be
independent of each other
2. The Sample Size Assumption – the sample size n must be large enough
Conditions:
1. Randomization Condition – if your data come from an experiment,
subjects should have been randomly assigned to treatments. If you
have a survey, your sample should be a simple random sample of the
population. The sampling method should not be biased and the data
should be representative of the population.
2. 10% condition – the sample size n must be no larger than 10% of the
population
3. Success/Failure Condition – the sample size has to be big enough so
that we expect at least 10 successes and at least 10 failures.
The last two conditions seem to contradict each other and as always, you would
be correct to say that a larger sample is better. It’s just that if the sample were
more than 10% of the population we’d need to use different methods to analyze
the data. Luckily, this is not usually a problem in practice.
A Sampling Distribution Model for a Proportion
We’ve simulated repeated samples and looked at a histogram of the sample
proportions. We modeled that histogram with a Normal model. We model it
because this model will give us insight into how much the sample proportion can
vary from sample to sample.
The fact that the sample proportion is a random variable taking on a different
value in each random sample and that we can say something specific about the
distribution of those values is a fundamental insight that will be used in each of
the next four chapters.
No longer is a proportion something we just compute for a set of data. We now
see it as a random variable quantity that has a probability distribution, and
thanks to Laplace we have a model for that distribution. We call that the
sampling distribution model for the proportion.
SAMPLING DISTRIBUTION MODEL FOR A PROPORTION
Provided that the sampled values are independent and the sample
size
is large enough, the sampling distribution of pˆ is modeled by a
Normal model with mean m( pˆ ) = p and standard deviation
pq
.
s ( pˆ ) = SD( pˆ ) =
n
Sampling models are what make statistics work. They inform us about the
amount of variation we should expect when we sample. The sampling model
quantifies the variability, telling us how surprising any sample proportion is.
Rather than thinking about the sample proportion as a fixed quantity
calculated from our data, we now think of it as a random variable – our value is
just one of the many we might have seen had we chosen a different random
sample.
Example: The centers for disease control and prevention report that 22% of 18
year old women in the US have a body mass index of 25 or higher – a value
considered by the national heart lung and blood institute to be associated with
increased health risk. As part of a routine health check at a large college, the
physical education department usually requires students to come in to be
measured and weighed. This year, the department decided to try out a selfreport system. It asked 200 randomly selected female students to report their
heights and weights. Only 31 of these students has BMI’s greater than 25.
Is this proportion of high BMI students unusually small???
Example: Suppose in a large class of introductory statistics students, the
professor has each person toss a coin 16 times and calculate the proportion of
his or her tosses that were heads. Suppose the class repeats the coin-tossing
experiment.
a) The students toss the coins 25 times each now. Use the 68-95-99.7 rule to
describe the sampling distribution model.
b) Confirm that you can use a Normal model here.
c) They increase the number of tosses to 64 each. Draw and label the
appropriate sampling distribution model.
d) Explain how the sampling distribution model changes as the number of
tosses increases.
End of Day 1: Complete pages 432-433 #2, 5, 6, 8, 9
Example: Suppose that about 13% of the population is left-handed. A 200-seat
auditorium has been built with 15 left seats, seats that have the built-in desk on
the left rather than the right arm of the chair. Question: In a class of 90
students, what’s the probability that there will not be enough seats for the lefthanded students?
What About Quantitative Data?
The good news is that we can do something similar when we talk about
quantitative data!
Simulating the Sampling Distribution of a Mean
Please read/consult pages 420-421 for the simple simulation of dice tosses and
their averages.
Remember that the Large of Large Numbers says that as the sample size gets
larger, each sample average is more likely to be closer to the population mean.
The shape of the distribution becomes bell-shaped – and not just bell-shaped –
it’s approaching the Normal model. But can we count on this happening for
situations other than dice throws? It turns out that Normal models work well
amazingly often.
The Fundamental Theorem of Statistics
The dice simulation may look like a special situation, but it turns out that what
we saw with dice is true for means of repeated samples for almost every
situation. And, there are almost no conditions at all!!
The sampling distribution of any mean becomes more nearly Normal as the
sample size grows. All we need is for the observations to be independent and
collected with randomization. We don’t even care about the shape of the
population distribution! This surprising fact is the result Laplace proved in a
fairly general form in 1810. At the time, Laplace’s theorem caused quite a stir
because it is so unintuitive. Laplace’s result is called the Central Limit theorem
(CLT).
Not only does the distribution of means of many random samples get closer and
closer to a Normal model as the sample size grows, this is true regardless of the
shape of the population distribution! It works better and faster the closer the
population distribution is to a Normal model. And it works better for larger
samples.
The CENTRAL LIMIT THEOREM (CLT)
The mean of a random sample is a random variable whose sampling
distribution can be approximated by a Normal model. The larger
the sample, the better the approximation will be.
Assumptions & Conditions


Independence Assumption: the sampled values must be independent of
each other
Sample Size Assumption: the sample size must be sufficiently large

Randomization condition: the data values must be sampled randomly or
the concept of a sampling distribution makes no sense

10% condition: when the sample is drawn without replacement, the
sample size, n, should be no more than 10% of the population

Large Enough Sample Condition: although the CLT tells us that a
Normal model is useful in thinking about the behavior of sample means
when the sample size is large enough, it doesn’t tell us how large a sample
we need.
The truth is, it depends; there’s no one-size-fits-all rule. If the population is
unimodal and symmetric, even a fairly small sample is okay. If the population is
strongly skewed, it can take a pretty large sample to allow use of a Normal
model to describe the distribution of sample means. For now you’ll just need to
think about your sample size in the context of what you know about the
population, and then tell whether you believe the Large Enough Sample
Condition has been met.
But Which Normal Model should we use???
For means, our center is at the population mean. What about standard
deviations though? Means vary less than the individual observations. Which
would be more surprising, having one person in your stat class who is over 6’9”
tall or having the mean of 100 students taking the course is over 6’9”??? The
second situation just won’t happen because means have smaller standard
deviations than individuals.
THE SAMPLING DISTRIBUTION MODEL FOR A MEAN (CLT)
When a random sample is drawn from any population with mean m
and standard deviation s , its sample mean, y , has a sampling
distribution with the same mean m but whose standard deviation is
s
. No matter what population the random sample comes from,
n
the shape of the sampling distribution is approximately Normal as
long as the sample size is large enough. The larger the sample used,
the more closely the Normal approximates the sampling distribution
for the mean.
So, when we have categorical data, we calculate a sample proportion, p̂ . The
sampling distribution of this random variable has a Normal model with a mean at
the true proportion p and a standard deviation of SD( p̂) =
model in chapters 19-22).
pq
. (we’ll use this
n
When we have quantitative data, we calculate a sample mean, y . The sampling
distribution of this random variable has a Normal model with a mean at the true
mean,
m , and a standard deviation of SD( y ) =
s
n
. We’ll use this model in
chapters 23-25.
These are standard deviations of the statistics p̂ and y . They both have a
square root of n in the denominator. That tells us that the larger the sample,
the less either statistic will vary.
Example: A college physical education department asked a random sample of
200 female students to self-report their heights and weights, but the
percentage of students with BMI’s over 25 seemed suspiciously low. One
possible explanation may be that the respondents ‘shaded’ their weights down a
bit. The CDC reports that the mean weight of 18-year-old women is 143.74 lbs,
with a standard deviation of 51.54 lbs, but these 200 randomly selected women
reported a mean weight of only 140 lbs.
Question: based on the Central Limit Theorem and the 68-95-99.7 Rule, does
the mean weight in this sample seem exceptionally low, or might this just be
random sample-to-sample variation?
Example: The Centers for Disease Control and Prevention reports that the
mean weight of adult men in the US is 190 lbs with a standard deviation of 59
lbs. Question: An elevator in our building has a weight limit of 10 persons or
2500 lbs. What’s the probability that if 10 men get on the elevator, they will
overload its weight limit?
Means vary less than individual data values. Averages should be more
consistent for larger groups. It’s harder to get the standard deviation down
because that nasty square root limits how much we can make a sample tell
about the population. This is an example of something that’s known as the Law
of Diminishing Returns.
The Real World and the Model World
BE CAREFUL!!! We have been slipping smoothly between the real world, in
which we draw random samples of data, and a magical mathematical model
world, in which we describe how the sample means and proportions we observe
in the real world behave as random variables in all the random sample that we
might have drawn. How we have two distributions to deal with. The first is the
real-world distribution of the sample, which we might display with a histogram
(for quantitative data) or a bar chart or table (for categorical data). The
second is the math world sampling distribution model of the statistic, a Normal
model based on the Central Limit Theorem. Don’t confuse the two.
For example, don’t mistakenly think the CLT says that the data are Normally
distributed as long as the sample is large enough. In fact, as samples get larger,
we expect the distribution of the data to look more and more like the
population from which they are drawn – skewed, bimodal, whatever – but not
necessarily Normal. You can collect a sample of CEO salaries for the next 1000
years but the histogram will never look Normal. The Central Limit Theorem
doesn’t talk about the distribution of the data from the sample. It talks about
the sample means and sample proportions of many different random samples
drawn from the same population.
Just Checking:
1. Human gestation times have a mean of about 266 days, with a standard
deviation of about 16 days. If we record the gestation times of a sample
of 100 women, do we know that a histogram of the times will be well
modeled by a Normal model?
2. Suppose we look at the average gestation times for a sample of 100
women. If we imagined all the possible random samples of 100 women we
could take and looked at the histogram of all the sample means, what
shape would it have?
3. Where would the center of that histogram be?
4. What would be the standard deviation of that histogram?
Sampling Distribution Models
At the heart is the idea that the statistic itself is a random variable. We can’t
know what our statistic will be because it comes from a random sample. This
sample-to-sample variability is what generates the sampling distribution. The
sampling distribution shows us the distribution of possible values that the
statistic could have had.
The two basic truths about sampling distributions are:
1. Sampling distributions arise because samples vary. Each random sample
will contain different cases and, so, a different value of the statistic.
2. Although we can always simulate a sampling distribution, the Central
Limit Theorem saves us the trouble for means and proportions.
Example: A tire manufacturer designed a new tread pattern for its all-weather tires. Repeated
tests were conducted on cars of approximately the same weight traveling at 60 miles per hour.
The tests showed that the new tread pattern enables the cars to stop completely in an average
distance of 125 feet with a standard deviation of 6.5 feet and that the stopping distances are
approximately normally distributed.
(a) What is the 70th percentile of the distribution of stopping distances?
(b) What is the probability that at least 2 cars out of 5 randomly selected cars in the study will
stop in a distance that is greater than the distance calculated in part (a)?
(c) What is the probability that a randomly selected sample of 5 cars in the study will have a
mean stopping distance of at least 130 feet?
Example: