- Hello.
I am professor Rick Jerz.
And I welcome you back to my week six reinforcement session
of business analytics.
We have learned about calculating the probability
of the population mean under several data scenarios.
In this session, I want to reinforce our methods
by doing a few more interesting problems.
I also want to take a little time
to show you the central limit theorem concept
by using another Excel modeling technique.
Just to help you appreciate the central limit theorem.
Remember that there will be times where you encourage you
to pause my video, try to solve one of these
problems on your own, and then restart the video
and watch how I solve the problem.
Let's get started.
My objectives are to review a few administrative items,
to review sampling theory,
to do some problems about sampling,
this is chapter eight, to do some problems
about confidence levels, this is chapter nine,
and to say a few words about our next session.
Administratively, I just want to remind you
that exam one is scheduled for this week.
Let me know if you cannot take it
on the scheduled date, I'm willing to move
this exam for you plus or minus two days
from the schedule date.
Also, these two chapters, eight and nine,
will not be on this first exam.
Recall from my lecture video
on sampling theory that there's this central limit theorem
that says that the distribution of sample means
is normally distributed.
This theorem is what allows us to estimate
the population mean from sample data.
Recall that in my lecture video,
I gave you this little demonstration Excel file
that shows on this tab that if we take random samples
from a uniform distribution and plot their means,
that the means of these samples
will be normally distributed.
I decided I would try to get
a little bit creative with this demonstration model.
This version of the model needs to have macros enabled
when you open it.
On this law of large numbers tab,
I am picking samples from the population data
and calculating their mean and standard deviation.
One needs to hit the F9 key to regenerate each pick.
Recall that I am showing an NA for those situations
where I have duplicate picks.
I didn't want to have duplicates in my data.
The problem with this model is that
we only get one sample mean
from this population.
I got to wondering how I could use Excel
to do this 500 times and retain these sample means.
So I created a little simulator.
This simulator is shown off to the right.
First, you need to go to developer, macros,
and reset this simulation.
Return to your macros,
and then let the simulation run
by picking random numbers.
This can take a little time
depending upon the power of your computer.
Just sit back and let Excel do its thing.
When finished, Excel has completed running the simulation
500 times and has picked the sample means
for each time.
If one is curious as to whether or not
these sample means really are normally distributed,
off to the right I show an example of hoy you might use
pivot table to generate a chart of these sample means.
Here, let me do it for you.
I will copy the values for one sample.
Do not include the NAs.
And then paste them into this column,
making sure that I do not have any left over data
here from before.
Then I will use pivot table,
just as we have seen before,
to generate a count of all of these sample means.
I really don't want a count
of each individual value.
You will see most of these counts being one.
But I can use the grouping feature of pivot table
to put these values into a group.
And then finally create a chart
based upon these group values.
Finally, you will notice that the sample means
do look normally distributed.
Well, this is a lot of work to prove this theorem.
But the central limit theorem is really
not a simple theorem.
It's very significant in what we can do
with sample data.
So sometimes I like to go through a little bit more
effort to show you that this theorem really
does seem to work.
Another important concept is that
for this sampling theory to work,
the samples must be randomly picked.
I made this point in the lecture video.
I'd like to share with you a real life situation
of this sampling theory being put into practice.
My wife works for the bureau of labor statistics,
and her job is to collect data from companies
about what they pay employees.
This feeds BLS employer cost index data
that you can find on the BLS website.
She is not allowed to pick her own samples.
This is done by the statisticians in Washington, D.C.
She shared with me this report from the BLS
that describes their sampling process.
In the appendix of this report,
there is a technical note that describes precisely
how their sampling method works.
They use a combination of clustered sampling
along with stratified sampling to pick their samples.
In the data reliability of this report,
sampling error is described.
They do make note that samples
from different parts of the population
may vary from each other.
Of course, the BLS has to use a sample
to get to this data.
Sure, I would prefer that they did try
to get population data.
This would be job security for my wife.
But the key point here being made by the BLS
is that they have to use this sampling method
accurately in order to use sample data
to predict population means.
The challenge for my wife is to get the companies
to give her the data.
The BLS is not a regulatory agency like the IRS,
so she has to convince the sampled companies
to provide information.
My wife is pretty good at doing this.
She's able to convince most companies
to give her data.
Her capability goes into her performance review.
So I'm happy that she is pretty good at what she does
because this translates into a bonus at the end of the year.
I am going to show you the sampling theory
using our M&M data.
Recall that we do have the weights for 100 M&Ms.
And let's consider this the population.
Let's see how we would predict the population mean
by taking a sample from this data.
For this experiment, let's use a sample size of, oh, 10.
I can use the random numbers worksheet
in my Excel model to generate 10 random numbers.
I will use the rand between function and technique.
You have to make sure that these 10 numbers
do not have any repeats.
So you might have to do this a few times.
I copy the random numbers off to the right
so that they are no longer randomly changing on me.
Then I sort this list.
And then verify that I have 10 unique numbers.
These 10 numbers represent the row of the data.
So I have to still pick the values associated
with these rows.
For now, I will do this manually.
I will put a row number next to the M&M weight data.
And then individually pick each of the rows
that are from my randomly generated sample.
Copy these.
And then paste them next to the row number.
I now have my sample.
In order to be able to use my normal distribution model
for sample data, I have to calculate
the sample mean and standard deviation.
I am going to use my descriptive statistics model
to do this.
I copy this sample data.
And then paste it into my descriptives statistics model.
I make sure to delete any data left over
from a previous problem.
And as we have seen in the past,
my descriptive statistics model calculates
the sample mean and sample standard deviation.
Now I will switch to my normal distribution model.
We can use this model to answer a variety of questions.
Let's say that we want to know the probability
that the population mean is between 0.9 and one.
And point these value into column D.
I am going to reference the sample size, mean,
and standard deviation of the sample
from the descriptive statistics model.
By referencing these values,
I don't have to worry about writing them down
and then typing them back into Microsoft Excel.
The model calculates that there is a 97% probability
that the population mean will be between these two values.
If I go back to the actual M&M weight data,
swipe through it, I see that the population mean
is 0.9147.
And, as you can see, this value is between 0.9 and one.
So this sampling theory does seem to provide us a way
of saying something about the central tendency
of the population.
Let me ask a slightly different kind of question.
Can we use this sample data to predict
the two values that the population mean should be between,
oh, let's say 95% of the time?
I encourage you to pause this video
and try to solve this problem on your own
before seeing how I solve this problem.
To answer this question,
I would use column F.
I copy the sample size, mean, and standard deviation
of the sample over to column F.
I enter 95% as my probability of interest.
And I see that 95% of the time,
the population mean should be between 0.899 and 0.950.
This is pretty powerful stuff.
We are using the results of a sample
to make a statement of the entire population.
That's what this sampling theory is all about.
Now let's do a few problems from our textbook.
The rent for a one bedroom apartment
in southern California follows the normal distribution
with the mean of 2,200 per month
and a standard deviation of 250 per month.
The distribution of the monthly costs
does not follow a normal distribution.
In fact, it is positively skewed.
What is the probability of selecting a sample
of 50 one bedroom apartments and finding the mean
to be at least $1,950 per month?
I encourage you to pause this video
and try to solve this problem on your own
before seeing how I solve this problem.
Since this question is seeking a probability,
that puts me into either columns D or E of my model.
There is nothing about a plus or minus in the question.
The at least 1,950 then suggest to me
that I am in column D.
I enter 1,950 as the low value,
something large, like 10,000 as the high value.
I enter the sample size of 50, the mean of 2,200.
And the standard deviation of 250.
The model calculates that the probability
of the population mean being greater than 1,950 a month
is 100%, meaning this is pretty certain.
If you're interested in seeing how this looks graphically,
put condition D into plot which condition,
and then just scroll down and look at the chart.
I think that you can see that Excel and my model
is making these calculations pretty easy.
We didn't have to read any tables
in the back of the book.
Nor did we have to make any intermediate calculations.
That the population data is not normally distributed
as stated in the problem is not important to us.
The central limit theorem says that the population data
can be any distribution.
So I think the author was just trying to be
a little sneaky and threw us off.
Let's do another problem.
Nike's annual report says that the average American
buys 6.5 pairs of sport shoes per year.
Suppose the population standard deviation is 2.1
and the sample of 81 customers will be examined next year.
A, what is the standard error of the mean
in this experiment?
We're not going to do this, a,
because it is an intermediate calculation.
B, what is the probability that the sample mean
is between six and seven pairs of sport shoes?
C, what is the probability that the difference
between the sample mean and population mean
is less than 0.25 pairs?
And, d, what is the likelihood is that sample mean
is greater than seven pairs?
I encourage you to pause this video
and try to solve this problem on your own
before seeing how I solve this problem.
I am going to use my model
to answer all of these questions pretty quickly.
Question b is a between question.
So that puts me in column d.
I enter six as the low, seven as the high.
I enter 81 as the sample size.
6.5 as the mean,
and 2.1 as the standard deviation.
And the model calculates that there is a 96.8%
probability that the population mean
is between six and seven.
For the second question with n.25,
I am going to use column E.
I copy over the sample size, the mean,
and the standard deviation.
And I enter 0.25 as the plus or minus error.
The model calculates that there is a 71.6% probability
that the population mean will be within 0.25 pairs
of the sample mean.
For the last question, greater than seven,
I am back in column D.
I enter seven as the low.
Oh, something big as the high, like 50.
And the model calculates that there is a 1.61% probability
that the population mean will be greater than seven pairs.
If I want to see this graphically,
I make sure to enter D in the plot which condition,
and then I can scroll down and look at the chart.
The area in the red is this 1.6% probability.
There are many business scenarios where this kind of
analysis would be important to Nike.
For example, if you know that it is very unlikely
that the average customer is going to buy
more than seven pairs of shoes per year,
you can use this information to help you determine
production levels and inventory levels.
Let's look at another problem.
Refer to the real estate data
which report information on homes sold
in Goodyear, Arizona, area last year.
Use statistical software to compute the mean
and the standard deviation of the selling prices.
Assume this to be the population.
Select a sample of 10 homes.
Compute the mean and the standard deviation
of the sample, determine the likelihood of a sample mean
this large or larger from the population.
I'm not going to do this problem,
because this is very similar to what I just showed you
about our M&M weight data.
We have to randomly select a sample,
use that sample to calculate the sample mean
and standard deviation, and then finally use
that information to calculate probabilities.
The author suggests using statistical software.
But I think we're seeing that Excel
can do this just fine.
You can explore this problem on your own.
And let me know if you have any difficulties with it.
Let's now switch our focus to chapter nine,
confidence intervals.
This chapter comes to us like a whole new idea.
However, you might have noticed that my Excel model
already includes a column to calculate confidence intervals.
And this is the column that we can use in chapter nine.
The one new idea that it is introduced in chapter nine
is that when you do not know the population
standard deviation, you should instead of using the normal
distribution, you should use this new t-distribution.
I provide a logic flow chart
to help you understand when to use the normal distribution
and when to use the t-distribution.
Additionally, I provide you a new Excel model
for this t-distribution that looks very similar
to our Excel normal distribution model.
One concept that students struggle with
relative to confidence intervals is what happens
to your confidence interval
when you want to be more confident?
A lot of students will think that
when you say you want to be more confident
that your predicted interval gets smaller,
meaning more accurate.
But this is not the case.
For example, if I want to predict the height
of the next student coming into the classroom
and I want to be more confident
that I am going to win this bet,
I would not use a range like 5 feet 6 inches to 5 feet 8.
If I want to win this bet
let's say 99% of the time,
I would use a wider interval,
such as between two and 10 feet,
to assure that I am going to win this bet
99% of the time.
So when your confidence interval percent goes up,
your confidence interval gets wider.
Another concept is that the t-distribution
confidence interval will always be greater than
the normal distribution confidence interval.
We use the t-distribution model when we do not
know the variability of the population.
Meaning we do not know the population standard deviation.
When you have less information,
you will always hedge your statement a little bit wider.
However, as your sample size increases,
the t-distribution confidence interval
approaches the normal distribution confidence interval.
An interesting note is that Microsoft Excel's
data analysis tool pack when providing a confidence interval
will always use the t-distribution.
I think that this is because Microsoft wants
to hedge a little bit and be more conservative,
especially if you're going to use Excel
to make real business decisions.
We don't have to worry about this
because we are not using the data analysis tool pack.
Earlier, when we took a sample of our M&M weight data
and calculate the confidence interval.
What we really should have done
was to use the t-distribution instead
of the normal distribution, given that we do not know
the standard deviation of the population.
Had we done this, the t-distribution would have calculated
the confidence interval to be 0.895 to 0.953
instead of 0.899 to 0.950.
Remember the t-distribution confidence interval
is always wider than the normal distribution
confidence interval.
Sometimes this difference may not seem
all that significant.
But it could be if you were trying to make
a billion-dollar business decision.
Let's do a couple problems from the textbook
to see how this works.
Warren County Telephone Company claims in its annual report
that the typical customer spends $60 per month
on local and long distance service.
A sample of 12 subscribers reveal
the following amounts spent last month
and we're provided the data.
A, what is the point estimate of the mean?
B, develop a 90% confidence interval
for the population mean.
And, c, is the company's claim that the typical customer
spends $60 per month reasonable?
Justify your answer.
I encourage you to pause this video
and try to solve this problem on your own
before seeing how I solve this problem.
To do this problem,
we need to calculate the sample mean
and sample standard deviation.
I again will use my descriptive statistics model
to calculate the sample mean
and sample standard deviation.
I have entered this data from the problem
into my model.
You might notice that we are not being told anything
about the population standard deviation.
So to solve this problem,
we have to use the t-distribution model.
In my t-distribution model,
I will reference the sample size, the sample mean,
and sample standard deviation
from the descriptive statistics model.
I enter the 90% confidence interval.
And I see that a typical customer should be spending
between $60.54 to $64.63.
This is the enter to part b.
Oh, by the way, we have entered part a.
The sample mean of $62.58
is our best estimate of the population mean.
But remember, a sample mean can have variation in it
based upon different samples.
This is why it is better to use a confidence interval
to predict the population mean.
To answer part c, is a claim of $60 reasonable?
I will look at the graphic for the sample data.
The claim of $60 falls outside
our 90% confidence interval.
Our data seems to suggest that 90% of the time,
this kind of claim is not reasonable,
because it's outside the sample's predicted range
for this population mean.
So I would say to whoever's making this claim,
this is not likely.
Let's do another problem.
A sample of 352 subscribers to Wired magazine
shows the mean time spent using the Internet
is 13.4 hours per week
with a sample standard deviation of 6.8 hours.
Find the 95% confidence interval
for the mean time Wired subscribers
spend on the Internet.
I encourage you to pause this video
and try to solve this problem on your own
before seeing how I solve this problem.
One thing that you should notice
about this question is that we are not being provided
the population standard deviation.
And instead are being given the sample standard deviation.
This suggests that we should use
our t-distribution model.
I enter this data into my t-distribution model.
And I calculate that the population mean
should be between 12.69 and 14.11 hours.
This doesn't seem all that hard.
But why is this question problem number 67?
Remember that high number problems are problems
that the author thinks are more difficult.
Well, it turns out that if we were
using the t-distribution tables,
this sample size of 352 is not on it.
So we would be struggling.
What the author wants you to think about
is that the t-distribution approaches
the normal distribution as our sample size increases.
So, we could have put this data
into our normal distribution model
and see that the calculated confidence interval
is nearly identical to what the t-distribution provides,
12.69 to 14.11.
While using Microsoft Excel for statistics,
we do not need to worry about limitations of tables.
Let's look at one last problem.
Refer to the real estate data which reports information
on homes sold in Goodyear, Arizona, during the last year.
A, develop a 95% confidence interval
for the mean selling price of the homes.
I believe what what the author wants us to do
is to treat this data as sample data.
I encourage you to pause this video
and try to solve this problem on your own
before seeing how I solve this problem.
This is a high numbered problem from this chapter
and it's high numbered because the average student
would have to calculate the sample mean
and sample standard deviation before applying
the t-distribution model.
I can do this very quickly with my descriptive
statistics model by changing the reference
for my data to point to the Goodyear, Arizona, data set.
Then I will go to the t-distribution model.
Once again, just as before, I will point the required
input data to the descriptive statistics calculations.
We can state that the mean selling price of homes
in Goodyear, Arizona, should be between $212,000
and $230,000.
Well, once again Microsoft Excel and my statistics models
are making these problems easy to solve.
And remember, in this session,
I have been doing high numbered problems
from our textbook.
The author considers these difficult problems.
And we are doing them very quickly.
I encourage you to do as many problems as you can
so that you can get comfortable using the models
to solve these problems.
Next week, we continue working with sample data.
And we explore another way of looking
at this kind of data.
The topic is titled test of hypothesis.
And it does tend to be a little bit more confusing topic.
I have developed some features in my Excel models
to help us address this topic.
Thanks for spending some of you time with me.
I hope you have enjoyed what I have shared with you.
If you ever have any questions or need my help,
please let me know.