(upbeat music)
- Hello, I am professor Rick Jerz,
and I welcome you to this session titled,
"Sampling and Sampling Methods."
In our previous session, we looked at
continuous population data and in particular,
population data that was normally distributed.
We saw that for this type of data,
when someone asks about the likelihood of some
population value being between two values,
we could calculate this likelihood, or probability,
by calculating the area under the curve of interest.
We learned how to use my Excel model for these calculations.
In this session, we switch our focus to situations where
we do not have access to the entire population data,
but we want to approximate the population mean.
We will learn that we can do this from a sample.
A sample, however, is not perfect.
So when we estimate the population mean,
we are going to hedge when we provide a value
for this population mean.
The interesting thing about samples
is that if we happen to take many samples
and calculate each sample's mean,
then create a frequency diagram for these sample means,
the distribution of sample means
is always normally distributed.
Hm, this is interesting.
Again, some form of data that is normally distributed.
This already sounds a little bit like our last chapter,
and we are going to learn that it is,
with just a few twists to it.
Okay, let's get started.
In this session, we have the following goals.
We want to explain why a sample is the only feasible way
to learn about a population.
We want to describe methods to select a sample.
We want to define and construct a sampling distribution
of the sample mean.
We want to explain the central limit theorem.
And we want to use the Central Limit Theorem
to find probabilities of selecting possible sample means
from a specified population.
Before we get into this topic,
let's review where we have been
relative to calculating probabilities.
Two sessions ago, we looked at the topic
of probability distributions and in particular,
discrete probability distributions.
In this unit, we explore
the binomial probability distribution,
which was a discrete probability distribution.
If we look at this graphic
describing the levels of measurement and types of variables,
discrete probability distribution can be applied
to either interval or ratio data and the type of variable
must be both quantitative and discrete.
In the previous unit on continuous probability distributions
everything remains the same,
except we have continuous quantitative data
instead of discrete data.
For both of these units, we also have population data.
This is especially important to realize
in our previous session on normal probability distributions.
We were dealing with population data.
Remember that in order to use
the normal probability distribution for population data,
when you collect this population data and plot it,
it has to appear normally distributed.
If it is not normally distributed,
for example, if it is uniformly distributed,
you can't apply the normal distribution.
In this session we switch our focus a little bit.
We are now going to explore the issue of
what do we do when we do not have access
to the entire population data?
Maybe you were thinking,
well that ends our statistics class.
But we are going to find that that is not so.
That even if we don't have access
to the entire population of data,
there are still ways to gather data
and make statistical statements about what we think
the population data looks like.
We are entering a topic that is often referred to as "inferential statistics."
How can we infer something about a population when we do not have population data?
Here are some reasons why we sample the population.
It would be great if for every business decision,
we had access to the entire population of data.
If we did, we could use the classical approach
to probability and know exactly what the probability is
of something happening.
But more typically, in a business environment,
it is physically impossible to check all items
in the population.
Sometimes, the cost of studying all the items
in a population is prohibitive,
or contacting the entire population might be time consuming.
Another problem is the destructive nature of certain tests.
If, for example, we had designed a new screwdriver
and we wanted to see at what pressure the tip would bend,
if we bent the tip of every screwdriver,
meaning the entire population,
we would have nothing left to sell.
We will learn that sample results, if done properly,
are usually adequate to describe the population.
The key to using a sample to make statistical statements
about the population is that you have to be careful
about selecting the sample.
A probability sample is a sample selected such that
each item or person in the population being studied
has a known likelihood of being included in the sample.
If, for example, we wanted to study the productivity
of people throughout a day,
we wouldn't want to select our sample from only,
oh let's say 8:00AM to 8:30AM.
This would be a "biased" sample
that wouldn't reflect what's going on throughout the day.
So if we were going to use a sample to say something
about the population, it becomes very important
to use the right method to select our sample.
Let's learn about some of these methods.
There are four methods of probability sampling.
The first is a simple random sample.
This is a sample formulated so that each item or person
in the population has the same chance of being included.
For this method, we use random numbers
to pick from the population.
In most statistics books, a table of random numbers
is provided at the back of the book.
We will use Microsoft Excel to generate random numbers.
The second method of probability sampling
is called systematic random sampling.
The items or individuals of the population
are arranged in some order.
Oh, maybe by age, for example.
A random starting point,
determined by using a random number, is selected
and then every k'th member of the population
is selected for the sample.
If, for example, we wanted to determine
the average height of a student in class,
we could start with a list of all students enrolled,
arrange this list alphabetically
from last name to first name, generate a random number
which might say to start with the,
oh, let's say sixth person,
and then we might pick every tenth person
from this point on the list.
The third method of probability sampling
is called stratified random sampling.
A population is first divided into subgroups, called strata,
and a sample is randomly selected from each stratum.
If, for example, we wanted to compare
average heights of students based upon sex,
we would first want to separate all students by sex,
and then pick samples from within these two groups.
The last method is called cluster sampling.
A population is first divided into primary,
most typically geographic units,
and then samples are selected from the primary units.
This technique is sometimes used for presidential elections.
Instead of going around to all the states
collecting our sample data,
we might instead pick individual states
randomly selected from all of our states.
In another session, I'll show you how to produce
random numbers from Microsoft Excel.
Remember that we want to use the sample
to estimate population characteristics.
For example, the population mean and standard deviation.
But since the sample is not the entire population,
it is very unlikely that the sample mean
would ever exactly equal the population mean.
Also, it is unlikely that the sample standard deviation
would exactly equal the population standard deviation.
This difference between what the sample statistic tells us
and the population parameters is called sampling error.
The sampling error is the difference between
a sample statistic
and it's corresponding population parameter.
Ideally, the sample mean should be exactly
what the population mean is.
Th sampling distribution of the sample mean
is the probability distribution consisting of
all possible sample means of a given size
selected from a population.
Let's look at an example.
Tartus Industries has seven production employees.
Let's consider these seven employees
to be their entire population.
The hourly earnings of each employee
are given in this table below.
What is the population mean?
What is the sampling distribution of the sample mean?
For samples of size, let's say two.
What is the mean of the sample distribution?
And what observations can be made about the population
and the sampling distribution?
Once again, considering these seven employees
to be the entire population,
we can find out what the population mean is
by just averaging their salaries.
This polulation mean calculates to $15.43 per hour.
If we were to pick a sample of two employees
from these seven and look at their hourly earnings.
Oh, let's just say we randomly picked two
that had $14 per hour earnings.
The average of course would be $14 dollars per hour,
which would not equal the
population mean of $15.43.
Our sample error for this simple sample would be $1.43.
But we only have one sample.
Maybe if we could pick more and more samples,
the mean of all of these samples might more accurately be
the population mean.
Let's conduct an experiment where we pick
every possible sample of two, from these seven employees.
Let's think about all the ways that we might select
two employees from these seven.
Hopefully you can remember back
to our session on probability and remember that
this is called a combination.
Our Excel model can be used to determine how many ways
we can select a sample of two from seven.
Turns out to be 21 ways.
This table tallies these 21 ways
of selecting two employees from seven.
It also calculates the mean of each sample.
Guess what?
If we average the mean of all of these samples,
it exactly matches the average for the population.
Normally, we don't have access to every possible sample
selected from a population.
But from this example, you might start appreciating that
sample means can reflect population means.
The more samples we take,
the more accurately we can predict population mean.
Let me take a moment and show you an Excel model
that illustrates this concept that
larger samples are better than smaller samples.
The data in this example represents
birth weights of children.
And I'm going to generate a population of 250 observations.
I label this as "sample number" in the first column.
In the second column labeled, "birth weights"
I generate a random birth weight
using the NORMINV() function within Microsoft Excel.
This function takes three arguments.
The first one is a random number,
the second one is the population mean,
and the third value is the population standard deviation.
I'm going to use a population mean of 7.04 pounds
with a standard deviation of 0.42 pounds.
In order to pick random values from this table,
I first generate a row number between four and 253.
I then pick my samples based upon these
referenced row values.
I don't expect you to understand how to do this
indirect referencing yourself, but if you want to learn
you can see how I do it.
In this table to the right,
I pick a sample of three and a sample of 30.
Down below, I calculate the mean
and standard deviation of these samples,
and also the mean and standard deviation
of the population data.
Finally, to the right I compare the sample means
and standard deviations with the population mean
and standard deviation.
You can regenerate this data by hitting the F9 key.
In this model, I couldn't figure out a good way
of avoiding picking duplicate rows.
So instead, what I do is I check for duplicates,
and if there are any, I show #N/A.
If this happens to you, just keep hitting the F9 key
until the #N/A's disappear.
In my summary table, shown in red,
what you should almost always observe is that
the error between the population mean and standard deviation
with the sample of 30 is smaller than the sample of three.
I show this graphically right above.
The blue bar should almost always be taller, or bigger,
than the orange bars.
Wow, this might be a complex model to illustrate
a very simple point.
Sampled data has been studied for oh, over 100 to 200 years.
A mathematical theorem was created relative to sample data.
It is called the "Central Limit Theorem."
It states of a population with a mean "u"
and a variance "sigma squared"
the sampling distribution of the means
of all possible samples of size "n"
generated from the population
will be approximately normally distributed.
Let me say this another way.
If we take a bunch of samples from the population
and plot these sample means,
the plot will look like a normal distribution.
The mean of the sampling distribution is equal to "u"
and the variance is equal to
sigma squared divided by sample size "n"
The standard error of the mean is
sigma over the square root of "n"
This "Standard Error" calculation is somewhat embedded in calculating
z-values and reading normal probability tables
in the back of the textbook.
We will use Excel, which doesn't require this "standard error" calculation.
"Standard Error" is a shorter name for
"Standard deviation of the sampling distribution of the sample mean,"
which gives us some information about the "dispersion" of the sample mean.
The standard error will play a little bigger role
with Confidence Intervals, a future topic.
So for right now, for us, it is just another statistic.
Most people just have to believe this concept,
but in my Excel model, I give you another worksheet
where I try to illustrate this concept.
I have 100 samples of data, and for each sample
I randomly generated 30 birth weight values.
Off to the right, I calculate the sample means
and I calculate the frequency for these means.
To the right of this, I plot a simple frequency diagram.
By hitting your F9 key, you can regenerate the data
and see how this frequency diagram changes.
There's not a lot of data in this diagram,
so the plot is kind of rough.
But as you hit the F9 key, you should observe
that the graph approximates a normal distribution.
So this model tries to prove that the means of samples
are normally distributed.
One of the important concepts of the Central Limit Theorem
is that it doesn't matter what the distribution
of the population data is.
When you pick samples from any of these distributions,
the sample means will be normally distributed.
In my last worksheet in this Excel model,
I try to prove this concept that it doesn't matter
what the population distribution is.
The distribution of the sample means
is always normally distributed.
What I do in this Excel model is
I generate 32 samples of data,
each containing 15 uniformly distributed data points.
Off to the right, I plot this data.
You should observe that this data
is approximately uniformly distributed.
Since we have randomly generated data,
the plot is not a perfect rectangle.
This is the nature of randomly generated data.
The chart below this illustrates the plot
of the sample means from this uniform distribution.
You should notice that the plot of this sample means
are approximately normally distributed.
Again, proving the Central Limit Theorem.
You can hit your F9 key to regenerate the data.
How closely these sample mean distributions
match a normal distribution will depend upon
sample size, as illustrated in this graphic.
Bigger sample sizes are better.
You might wonder why do we go through all of this trouble?
The reason is that the normal distribution
is something that we know a lot about,
so if we want to make some statements about the population,
and all we have access to are some samples,
we can use our knowledge that we learned
in a previous session about the normal distribution
to make statements about the population.
We saw the concept of a Z statistic in our previous session.
Although I prefer to use my Excel model
to calculate the probability of a range of values
given population data,
the concept is nearly identical for sample means.
The only adjustment we need to make
is to use sigma divided by the square root of sample size
instead of sigma by itself.
In my Excel model, for the normal distribution,
I provide a second table that can be used for sampled data.
Let's look at an example.
The Quality Assurance Department for Cola Incorporated
maintains records regarding the amount of cola
in it's Jumbo bottles.
The actual amount of cola in each bottle is critical,
but it does vary by a small amount
from one bottle to the next.
Cola Incorporated does not wish to under fill the bottles.
On the other hand, it cannot overfill each bottle.
It's records indicate that the amount of cola
follows the normal probability distribution.
The mean amount per bottle is 31.2 ounces
and the population standard deviation is 0.4 ounces.
At 8 A.M. today, the quality technician randomly selects
16 bottles from the filling line.
The mean amount of cola
contained in the bottles is 31.38 ounces.
Well certainly, 31.38 is not
the 31.2 that we were hoping for.
And you might immediately conclude without statistics
that there is a problem.
But the question is, statistically, is this unusual?
Is it likely the process is putting
too much soda into the bottles?
To put this another way,
is the sampling error of 0.18 ounces unusual?
The author, of course, will try to calculate probabilities
using "z" values, but more importantly
if you look at this chart depicting what has just happened
you will notice that the probability of getting 31.38 ounces
in a sample is .0359%.
Now, is this .0359% good or bad?
It really depends upon our own tolerance
of what is allowed in this process.
Here's what we can conclude.
It is unlikely, less than a 4% chance,
we could select a sample of 16 observations
from a normal population with a mean of 31.2 ounces
and a population standard deviation of 0.4 ounces
and find the sample mean equal or greater to 31.38 ounces.
We can conclude the process is putting
too much cola into the bottles.
This is a pretty strong statistical statement
about the process.
Using statistics to look at this situation,
instead of just, oh let's say guessing at
whether or not we're overfilling, or maybe worse yet,
getting a committee together to decide,
"Gee, does it look like we're overfilling?"
Using statistics is a much better approach.
In this example we had to use a sample.
We couldn't pick every bottle
and check how many ounces were in it,
because doing so would destroy all of our product.
Our only way of approaching this was to take a random sample
of the production process.
I hope you're getting a sense that this sampling method
and the statistics behind it
provides a powerful business tool.
In another session, I will show you how we could use
my Excel model to perform these calculations.
But for now, this is enough.
(instrumental music)
Thanks for taking time to spend some moments with me.
I hope that you have enjoyed what I have shared with you.
If you ever have any questions or need my help,
please let me know.
(upbeat music)