The Standard Deviants present
the high-stakes world of statistics,
history, Chaz Mastin, Alyssa Rosen, Kalila Asrads,
and Michael LaForte.
Whoa, oops.
Sorry.
Hi, I'm Chaz Mastin.
And welcome to the high-stakes world of basic statistics.
Over the next two hours, we'll do our best
to supplement and enhance your learning of the principles
and concepts of statistics, including statistical problems
in data sets, probability in distributions, sampling,
and much, much, much more.
Realize that we move very fast, so you'll
have to pay attention.
But keep in mind that we're on video,
so you can pause the tape, stop the tape, or rewind the tape
at any time you want.
If you miss something or need to go over something again,
don't freak out.
It's not like class.
You can just rewind and watch us again.
Also, because we'll be covering a lot of statistics,
we've included a video outline at every large break
in our three-part program, detailing exactly what's
in each of these sections.
Well, we're on our way.
But before we move on, here's a word
from a concerned member of our community.
Hey, although the deviants have done a wonderful job putting
this very thorough statistic supplement together,
you must take a few things into account.
To truly learn statistics, you must attend your classes,
read your books, and most importantly,
listen to your professors.
So all I'm saying is just go to class.
I get no kicks from statistics.
Those little wacky numeric values always get me down.
Down, down, down, down.
Yeah, I'm sorry.
It gets me down.
Yeah.
Part one, statistical problems and data fair.
Section A, what is statistics?
Statistics is a lot like cooking in a microwave.
And stat problems are a lot like a microwave pizza
or microwave popcorn, whatever you like.
Just like when you need to nuke your food in the microwave,
when you need to solve a stat problem,
you just follow the directions and plug in the numbers.
In order to do the right plugging,
you'll need logical thinking and a good understanding
of the basics of statistics.
Hopefully, we can all add, multiply, and divide, right?
Well, that's a great start.
You don't have to have stat phobia.
We'll supply you with the right directions and explanations,
and then you'll be on your way.
Yeah, by the end of this video, instead of hearing this
when we say statistics,
you'll hear this.
There's a couple ways for us to look at the word statistics.
First, let's look at statistics in the big picture.
In this way, statistics refers to the overall science
of extracting information from a group of numerical data
and using the information you found
to make inferences about that larger group of data, which
leads us to our second way of looking at statistics.
Now, this broad science of statistics
is comprised of individual numeric values
that you use in the first place to investigate the data.
Now, these values are statistics themselves.
It's a classic case of the part making up the whole.
So our angle is to make you comfortable
with the most basic individual statistics
so that you can put them all together
and build an understanding of the bigger picture.
Our video is a progression from the most basic statistic
itself to far more elaborate ways
of solving statistical equations and tests.
Everything we use in the beginning of the video
we use in the end, just a couple different directions.
But if you build with us as we move along,
you'll eat them alive.
Section B, statistical problems.
This science of statistics is ultimately
concerned with examining statistical problems.
And a statistical problem exists when
there's something unknown about your population.
A population is exactly that, a population.
But to be a little more formal, it's
a set of values representing all of the measurements
that you're interested in.
Meet five-card Charlie.
Now, Charlie is a notorious entrepreneur
who has a reputation for products that flop miserably.
But he presses on for the one cutting-edge product
that will allow him to open his own little boat casino.
Well, Charlie's latest product is a soft drink
called Fuzzy Dice Cola.
And he wants to know whether or not
it will meet with the same fate as many
of his previous products.
If Charlie wants to know how much of a success his cola is,
he's faced with a statistical problem.
You see, he wants to know the percent of the population that
enjoys the taste of his cola.
What he needs to do is to make inferences using
data he collects from those who have
tasted his new product, Fuzzy Dice Cola.
Oftentimes, the statistician, or Charlie in this case,
is faced with a population that's virtually immeasurable
due to extreme size.
The solution is often to take a sample
from the entire population and use this sample data
to make inferences about the population.
We'll go into sampling later on in the video.
But for now, let's deal with five-card Charlie.
To collect the data, Charlie has to perform experiments
on his population of Fuzzy Dice Cola drinkers.
Or he may use a sample.
A sample is a part or subset of the population.
In this case, a subset of the population of drinkers.
In order to solve statistical problems like Charlie's,
there are certain things that must be established.
These are known as the elements of a statistical problem.
Here's the first element.
You have to know exactly what you're trying to find out.
In other words, you have to know the question you want answered.
What we mean by this is you must identify the question itself
and the population for which it exists.
Secondly, you need to know the method you'll
use to get information from your sample.
This means you have to establish the design of your experiment
or how you'll choose the sample from the population.
Third, you must determine the way you'll
analyze the data you collect.
As we'll learn, there are a lot of ways to analyze your data.
Fourth, after you analyze your data,
you need to follow a procedure to make specific inferences,
predictions, and decisions concerning
your statistical problem.
There are often varied methods of interpreting data,
and we'll describe them as we move along.
The fifth and last step is obvious.
You want to make sure that the work you've performed
is correct or at least heading in the right direction.
So as you'll see later on in our presentation,
you'll need to set up certain measures of reliability
to assess the accuracy of the inferences and conclusions
you've made about the population you're working with.
These five elements are the basic steps
to solving a stat problem.
You need to be clear on these steps
so you're not totally lost as we work through them
during the video.
So let's go over them again quickly.
One, identify the question being asked
and identify the population.
Two, determine the design of the experiment
or the sampling procedure.
Three, establish the method you'll
use to collect and analyze your data.
Four, choose a procedure for making inferences
about the population based on the data.
And finally, five, find a measure of reliability
for whatever inferences you've made.
OK, great job.
We're on our way.
Bolo, cali, bolo.
Baby needs a Vienna sausage.
11 winner.
Woo!
Bring it on, baby.
Bring it down here.
All right.
You're working with that shit.
Section C, data sets.
Before going any further into the intricacies
of statistical problems themselves,
you need to know more about how to present the data collected
from the sample or population.
You should understand right now that over and over,
we'll be looking at presentations
of collected data or what we call distributions of data.
As we move through the material, our ability
to work with them and analyze them
will increase dramatically.
So let's keep on trucking.
After statisticians collect all their data,
they're left with a set of numbers
representing some sort of information
from the sample or population.
These are called our data sets.
Here's an example of a data set.
Suppose that these numbers represent the grade point
averages collected from 12 statistic students.
We'll use this data set as we continue
to explain different ways of presenting data.
The first way to present the data from your data set
is to do so graphically.
And the type of graphical method we'll use now
is called a relative frequency histogram.
These are really easy to construct,
so we'll do it quickly.
Due to the graphic nature of the relative frequency histogram,
this graphic has been rated E for explanation.
Parental guidance is suggested.
A relative frequency histogram divides the data
you've collected into subintervals or classes.
This method provides us with a way each measurement
or each individual value in the data set can be classified.
All it means for a measurement to be classified
is that it falls into one and only one class
based on its value.
This way, it's placed specifically where it belongs.
Normally, when statisticians use this type
of graphic representation of data,
they use between five and 20 classes,
depending on how many measurements the data set
contains.
Basically, a histogram sorts out the data
into orderly classifications, which oftentimes simply
make the best sense.
So now we know that after measurements of a data set
are collected, they're categorized according to class
and represented graphically.
It's important that you understand the histogram now
because of its relationship to the distribution curve.
The distribution curve is what statisticians use most often
when representing data.
There we go.
That's the strangest thing.
One day, I was just walking through the woods,
just like a little boy does.
And I was looking at fossils and little beaver dams and stuff
and sticks and jugs and floats through the rivers.
And all of a sudden, I got this very strange feeling
that came around me.
And I was, oh boy, here it goes again.
Da da da da da da da da da da da da da da da da da.
Step boy, here it goes, it's happening.
Da da da da da da da da da da da da da da.
Step boy.
Da da da da da da da da da da da da da da da da.
Step boy, step boy, step boy.
Let's show you how to build a histogram for yourself.
Of course, there are a few rules you'll
have to follow when constructing a relative frequency histogram.
And here's Alyssa to tell you what they are.
The number of classes will, again,
depend on the total size of your data set.
If there aren't enough classes, then important characteristics
of the data may not be accurately
represented by the histogram.
If there are too many classes, empty classes may result.
And the presentation of your data or your distribution of data
won't be as useful.
To be honest, the particular number of classes
is fairly subjective, so sometimes you'll
just have to wing it.
Second, you need to determine the width of your classes.
Now, when determining what the class width will be,
the difference between the largest and the smallest
measurement should normally be divided
by the number of classes you want.
After you divide these numbers, you
can round the quotient up to the most convenient figure
if it's helpful.
We'll do an example in just a second.
All classes are usually of equal width,
allowing for equal comparison of your measurements.
In some cases, though, classes of equal width
are not always suitable to accurately represent
the distribution of the data.
For example, when summarizing income data,
some income brackets may require a larger range than others.
You know, $25,000 to $50,000, $50,000 to $125,000,
these are obviously not equal.
They do this so that each bracket or class width
has a similar amount of measurements in each bracket,
say for taxing or other reasons.
Of course, this income data example
is a very practical application of this material
in the real world.
When we do our GPA example that we started,
we'll use a more textbook method of determining the class width.
Here's rule number three.
When locating the classes, the lowest class
must at least contain the smallest measurement.
The remaining classes are determined
by adding the class width we just discussed
to the upper boundary of the previous class.
The boundaries are often set so that no measurement can
fall on or be equal to any class boundary.
It's possible, just so you know, to classify your data
in a less strict textbook fashion
by simply establishing more broad and reasonable intervals.
In the case that a measurement falls on a boundary,
you can simply make the decision to include that measurement
in an upper class.
It's almost like rounding.
In keeping, though, with our more traditional procedure
to get the basics down pat, we'll
make sure that none of our measurements
fall on a class boundary.
In our grade point average example,
the GPAs range from 2.0 to 3.2.
This means that the total span is 1.2.
And according to our calculation,
the number of classes would be 6,
meaning that each class would span about 0.20.
However, as we said before, this choice
is a fairly subjective one with the standard usually
between 5 and 20 classes.
Because we don't want any empty classes or classes that
are too full or measurements that fall on the class
boundaries themselves, 7 classes is a more convenient choice
because it ensures us that all of these conditions
will be fulfilled.
In order to ensure a greater degree of accuracy,
we'll use an additional significant digit
to define the classes.
We'll begin the first class at 1.95.
Once again, it's important that the classes are divided
so that each measurement falls into only one of the classes.
Now, we can set up our intervals according to the rules
we've just established.
The first class contains GPAs between 1.95 and 2.15,
the second class between 2.15 and 2.35,
the third between 2.35 and 2.55, and so on.
These are the classes we'll use as intervals in order
to present our data.
Now that we have our classes and boundaries determined,
we can categorize the measurements in the data set.
The first GPA measurement of 2.0 falls into the first class,
the second measurement of 2.4 falls into the third class,
and so on and so on.
In this graphic, we see the tallies of the different GPAs
and the classes in which they fall.
We also show the class frequencies,
or how frequently the measurements fall
into each class, and the class relative frequencies,
or the proportion of the number of measurements falling
into each class.
This simple arrangement of data set information
is what we use to form our histogram.
To create this graphic representation,
rectangles are constructed over each interval or class.
Because the class widths in our example are equal,
the height of each rectangle in the histogram
is the frequency of that specific class.
Thus, the relative frequency histogram for our set of GPAs
looks like this.
Our graph also shows how the measurements in the data set
are distributed along the horizontal axis
of this typical statistical graph setup.
Don't forget again that if you need to go back and look
at any example, or if you're just a little bit confused,
you can very easily press Rewind on your VCR.
Also, realize that a lot of what we're covering right now
is laying the groundwork for the rest of our material.
So make sure you have these basics covered.
Let's go back to our original data set of the 12 GPAs
for a second and then move right along.
Suppose we wrote each GPA measurement on a slip of paper
and put it in a hat.
The chance that we would pull out a slip of paper
with a GPA in a specific class is
equal to the relative frequency of that class.
To put it another way, the relative frequency
of the third class, for example, is 2 divided by 12,
or about 17%.
Therefore, there's a 17% chance that,
of the 12 slips of paper in the hat,
we would choose one with a GPA between 2.35 and 2.55.
With this quick example, we started
to touch upon the ever so popular subject
of probability.
Now, we just wanted to wet your whistle a little bit
to show you how all this material is
so intricately intertwined.
But before we go full force into probability,
we need to go over some other material
to cover all of our bases.
Now, it's time for another graphical method
of presenting the data set of measurements.
This new method is something called a stem-and-leaf display.
OK, class, is everyone ready to show their stem-and-leaf
display?
Alyssa?
Wonderful.
Chucky?
That's nice.
Michael?
Well.
The stem-and-leaf display is somewhat similar
to the relative frequency histogram
in that it serves as a picture-like representation
of your data.
But unlike the relative frequency histogram,
the stem-and-leaf display allows you
to retain the actual observed values or measurements
of the data set.
You don't create any classes or change the numbers at all.
Let's use this data set in order to construct
a stem-and-leaf display.
This table of values shows the top 40 colleges and universities
by percentage of student population
expelled for various reasons over a four-year period.
OK, when you're creating a stem-and-leaf display,
each measurement of the data set is divided into two parts,
the stem and the leaf.
In our current example, we can separate each value
at the decimal point so that whatever
comes before the decimal point is the stem
and whatever comes after the decimal point is the leaf.
This is easy stuff, right?
Here's a graphic to help you visualize it.
For example, the first value of 7.98
is separated so that 7 is the stem and 98 is the leaf.
There may be several ways to break up the values,
but the easiest way to break up the values in our example
is to use the decimal point.
For other data sets, it's your choice
to break up the stems and leaves at any significant digit
in the measurement, which best displays
the distribution of the data set.
A stem-and-leaf display is constructed
by vertically listing the stems of the data set
in ascending numerical order from top to bottom,
then horizontally listing each corresponding leaf
in ascending numerical order from left to right,
placed in a line to the right of each corresponding stem.
The stems are like classes, and their leaves
show a picture-like representation
of the data measurements.
This picture is defined by the number of leaves per stem.
Also, it's practical to include a key which
shows how the measurements were broken up.
The result of our stem-and-leaf display of the top 40 colleges
to boot students looks like this.
The key at the bottom indicates that we've
separated the measurements at their decimal points
by providing an example.
The purpose of the key is to allow
you to reconstruct the data set by just looking
at the stem-and-leaf display.
With this graph, it's easy to see the representation
of the data set in this form.
Great job, you crazy statisticians.
Here's a quick recap on how to construct
a stem-and-leaf display.
First, decide how to break up the measurements
into stems and leaves.
Next, list the stems in a vertical column
in ascending numerical order.
Then, list the leaf for each measurement
in ascending order horizontally in a row
next to the corresponding stem.
Finally, provide a key in order to decode the stem-and-leaf
display if needed.
Remember, we're showing you the simple ways
we start our stat problems, so let's move on.
The different display methods for our data
we've just gone over obviously use graphics to do the job.
However, the methods we're going to explore right now
don't rely on any sort of visual representation of the data.
We're now going into numerical methods,
and they're used just as easily to describe sets of data
as they rely on numeric values to describe
samples and populations.
Oh, jeez, listen, it's obvious that I'm a statistic genius.
We got that established, but I'm also
got some more good state power.
Oh, boy.
These numerically descriptive measures
are given names that really make a lot of sense.
Now, throughout your investigation
into the broad body of data, if you
get a number that's calculated from a population,
it's called a parameter.
And if you get a number that's calculated from a sample,
it's called a statistic.
Everybody together, please.
Statistic.
Great job.
Ha ha ha.
OK, here's a joke for you, all right?
Statistic walks into a bar.
Bar tender says, hey, hey, sorry.
We don't serve statistics here.
Statistics says, hey, I was counting on that.
Thank you, thank you, my mother wrote that one.
In terms of statistics, the first type
of numeric representation is measures of central tendency.
These measures describe and locate
the center of the distribution of data
that you're working with.
A common measure of central tendency
is called the arithmetic mean, or just plain and simple,
the mean.
Now, the mean is nothing more than the arithmetic
average of the set.
Let's keep it simple.
Here's the definition.
The mean of a set of measurements
is equal to the sum of all the values in the set
divided by the number of values in the set.
It's really easy, so don't make anything more out of it
than it is.
It's an average.
There's a distinction, however, in the symbols
we use to identify the mean for a sample
and the mean for an entire population.
The mean of a sample is identified as x bar,
and it looks just like this.
On the other hand, the mean of an entire population
is identified as the Greek lowercase letter mu.
Mu.
So how does a statistician display
the method for calculating the mean?
Take a look.
The procedure for calculating the mean
is represented by a formula, as are most statistics.
Here is the statistical formula for the arithmetic mean
of a data distribution.
The large E-shaped character is the Greek uppercase letter
sigma, which indicates that you should take
the sum of a set of values.
We'll be using this throughout the video, so learn it now.
Sigma equals summation.
The lowercase i equals 1 indicates
that you begin the sigma summation with the first value
of the set.
If the formula were to say i equals 2,
then we would begin the summation
with the second value of the set.
The lowercase n indicates the number of values in the set.
X sub i represents the values to add up.
Then, to find the average of the data set or the mean,
divide the summation by the number of values in the set.
Remember, the summation begins with the first value
of the set and ends with the nth value.
Let's go back to the example of the 12 STAT students
and their GPAs.
We'll calculate the mean of that set of data.
If we add up the 12 GPAs and divide that total by 12,
we get a mean of 2.66.
In this case, n is equal to 12 because there
are 12 GPAs in the data set, and the average of those 12
is about 2.66.
Talk to me.
Oh, hi, mom.
Yeah, mom, how you doing?
Good.
Well, you know, did I, I don't know, a little below average.
All right.
After the mean, the next measure of central tendency
is known as the median.
The median is identified as the lowercase m
and is the measurement that falls in the middle position
when the data is ranked in order from smallest to largest.
Remember, median equals middle.
Take a look here.
Here are five salami sandwiches in a row,
from the largest sandwich, a big sandwich,
to the smallest sandwich, a very smallest sandwich.
Now, I know these aren't numbers, but this is just to illustrate a point.
Now, the median of these salami sandwiches is this sandwich
because it's the middle sandwich in the third ranking.
Here are a couple of good rules to follow.
If a total number of measurements in a set is odd,
then the median is the rank position of n plus 1 over 2.
If the total number of measurements in a set is even,
then the median is halfway between the two middle measurements
or between the measurements ranked n over 2 and n over 2 plus 1.
For our GPA example, the median would be
between the sixth and seventh values of the set,
both of which happen to be 2.7.
Therefore, the median of this data set is 2.7.
In addition to these two measures of central tendency,
the mean and the median,
there's another very common statistic called the mode.
The mode is a really simple statistic which determines the measurement
that occurs most often in a set of data.
As you can see here from our original data set,
there are two measurements which occur most often.
They're the measurements 2.7 and 2.8.
So our data set is a bit different in that it has two modes
or, again, two specific measurements
that occur more often than the rest.
Okay. If you're given any data set,
you should be able to figure out the mean, median,
and mode right away.
Now on to other types of numerical representations.
Another type of numerical representation of your measurements
is a measure of variability or dispersion of the data set.
The most simple measure of variability is the range of the data set.
To find the range is really easy.
All you need to do is find the difference between the largest
and smallest measurements in the set of data.
That's it.
Remember earlier on in the GPA example, we used the term total span
to describe the difference between the highest and lowest measurements?
From now on, this will officially be known as the range.
Our next numeric representation is the deviation.
Like the range, the deviation measures variability.
So what's the deviation?
Deviation is the numeric distance a measurement is from the mean
or, again, the average of the set.
The formula for calculating the deviation of measurement x
in the following example is x minus x bar.
Don't forget that x bar represents the mean of a sample data set.
So it's our x measurement minus the mean.
Take a look at this graphic to help you calculate deviation.
For our example, the deviations of each of these randomly selected
measurements are as follows.
Remember that the mean, or x bar, is equal to 2.66.
Also, make sure you note that the sum of all the deviations
from the mean is 0.
Why is this?
Because measurements that are less than the mean
have a negative deviation, and those that are greater than the mean
have a positive deviation, thereby canceling each other out to a total sum of 0.
In order to avoid the difficulty of working with negative deviations,
statisticians developed another statistic called the variance.
We have to be honest.
The variance is a little bit strange, but it serves a very specific purpose
and will be extremely helpful later on when we'll need to compute other
statistics, such as standard deviation.
To continue, the variance of a data set is equal to the sum of the squares
of the deviations.
Dealing with the squares of the numbers ensures us that they'll all be positive.
However, just as the mean had different symbols for both populations and samples,
the variance of a population and of a sample are represented
in slightly different ways.
For a population, the variance is designated by the Greek lowercase
letter sigma squared and is equal to the sum of the squares of the deviations
divided by the number of measurements in the set.
So it's the average squared deviation.
Remember that the Greek lowercase letter mu is the mean of a population.
Most of the time, measurements of a population are not able to be obtained,
so you'll have to use the measurements of a sample.
In this case, we need to use a slightly different formula.
Here's the formula for the variance of a sample.
Lowercase s squared is equal to the sum of the squares of the deviations
divided by 1 less than the number of values in the sample.
To help grasp the idea of variance a little bit better,
let's assume that our set of GPAs is a sample from a larger population.
In this case, we'll use the sample variance formula
to find the statistic we're looking for.
As you know, to find the variance, we need to find the deviations.
Then we need to find the sum of the squares of the deviations.
You'll notice that this sum is positive because we've
eliminated the negative signs by squaring them.
We then divide by 1 less than the number of measurements in the set, or 11.
This table shows the squares of the x minus x bar values
and their sum of 1.3188.
The variance of the sample is calculated by dividing 1.3188 by 11.
After the calculation, we find that the variance is 0.1199.
One more time.
The variance of a sample is equal to the sum
of the squares of the deviations divided by 1
less than the number of sample values.
That's all for calculating variance, but it's
pivotal in determining our next statistic, the standard deviation.
Here to describe the last measure of variability
will be, oddly enough, the standard deviance.
Good evening, and thank you.
I present to you now the standard deviation.
The positive, the square root.
Of the variance.
Thank you.
OK, anyway, the standard deviation is equal to a positive square root
of the variance.
So for our GPA example, it's equal to the positive square root
of 0.1199, which was our variance, or about 0.3463.
Listen carefully.
When you're finding the standard deviation of a distribution,
remember that the number is always positive.
But what is the standard deviation, aside from a statistic that
uses a lot of other statistics to find it?
Well, basically, the standard deviation
represents where measurements lie in the data set in relation
to the mean of the set.
Those measurements that are very large or very small,
with respect to the mean, may be several standard deviations
from the mean, while those measurements that
are close to the mean on either side are within one or two
standard deviations.
The standard deviation is a good measure of variability,
because it indicates where measurements
lie on the x-axis.
Due to its nature, the standard deviation
does determine the flatness or concentration
of the distribution curve once it's eventually drawn,
meaning that the smaller the standard deviation,
the more narrow the curve.
And the larger the standard deviation,
the wider the curve.
You should realize what an important statistic
the standard deviation is, and they
will be using it in our more detailed work
later on in the video.
Time for a deep breath.
Now that you know how to calculate all these statistics,
you know, mean, range, deviation,
we can start showing you how they're
used in more practical ways.
All right, box cards, baby.
Let's go.
Come on.
Come on.
Get it.
Set.
Hey, you're five-card Charlie, aren't you?
Yeah, what's it to you?
My name's Caesar, you know, like the salad.
I couldn't help but noticing you are calculating ferociously.
Yeah, I'm working out a relative frequency histogram
for this game.
I think I have something that can help.
I use it whenever I have a problem.
I think we should bet it all, baby.
Charlie, I think our luck's beginning to change.
Let's think way back to the relative frequency histogram
and how it graphically shows the distribution
of the measurements of your set of data.
Well, we've got some news for you.
Graphical representations of most population distributions
end up as curved lines.
These distributions often represent
extremely large populations or approximations of populations.
Distribution graphs for these larger data sets
take on the shape of a curve because the frequency
of each measurement is so large that each single measurement
could conceivably be a single class.
Because there are so many individual measurements
in classes, the distribution must
be represented by smoothing out the data into a curve.
The frequency on a distribution curve
is shown along the vertical axis,
while the possible values for x are
shown along the horizontal axis.
From now on, when we discuss and represent our data
and subsequently do our analysis,
it will be in the form of a curved line.
Typically, these graphs look like this.
This graph in particular illustrates
what's referred to as a normal distribution or one
that's shaped like a bell.
We'll discuss this specific type when we get into our distribution
section.
But for now, we'll show you how to use the mean
and the standard deviation with respect
to this example of a bell curve.
In a normal distribution like the one we just saw,
68% of the measurements are within plus or minus 1
standard deviation from the mean of the distribution.
Another way of stating this is that 34% of the measurements
are within 1 standard deviation to the left and right
of the distribution's mean on the graph.
But realize that this does not indicate
a negative standard deviation.
To continue, 95% of the measurements
are within plus or minus 2 standard deviations
from the mean.
And almost all of the measurements
in this type of distribution are within plus or minus 3
standard deviations of the mean.
This is a very important result in statistics,
and it's known as the empirical rule.
Remember, the empirical rule states
that nearly every measurement is within 3 standard deviations
from the mean, unless it is rebel scum known as an outlier.
Don't underestimate the dark side of the empirical rule.
He, he, he, he, he, he, he.
Again, the empirical rule states that nearly every measurement
in your distribution is within plus or minus 3 standard
deviations from the mean, unless, of course,
it lies dramatically outside of this area
and is given a special determination as an outlier.
We'll use this quite often throughout the video,
so keep it in mind as we continue.
There are a few statistics that describe where measurement
lies with respect to other measurements
in the distribution.
This is known as relative standing.
The first measure of relative standing
is called the z-score.
I am z-score, z-a-b-a, a measure of relative standing.
I am not your relative, and I am no longer standing.
The z-score measures the distance a specific measurement
is from the mean in terms of the standard deviation.
In other words, the z-score tells us
how many standard deviations a measurement is away
from the mean of the distribution.
Here's the z-score formula.
The z-score equals x minus x bar divided
by the standard deviation.
Once again, the z-score identifies a measurement
along the x-axis in terms of the variance
and standard deviation of the distribution of data.
For example, suppose that for a certain stat course,
there are two different sections of the class, section 1
and section 2.
On the midterm exam, the distribution of the exam scores
for section 1 has a mean of 75 and a standard deviation of 5.
The scores of section 2 also have a mean of 75,
but the standard deviation is 7.
Now let's consider a score of 85 on the exam
as a measurement on the distribution curve.
In section 1, the z-score for an 85
is 2.0 using our z-score formula.
And in section 2, the z-score for an 85 is 1.43.
As you can see, an exam score of 85 in section 1
is relatively higher than the exact same score in section 2.
This is shown by the z-scores of the same measurement
in the two different data distributions.
The higher z-score of 2.0 in section 1
indicates a better exam score relative to the rest
of the scores in that section.
According to the empirical rule, most or all
of the measurements of any distribution curve
should be within three standard deviations of the mean
on either side, or in other words,
have a z-score of 3 or less.
A measurement with a z-score greater than 3
is called an outlier.
Obviously, we call it an outlier because it
lies outside of the normally expected area.
It may help you to understand that an outlier is either
very large or very small in relation
to the other measurements.
This accounts for its far out positioning.
Another measure of relative standing
that you should be aware of is called the percentile.
This is an easy concept because everyone
knows what percentiles are.
And all a percentile does in statistics
is place a measurement in the distribution
as a percentage ranking in relation
to all other measurements.
For example, if Alyssa here gets a score on her stat exam
that's higher than 70% of all the other scores but lower
than 30% of all the other scores, then that exam
is said to be the 70th percentile in the distribution
of scores.
Statisticians commonly use the 25th, 50th, and 75th
percentiles, which are usually referred to as quartiles,
of which there are three.
These three quartiles separate the data into four quarters.
Let's say that under this curve, areas A, B, C, and D
are all equal.
By this, we mean that they contain the same number
of measurements.
The dotted line on the left represents
the boundary of the first quartile
because it's greater than 25% of the other measurements.
The median represents the second quartile
because it's greater than 50% of the other measurements.
The dotted line on the right represents the upper quartile
because it's greater than 75% of the other measurements.
And the measurements line in the portion marked D
are percentiles greater than 75.
you should know that in terms of the distribution curve,
the z-score is very important because it's
used to determine the area under the curve between two points.
The determined area shows the percentage
of measurements in that area.
An example of this is in one of the quartiles
in the previous graphic.
Each quarter, A, B, C, and D, has
an area of 0.25 or 25% of the total amount of measurements
in the set of data you're working with.
Good morning, my little bearings.
You are so helpful in calculating standard deviation.
Time for morning exercises.
Ow, my eyes.
Let's go, my little bearings.
Come on.
Before we move into a more detailed discussion
of distributions and probability,
make sure you understand everything.
As we said in the beginning of the video,
this science of statistics is cumulative,
and a solid understanding of the most basic concepts
is vital to much more challenging work
in statistics.
So again, rewind if you need to.
Thank you.
Hey, what's the matter, buddy?
You in debt?
Ha ha ha.
Woo hoo!
Ah!
Part two, probability and distributions.
Section A, probability.
When we look at probability, let's keep in mind
that we're dealing with the base root of the word,
which is probable, referring to the chance
that something may happen.
Now, there's a really easy way to explain probability,
and practically everybody uses it.
All you have to do is flip a coin.
This side's heads, this side's tails.
Blue Captain, call in the air.
Heads!
If you flip a regular balanced two-sided coin,
there's a one-in-two chance that the coin will land heads up.
Therefore, the probability that heads will appear face up
is one in two, or as we just learned with percentiles,
a 50% chance.
Probability is extremely useful in statistics
because it helps us identify characteristics of a sample
from its greater population in repeated sampling situations.
In order to obtain measurements for a sample,
statisticians usually perform controlled experiments.
Each time an experiment like this is performed,
it basically draws a measurement or a probable result
from the population for the sample.
So we have to use the sample to determine the probability
that the sample is an event.
So we use the sample to determine the probability
that the sample is an event.
And when we're dealing with probability in statistics,
we use the sample to describe the population.
However, with the use of probability,
we instead move from the entire population
to the sample itself.
Also, keep in mind that when dealing with probability,
we're working with populations, so we
need to use the correct population formulas.
Good morning and welcome to Cathy in Kitchen where this morning I'm preparing for a wonderful
event. I'm preparing eggs penado with hollanda swiss and little Englishman fans. I'm preparing
a wonderful event. A possible event of rolling a six sided die would be to roll a four for
example. Another possible event could be to roll an odd number. Because these are possibilities
of the experiment of rolling a six sided die they're called events. Events aren't always
measured in numbers. Let's say for example that the experiment takes the form of asking
a voter what party they favor in an election poll. Each party type is a different possible
event within the sample space of the population of voters. So in this case a possible event
could be that the voter answer is Republican or Democrat or even in my case Communist.
Again probability is extremely helpful because it allows the statistician to identify the
nature of a sample without having to actually perform the experiment over and over by choosing
measurements from the population. Okay now Galila will begin to discuss how we classify
the different types of events that occur in statistical experiments. There are two ways
to classify events and it's really pretty easy. Events are said to be either simple
or compound. Let's take the first one. A simple event is one that cannot be broken down any
further. In other words there's only one possible outcome for that event and it cannot be identified
with any other simple event. A compound event on the other hand is a combination of two
or more of these simple events we just described. This is caffeine kitchen still and I'm preparing
for another wonderful event. But this time I'm making poached eggs, poached eggs and
poached eggs. How about if we go back and roll some dice okay? A simple event would
be to roll a five. This is a simple event because it defines only one possible result
of the experiment. A compound event would be to roll a number less than three. This
is considered a compound event because it contains several single possible outcomes
of the experiment all under the heading of rolling a number less than three. Each one
of these single possibilities which make up the compound event is itself a simple event.
Therefore the compound event of rolling a number less than three is comprised of two
simple events. The simple events in this case are rolling a one and rolling a two. Look
at me I'm making sandwiches here. I'm a sandwich maker. Rolling on baby. Three craps. Section
B probability of an event. We already have a good idea of what probability is right?
It's simply the chance that something may happen. Now in statistics we talk about the
probability of an event. Now this is no more than the chance of an event occurring when
we perform an experiment like rolling a certain number with a die. A way of determining the
probability of an event is to perform the experiment a large number of times and then
record how many times the event occurs. The statistical notation looks just like this
and it is read the probability of event A is equal to the number of times event A occurred
represented by a lowercase n divided by the number of times the entire experiment was
performed represented by an uppercase n when n is very large. As uppercase n or the number
of times the experiment is performed becomes much much larger the calculated probability
of event A becomes much more of a specific and precise value. This formula as easy as
it is describes the relative frequency concept of probability. To put it even more simply
how many times and how probable it is that an event will occur is in relation to how
many times it is tested. For example a selection of one card from a deck of cards is itself
an experiment with 52 simple events with predetermined outcomes. Each denomination is represented
four times one in each suit. Therefore the probability of drawing an ace is four divided
by 52 or.077. The probability of drawing a face card is 12 divided by 52 or.23. This
is easy probability. When calculating probability you'll realize that the probability of an
event will always be a value between one and zero. The closer it is to one the more likely
the event is to occur and the closer it is to zero the less likely it is to occur. Remember
earlier when we told you the difference between simple and compound events? Well whether an
event is simple or compound is determined by the events composition. You know what it's
made of and when we determine the probability of an event it also depends on you guessed
it the events composition. The composition of simple events is the one possible outcome
that the simple event describes. This means that it's composed of only one thing or one
possibility. Therefore it's not possible for two simple events to occur at the same
time meaning on the same performance of the experiment. In this kind of relationship simple
events of the same experiment are mutually exclusive because when one simple event occurs
no others can. However simple events are not the only events that are mutually exclusive.
Regardless of whether or not they're simple or compound if two events of the same experiment
have absolutely no results in common then they're also mutually exclusive. For example
the event of rolling an odd number and the event of rolling an even number are not simple
events. However they're mutually exclusive because none of their results are the same.
If we look at all the possible events as being contained in a two dimensional shape say a
circle then this circle encompasses every result the experiment may have. Inside the
circle are all of the events. This circle is called a sample space because it contains
all the possibilities for our sample. Remember the goal of probability is to determine the
likelihood of choosing a certain measurement for a sample from the population. Looking
at our sample space we see how events can be mutually exclusive. Each one of the smaller
circles represents one of the simple events of rolling a six sided die. In this diagram
none of these circles cross each other because the results of their events have nothing in
common and are therefore considered mutually exclusive. If you toss a coin it will come
up either heads or tails. Both of these results are simple events and are mutually exclusive.
Therefore if on one toss of the coin heads comes up then it's impossible that tails
can come up on the same toss. This can obviously go for more than just coins. Think about this.
An experiment can have many simple events as possible outcomes and remember simple events
can't be broken down any further. Now if you take the sum of the probabilities of the simple
events of the experiment they must equal one. Once again if you toss a coin the probability
of the simple event of heads coming up is one in two or one half. The probability of
a simple event of tails coming up is also one in two or one half. The only two events
that can possibly occur in a coin toss experiment are heads or tails. Therefore the sum of the
probabilities of these two simple events equals one. Alright good work. Let's roll into the
event composition of compound events. The composition of a compound event depends again
on the simple events that make it up. If we want to find the probability of a compound
event occurring we have to add the probabilities of the simple events that make it up. This
should be fresh in your mind. The probability of a compound event is equal to the sum of
the probabilities of the simple events. Keeping things simple let's go back and roll some
dice. The probability of rolling a number less than four which is a compound event is
equal to the sum of the probabilities of the simple events of rolling a one two or three.
The probability of rolling a one is one in six because there are six possible simple
outcomes to the experiment and rolling a one is one of them. Therefore the probability
of rolling a one is one sixth. Now this is the same for the probabilities of rolling
a simple event of a two or a three. Okay the probability then of rolling a number less
than four is equal to one sixth plus one sixth plus one sixth which is three over
six or one half. If you want to deal in percentages we'll find there's a fifty percent probability
of the compound event of rolling under a four occurring. There are two new events to learn
which can be created from our simple and compound events. They are the intersection and the
union. Let's begin with the intersection. If we have two events let's call them A and
B then the intersection of events A and B is whatever simple events A and B have in
common. The intersection of events A and B is shown as either A intersection B or more
often shown as simply AB. For intersections of any two events say events A and B again
the intersection occurs where both compound events find a simple event that they have
in common. So the key word for an intersection is and. For event A we'll use rolling a number
greater than three as our event and for event B we'll use rolling an odd number. The intersection
of these two events is the simple events they have in common. When we break down both compound
events into their simple events we see this more clearly. Event A consists of rolling
a number greater than three or in other words rolling a four five or six. Event B consists
of rolling an odd number or in other words rolling a one three or five. Again event A
is a four five or six and event B is a one three or five. The simple event that both
A and B have in common is the event of rolling a five. Therefore the intersection of rolling
a number greater than three and rolling an odd number is rolling a five. This intersection
is represented by the shaded circle in this diagram which is the simple event of rolling
a five. Getting back to the key word and the intersection occurs where it's both greater
than three and an odd number. Again as you see here it occurs with rolling a five. Let's
take the same two events A and B and find the union of the two sets. A union consists
of any simple events in A, B or both. Because of this stipulation the key word for a union
is or. This is simply due to the fact that a union exists where any of the simple events
of one compound event or any simple event of another compound event occur. In this graphic
we see that the union of rolling a number greater than three and rolling an odd number
is the combination of all the simple events in both event A and event B. The union is
equal to rolling a one three four five or six. Rolling a four five and six are all greater
than three while one three and five are all odd numbers. This union is represented by
the shaded areas in the diagram. Though it's possible to roll a two with the die it's not
part of either compound event and therefore it is not a part of the union of the two.
Try to remember when events are mutually exclusive they don't intersect at all because it's
impossible for one result to occur at the same time as the other and their union is
simply the sum of the two events occurring. Before we move on to our next topic here's
a quick FYI on something called a complement in case you face it in your next class. The
complement of an event is equal to all the simple events of the experiment that are not
contained in that event and because the sum of all the probabilities of a simple event
in an experiment equals one the sum of the probabilities of an event and its complement
is always one. Alright let's wrap this up. An event and its complement are mutually exclusive
because it's impossible for an event and its complement to occur at the same time. In terms
of our two types of compound events the intersection of an event and its complement has zero probability
and its union has a probability of one because an event and its complement encompass all
of the simple events of an experiment. Sometimes the probability of an occurrence of an event
changes, often dependent upon whether or not another event has occurred. This leads us
to finding the probability of an event when conditions are applied to it and guess what
it's called? Conditional probability. Conditional probability is kind of like revised probability
in the sense that it's probability under the advisement of a little extra information
or stipulations attached. For instance, let's take for example a person playing blackjack.
Suppose the hand dealt to her is comprised of a nine and an eight for a total of seventeen.
She's faced with the question of whether or not to draw another card to either stay under
twenty one or hit twenty one right on the money. So her dilemma is written as the probability
of selecting a card less than or equal to four or the probability of selecting an ace,
two, three, or four. Because there are four cards and four in each suit the probability
of selecting one of these cards is sixteen over fifty two. After the calculation the
probability is point three zero seven or about a thirty one percent chance of selecting one
of these cards if nothing else is known. The point here is that we assume our player knows
nothing else and therefore this probability is unconditional. But now suppose our same
player is counting cards as she moves along and has noticed that twenty cards have already
been played. She noticed fifteen of the cards were greater than four and five were less
than or equal to four. Because of her additional knowledge of the situation her probability
has changed. Now the probability of selecting a card less than or equal to four is eleven
over thirty two or point three four three about thirty four percent. So her probability
has increased a bit. This is an example of a conditional probability problem. Conditional
probability can be calculated using a formula. Let's work through a conditional probability
problem. The conditional probability of event A given that event B has already occurred
is shown like this and is read just like a regular probability formula but you add the
condition which here is event B. So with conditional probability problems you say the probability
of A given B. Let's go on with the rest of the equation. Continuing on with our established
events A and B the equation for a conditional probability problem is as follows. The probability
of A given B is equal to the probability of the intersection of A and B or A B divided
by the probability of B. In this equation the probability of B is given the qualification
that it can't equal zero because well you just can't divide anything by zero. Okay let's
work one out. For our example of events A and B the probability of A given B is the
probability of rolling a number greater than three given that the number is odd. Using
the formula for conditional probability the probability of intersection A B divided by
the probability of B is equal to the probability of rolling a five divided by the probability
of rolling a one three or five. Numerically this is one over six divided by three over
six or one half which calculates to one third. Therefore the probability of rolling a number
greater than three given that it's an odd number is about one third or about thirty
three percent. Often times conditions are applied to your stat problems and these conditions
are figured into the probability of events occurring. Let's roll on. What are the chances
of rolling a seven shall we? I don't know. All right. Baby needs a new pair of retreads.
Seven winners. It's really helpful in conditional probability problems to think of events as
either dependent or independent. We've given you an example of how the probability of an
event is dependent on another condition. However two events can be independent of each other
as we saw with our blackjack player. If we have two events the probability of event one
given event two is simply equal to the probability of event one. In this case the independent
occurrence of event two does not change the chance of event one occurring. Now we're
going to test you in calculating probability using everything we just learned. Let's say
that the experts at the casino are analyzing their clientele in terms of the games customers
play and the customer's composition. So the experiment conducted by the experts is randomly
selecting customers from their casino. The three games are blackjack, craps and roulette.
Each customer plays only one game and there is no crossing over. And playing the games
are men, women and horses. That's right horses. The casino experts want to determine the probability
of what a single customer is playing in terms of whether that customer is a man, a woman
or a horse. Here's the table which provides us the information. This type of table is
known as a two-way classification because it classifies the customer game and biological
makeup. The games are listed along the top and the player description is listed along
the left side. Let's now take two possible events from our table of information and calculate
their respective probabilities and the probabilities of their intersection, union and also a conditional
example. Event A is that the customer is a horse and Event B is that the customer is
playing blackjack. Both of these are compound events. The probability of Event A, a horse
winning, is the sum of the probabilities of its three simple events. A horse playing blackjack
is 0.13, a horse playing craps is 0.03, and a horse playing roulette is 0.04. Thus the
probability of any randomly selected customer being a horse is 0.20 or 20 percent. The probability
of Event B selecting a customer playing blackjack is the sum of the probabilities of its three
simple events. A man playing blackjack is 0.17, a woman playing blackjack is 0.10, and
a horse playing blackjack is 0.13. Thus the probability of a randomly selected customer
playing blackjack is 0.40 or 40 percent. Now we'll need to find the probability of the
intersection of A and B, the probability of the union of A and B, and the probability
of B given A. The intersection of A and B is that a horse playing blackjack is selected.
This is only one simple event, and its probability is 0.13, as you can see from the graph. The
union of A and B is that our randomly selected customer is a horse, is playing blackjack,
or both. Looking at the table, five simple events make up this union. The total of their
probabilities is 0.47. Lastly, we can calculate the probability the customer is playing blackjack
given that the customer is a horse. Recalling our formula for conditional probability, we
see that there's about a 65 percent chance that the customer is playing blackjack given
also that the customer's a horse. Well, how'd you do? I'm sure you did great. You can go
over that section again if you need to. Now, let's move on to distributions. Section
C, distributions. Now that we can consider ourselves statisticians, when we perform experiments,
we use some sort of measurement. This measurement is called a random variable or our X variable
because its outcome is unknown before the experiment occurs, and the outcome will differ
with each performance of the experiment. So a random variable is a quantity whose value
depends on the outcome of a random experiment. Alright, there are two types of these random
variables, discrete random variables and continuous random variables. With discrete random variables,
you can assume a countable number of values, or basically, you can count the values. Whereas,
with continuous random variables, you're dealing with a value that comes from an experiment
with an unlimited or uncountable number of possibilities. Think of it this way, discrete,
countable, continuous, uncountable. Here's an example to help you understand this point.
An example of a discrete random variable would be the number of people who, let's say, live
in a randomly selected dorm room. You know, zero, one, two, three, four. You just can't
have 2.5 or 3.6 people in a dorm room. This is a measurement that can be taken from a
countable number of choices. An example of a continuous random variable would be the
exact weight of a randomly selected newborn baby. This population is continuous because
the weights of newborn babies have an infinite range to choose from. With tenths of ounces
of pounds, there's so many possibilities. There are obviously different probability
models for these two types of random variables, so it's really important to be able to distinguish
between the two. Remember that discrete random variables are countable, and continuous random
variables are uncountable. Okay, we need to get acquainted with exactly what a probability
distribution is before learning the specific types for each of our new random variables.
Let's take probability distributions in the most general terms for discrete random variables.
A probability distribution is the representation of the probability associated with each value
of the random variable. We'll assign a lowercase x to denote our random variable. Therefore,
according to what we've just gone over, a probability distribution for x shows the values
x can take and the probability for each value. Again, in order to have a probability distribution,
a few things must be established first. The probability of each value must be between
0 and 1, or 0% and 100%, as we said earlier, going over probability. And the sum of the
probabilities of all possible values of the variable x must be equal to 1. Suppose we
toss two coins and we want to measure the number of heads that appear. This means that
our random variable, or variable x, now represents the number of heads we observe. The simple
events and their probabilities of this experiment, designated by uppercase E1 through 4, are
shown here in this table. The first possible event is to toss a head on both the first
and second coin. The next possible event would be to toss a head on the first coin and a
tail on the second, and so on. For each event, the value for x is also listed. The probability
for each of these simple events is one-fourth, because the coins are fair and all outcomes
are equally likely to occur. In order to form the actual probability distribution for this
example, we have to list all the possible values and the probability of variable x for
each of its respective values. When our variable x is 0, or no heads have appeared, E4 has
occurred, and the probability of E4 is one-fourth. When x is 1, or one head has appeared, either
E2 or E3 has occurred, and the probability is equal to the sum of the probabilities of
these two events, or one-half. When x is 2, E1 has occurred, and the probability of E1
is one-fourth. Therefore, the probability of x being 0 is one-fourth, of x being 1 is
one-half, and of x being 2 is one-fourth. Now that we have our probabilities listed,
and as you might have guessed from the heading of this section, we need to create some type
of representation of our findings. We can do this in the form of a histogram, or draw
out the distribution to represent the data we found. When working with probability distributions,
the mean, or average, of the distribution is known as the expected value. It's represented
by the symbol mu, because we're dealing with a population, not a sample. The mean of a
probability distribution, or expected value, is equal to the sum of each x value multiplied
by its probability. So, the formula looks and sounds like this. The summation of x times
the probability of x. Remember that the uppercase sigma, that big mean E-looking thing, means
summation of the whole thing. In our example, you multiply the number of times that heads
occurs by the probability of that number of heads occurring. So, using this expected value
formula for our example, the summation of x times the probability of x would be, get
ready for this, 0 times one-fourth plus 1 times one-half plus 2 times one-fourth, which
equals one. Therefore, we find that after doing the calculation, that the mean, or expected
value for x, is equal to one.
Hey, how's it going? I just got in from the grocery store. The woman there says, hey,
you want paper or plastic? I said, what the f*** do you think, okay? Look at my head. Alright,
it's not that like I'm really tired and I got bags under my eyes or something, okay?
I am a bag. Look at my head.
Let's move back to our old friends, variance and standard deviation. The variance and standard
deviation can also be calculated for our random variable, x, in this example. Recall our formula
for variance in the previous section. Remember to rewind if you need to. Oh, there it is.
In the case of probability distributions though, we use a slightly modified formula to find
the variance of x. Let's take a look.
For a probability distribution, the variance, lowercase sigma squared, is equal to the summation
of the squares of the deviations, times the probability of x. In this case, the lowercase
mu is still the mean, but it's now the mean of the probability distribution, or what we
now call the expected value, which for our last coin tossing example, was one. The standard
deviation for a random variable, designated by lowercase sigma, is simply the square root
of the variance. We've graduated a bit in difficulty and jargon, but the meaning is exactly
the same.
Moving right along with our double coin tossing example, recall that our expected value is
one. Now we'll use our new formula for variance, the three x values and each of their probabilities.
If we insert the three values for x, zero, one, and two, and their respective probabilities
into the formulas, and add them together, we end up with a variance of one half. And
the standard deviation, or the square root of the variance, is roughly.707.
Let's go over the results of our experiment again. When we tossed the two coins and observed
the number of heads as the variable x, we found the expected value is one, the variance
is one half, and the standard deviation is.707.
Keep in mind as we move through our material that all we've done thus far is employ the
basic statistics we've learned from the beginning of the video, starting with datasets. We've
taken these stats and simply applied them to the more complex probability distributions
for our variable x. As we continue to move, keep in mind that we're building on the principles
we've already learned. Now, we'll continue to build on our understanding of probability
distributions.
In the high stakes world of statistics, there are several types of probability distribution
models that are used, and a great deal of what we'll cover in the remainder of the
video is frequently used in physical science, social science, business, and economics.
Yes, that is correct. The specific types of probability distribution models that we will
be discussing, of course, are particular to the discrete random variables known as binomial
to binomial probability distributions. Of course, not related to my cousin, Biafrid.
It does sound a little bit intimidating, I'll give you that, but don't sweat it. Just follow
along. A binomial probability distribution describes the probabilities of the possible
results of a binomial type experiment. This is just a specific type of probability distribution
and involves binomial coefficients, which you'll recognize in just a little bit. All this means
is that one of the elements we'll use to calculate probabilities in this type of distribution
or experiment comes from this particular type of algebraic function.
I use this material all the time in my line of work as a professional economics nebbish,
and perhaps you should use it too. I like you, and I am attracted to your statistics.
We already know that this type of probability distribution is for a discrete random variable,
but there's a whole lot to this model. A binomial experiment has the following characteristics.
One, the experiment has a certain number of trials, n, and all the trials are performed
in exactly the same manner. This is effectively the sample size of the experiment. Two, each
trial has one of two outcomes. The outcome will either be called a success, s, or a failure,
f. Three, the probability of observing a success for a single trial is the same for each and
every trial, and is shown as a lowercase p. The probability of observing a failure is
also the same for every trial, and is shown in our formula as a lowercase q, which is
equal to 1 minus p. Four, each single trial is independent of all others, meaning that
the outcome of each consecutive trial of the experiment does not depend on the results
of the previous trial, nor any other trial in the experiment for that matter. And five,
the experiment observes the number of successes denoted by the variable x in n trials. Let's
summarize a little here. In a binomial experiment, which we've just described, we have the following
elements, n, which equals the number of trials, x, which equals the number of successes, p,
which equals the probability of a success, q, or 1 minus p, which equals the probability
of a failure, s, which signifies a success, and finally, f, which signifies failure. The
binomial probability calculates the probability of observing x number of successes all the
way up to the specified n, which is the number of trials in the experiment or the size of
the sample. All right, here's our formula. The formula for observing x number of successes
in n trials is as follows. The probability of x equals the quantity n factorial divided
by x factorial multiplied by the quantity n minus x factorial multiplied by p to the
x power multiplied by q to the n minus x power. Just as a quick reminder, when we say n factorial
represented by n exclamation point, as in the previous formula, we mean n times n minus
1 times n minus 2 all the way down to 1. Oh, and just another quick reminder in case your
algebra is a little rusty. Zero factorial always equals 1. Therefore, 4 factorial, as
you can see here, is 4 times 3 times 2 times 1. After doing the multiplication, you'll
see that 4 factorial is equal to 24. With the information we have, let's go on to
another coin tossing example. Suppose we're tossing one coin 10 times. Therefore, n equals
10, so our experiment has 10 trials. This time, let's observe the number of tails.
That means that x, our variable, now represents the number of tails that we observe in 10
trials. The probability of observing a tail on a single toss of the coin, as we know,
is one half, or.5. Therefore, p, the probability of a success, equals.5, and q, the probability
of a failure, or 1 minus p, also equals.5. With that stated, now we need to find the
probability of observing 0 through 10 tails in 10 trials. What we're basically doing
is finding the probability of how many tails you'll see in 10 tries, meaning the probability
of 0 tails in 10 tries, the probability of 1 tail in 10 tries, and so on. We need to
evaluate the probability of x, finding a tail, for all its possible values in 10 tries.
Plugging in our formula for a binomial problem, the probability of observing 0 tails in 10
trials would mean that x, our variable, is 0. Here's what we get if we plug our numbers
into the equation. The probability of 0 equals the quantity 10 factorial divided by 0 factorial,
multiplied by the quantity 10 minus 0 factorial, multiplied by.5 to the 0 power, multiplied
by.5 to the 10 minus 0 power. After we do the math, we end up with the probability of
observing 0 tails, x, in 10 trials, n, being.001, or.1%. Using the exact same formula,
we can calculate the probabilities of all the other possible values of x, and end up
with the following values.
So, we have the probability of observing 0 tails in 10 trials, n, being.001, or.002,
10 tails equals.001, or.1%. Suppose in this experiment that we wanted to find the probability
of observing a number of tails less than 5. This is, of course, a compound event, and
it's calculated by adding the individual probabilities of observing 0, 1, 2, 3, and 4 tails. Thus,
the probability of observing a number less than 5 would equal.205 plus.117 plus.043
plus.010 plus.001, which equals.376. Our results show there's almost a 38% chance
of observing a number of tails less than 5 in 10 trials.
The mean, which is a lowercase mu, because we're dealing with a population, is equal
to the number of trials n multiplied by the probability of a success on a single trial
p. The variance, shown as a lowercase sigma squared, is equal to the number of trials
multiplied by the probability of a success multiplied by the probability of a failure.
And the standard deviation, shown simply as a lowercase sigma, is calculated by easily
taking the square root of the value you found for the variance. These formulas follow the
same algebraic principles as the ones we used earlier, but they are adjusted to the binomial
model. Let's plug some numbers in and go through the calculations. The mean of our most recent
example is equal to 10 trials times.5, which equals 5. The variance is equal to 10 times
.5 probability of success times.5 probability of failure, which equals 2.5. The standard
deviation equals the square root of the variance, or 1.58.
I did not think so. Now listen up as I enlighten you.
We have the relative frequency histogram and the statement leaf display. With a numeric
representation, we have measures of central tendency, variability, and relative standing.
Central tendency leads us to mean, medium, or mode. I want to hear you say it. Good.
I want to hear you say sir. Good. Now variability leads us to range, deviation, variance, or
standard deviation. Let me hear you say that. You are idiots, and you are no longer in your
brother's kitchen making sandwiches. You are here. Now, relative standing leads us to z
score and percentiles, but all of this is only important in terms of probability. Probability.
Probability. Do you understand probability? Probability leads to events, simple events,
or compound events, and eventually to the binomial probability distribution. Do you
understand what I've been talking about? No sir. Well then let's dance.
The binomial distribution also applies to urn models, which conceptualize several practical
sampling experiments. Urn models are basically descriptions of random sampling using the
analogy of balls in an urn. Let's first consider sampling with replacement. In other words,
each ball chosen for the sample is thrown back in the urn before the next ball is chosen.
In our urn, we have 30 balls, 12 red and 18 blue, such that the probability of randomly
selecting a red ball is 12 and 30, or.4, and a blue ball is 18 and 30, or.6. If we
want to find the probability of selecting a certain number of blue balls from the urn
using replacement and a sample size of 5, then we can use the binomial model. In this
case, n, the sample size, is 5. X, the discrete variable, is the number of blue balls in the
sample. P, the probability of success or selecting a blue ball, is.6, and the probability of
failure is.4. The example of the binomial distribution we just went through follows
the five rules of the binomial to the T. However, there's another situation where an experiment
is treated as a binomial, though it does not strictly follow the five rules and regulations
of the binomial. This situation occurs where sampling without replacement is performed
on a population that's extremely large. For sampling without replacement, of course, the
probability of selecting a blue ball changes as the number of balls left in the urn diminishes
due to our sampling. Therefore, the probability of selecting a blue ball the first time is
12 and 30, but the probability of selecting a blue ball the second time is 11 and 29 because
we have just taken one blue ball from the urn. However, when the population is extremely
large, the removal of one item changes the probability of success only slightly. Ignoring
the slight variation, we assume that the probability of success remains the same, and we can use
the binomial model to describe the distribution of the population. Suppose, however, that
we're sampling without replacement in a population that's not so large and we cannot use the
binomial model. Well, here's an example of what this type of distribution is all about.
Let's say we have a room with 30 people, all of whom have either blonde or green hair.
The distribution of these colors is 20 blonde and 10 green. Our sample size is 3 people.
Therefore, in our sampling, the probability of choosing a person with a certain color
hair is equal to the number of people in the room with that hair color divided by the total
number of people in the room. Remember that we're sampling without replacement, so every
time we choose a person for the sample, the population goes down by one, and the probability
of choosing each hair color changes. So for this sample, we need to calculate the probability
of getting a certain number of blonde heads and a certain number of green heads. We'll
calculate the probability of selecting an n equals 3 sample of blonde, blonde, green
in that order. So our sample consists of two blonde heads and one green head. The specific
order indicates that we're using conditional probability, because each new selection depends
on the result of the previous selection. We calculate this by multiplying the probability
for each event before we choose the new measurement from the sample. Thus, the probability of
choosing a sample of blonde, blonde, green is about.16 or 16 percent. We've just gone
over sampling without replacement to explain not only this type of sample, but to illustrate
a discrete random sampling experiment that does not use the binomial. Before we go on
to our other X variable, realize that we've just covered probability distributions for
discrete random variables. What are they again? Variables that have a clear, countable number
of values. As we said earlier, continuous random variables can take any value in an
interval or any degree of accuracy, so the number of possibilities is uncountable. The
probability of any specific value is 0. The total area under the curve, however, is 1,
and the area under the curve over the two points here of A and B is the probability
that the random variable will take value between A and B. Well, we are on our way to lunch,
but first I pose a very critical question. How many standard deviation between my left
hand on the steering wheel and my right hand on the steering wheel? Approximately 17 standard
deviations, give or take one or two or three, but depending on what I'm having for lunch.
But I'm quite excited, so let us, we're going to be in a car accident. There are several
types of distributions for continuous random variables, and the first of the two types
we'll cover is called the uniform distribution. In this distribution, the variable can assume
any value between two points on a line, say A and B. There's an equal probability of
drawing any measurement on the line. Thus, the difference between B and A is the range
in which the measurements lie. Say, for example, that this difference is 10. Then the height
of the curve is 1 over 10, which is equal to 0.1, as the total area equals 1. It should
be clear that in a uniform probability distribution, there's an equal probability of drawing any
measurement on the line, but the measurements on the line do vary, but they don't vary
any further than the two points on the line will allow. Equal probability, varied measurements.
Yes, and how many measurements can occur on a line between point A and point B? Well,
the answer, my friend, is uniform distribution. The answer is uniform distribution. This means
that the distribution is a straight line between A and B at the height of 1 over B minus A,
and the shape of this distribution is a rectangle. Because of the nature of the measurements,
it's clear to see where the term uniform comes from. It's also very easy to calculate probability
using this model because we're dealing with the areas of rectangles and not curves. These
are the formulas for the mean and standard deviation in a uniform probability distribution.
The mean is equal to A plus B divided by 2. The standard deviation is equal to B minus
A divided by the square root of 12. This formula is a constant in determining the standard
deviation for a uniform distribution. An example of this kind of distribution is a person's
waiting time for a bus that arrives every 10 minutes. The waiting time could be anywhere
from 0, 1 point, to 10 minutes, a second point, depending on when the person arrives at the
bus stop. The possibilities of arrival times are virtually uncountable in a statistical
situation like this and are perfectly suited to its nature as a continuous random variable.
Using our explanation of the uniform distribution, because this is a continuous random variable,
there's an enormous amount of possible measurements from 0 to 10 minutes. And although the bus
has an equal probability of arriving any time in that period, the measurements are vast.
The bus could possibly arrive at any tenth or hundredth of a second in that 10-minute
span of time. The probability that the value of X will fall between two specific measurements
is equal to the area of the rectangle formed by those two measurements on the X axis. For
example, the probability that a person will wait between three and five minutes for the
bus is equal to the area of the sub-rectangle formed by three and five in our uniform distribution.
As we said earlier, most commonly used probability distribution curves for continuous random
variables end up as smooth curves, usually in a bell shape. The most important of such
curves is known as a normal probability distribution, which was also the first curve we showed in
the video. It's the second type of probability distribution in our treatment of the continuous
random variable. A normal probability distribution curve has an area underneath it of 1. This
is true for all continuous distributions. It's symmetric about its mean, lowercase mu, and
is also determined by the population's standard deviation, lowercase sigma. The larger the
value of the standard deviation, the shorter the height of the distribution curve. Since
the curve is symmetric about its mean, half the area under the curve, or.5, lies both
to the right and to the left of the mean. Recall one of our measures of relative standing,
the Z-score. Sometimes when Z-score is waiting for his food, he thinks about when he was
a Z-score, measuring little itty bitty things, like stirring carrots and peas and rattles
and other things he got from his mummy. But now Z-score is all alone, ah, cursed to measure
standard deviations across the land. But he will get over it. Z-score is strong. Once
again, the Z-score denotes the distance a measurement lies from the mean in terms of
the standard deviation of the distribution. Therefore, a Z-score of zero means that the
measurement is equal to the mean. Also, a measurement with a positive Z-score lies to
the right of the mean, while a measurement with a negative Z-score lies to the left of
the mean. In this vein, a Z-score of one means that the measurement is one standard deviation
away from the mean. For example, if we calculate that a measurement from a distribution has
a Z-score of one point five, then that measurement lies one and a half standard deviations away
to the right of the mean of the population. If we show the placement of the Z-score on
the graph, we see that the area under the curve between the mean and the Z-score, denoted
as capital A, is the percentage of the measurements that the Z-score of one point five encompasses
between it and the mean. The value of this area is also the probability of choosing a
measurement with a Z-score between zero and one point five. Whether the area lies to the
right or left of the mean depends on whether the Z-score is positive or negative. There's
a table which helps us to determine this area. This area actually represents the probability
of choosing a measurement with a Z-score between zero and some Z-score value. Rather than have
to calculate this, statisticians use a prefabricated table, which you should be able to find in
your statistics book. The left hand column lists the Z-score in increments of one tenth.
The top row lists further increments of one hundredth. The numbers inside the table are
the corresponding areas for each Z-score in a combination of its measurement to the tenth
and to the hundredth. If we wanted to find the area for a Z-score of point two one, we
would look down the left column for point two and across the top row for point zero
one and then find the area at which they intersect. This area is point zero eight three two and
is the probability of choosing a measurement with a Z-score between zero and point two
one. Be careful here. Point zero eight three two is not the probability of choosing a measurement
with a Z-score of point two one, but of choosing a measurement with a Z-score between zero
and point two one. As an example, suppose we have a normal distribution with a mean
of ten and a standard deviation of two. What if we needed to find the probability of choosing
a measurement of X between nine and twelve? First, we'll need to find the Z-scores for
nine and twelve. Remember that the Z-score is equal to X minus the mean and we take that
quantity and divide it by the standard deviation. So for nine, we take nine minus the mean of
ten divided by the standard deviation of two and get a negative point five. For twelve,
we take twelve minus ten divided by the standard deviation of two and arrive at one. So we
get a Z-score of negative point five for nine and of one for twelve. To find the probability
associated with these Z-scores, we'll have to use the table. Because of the symmetry
of the normal distribution, the area associated with a negative Z-score is equal to the area
associated with the positive value of that Z-score. Therefore, we'll need to find the
area between zero and point five and between zero and one. These areas according to the
table are point one nine one five and point three four one three respectively. Now that
we have these two areas from the table, we must add them together in order to find the
total area between the Z-scores of negative point five and one. By adding point one nine
one five and point three four one three, we get a total area of point five three two eight.
This total area represents the probability of choosing a measurement between nine and
twelve in our distribution. This area of point five three two eight also represents the percentage
of measurements in the data set that lie between nine and twelve. In other words, 53.28 percent
of the measurements in the data set are between nine and twelve. I'm in trouble. I don't
see what the problem is. Why don't you guys understand this stuff? I don't get it. I don't
even have a blue book. Hey, come on you guys. It's not so bad. Let's just go over this again.
Yeah, Chaz, would you explain it just one more time? Alright, just listen up and I'll
explain it to you. Beginning with probability or the chance that an event will occur, we
discussed events, event composition, and conditional probability. An event's composition, whether
it's simple or compound, determines how to calculate its probability. Simple events are
mutually exclusive of each other and cannot be broken down into smaller events. And compound
events are made of several of these building block simple events. The union of two events
means that either or both events can occur. An intersection of two events is only the
simple events which the two have in common. And don't forget about conditional probability.
Events can be either dependent or independent. Then we moved on to our random X variables,
which are either discrete or continuous. Discrete random variables can take on a countable number
of values, while continuous random variables can take unlimited or uncountable number of
values. In statistics, we're concerned with distributions or ranges for the values of
a certain variable in a population. For discrete random variables, the probability distribution
of a population shows the likelihood of the variable taking on a certain value within
the population. We also calculated the expected value, or mean, the variance and the standard
deviation, which are common things you need to calculate when you do probability distributions.
Don't forget the importance of the binomial probability distribution. If you remember,
it's one type of model for discrete random variable probability distributions, which
deals with the probability of successes and failures, or whether a certain event occurs
or doesn't occur. Michael?
Right. Moving right along with distributions for continuous random variables, a probability
is represented as the area under a curve. To determine areas under the normal distribution
curve, we use the standard deviation and the z-score of measurements. We also discuss the
uniform distribution, where there's an equal likelihood for all the possible measurements
of the population. Remember, too, that the normal probability distribution is distributed
so that the graph is a bell curve, and the population follows the empirical ruler.
Also, don't forget that we learned even more about our favorite measure of relative standing,
the z-score.
There's a term that you all know I am z-scoring, and now I am going to determine how many are
standard deviations between these sockets, and this little...
In terms of the z-score, we not only learned where a measurement lies in relation to the
mean, but also how to get the probabilities of certain z-scores with the z-score table,
which brings us up to where we are now. Hopefully, you're feeling comfortable and caught up,
and we can move on to part three.
In the immortal words of my dog, Varine's, woof.
Part three, sampling, and sampling, and sampling, distribution.
Section A, a look at sampling distributions.
As we've said over and over, when populations are too large to measure, or too large to
determine anything from them, statisticians analyze samples in order to make inferences
about the population.
In order to assess the reliability of inferences about the population based on a particular
statistic from the sample, statisticians look at the sampling distribution of that one statistic.
Here's a nice way to visualize how a sampling distribution is created.
Let's imagine that there are bins set up along the x-axis of the graph of the sampling distribution.
We've created hypothetical numbers for the calculations, so you don't have to worry about
the specifics. But every time the mean of a sample is calculated, a bin is dropped into
one of the bins in a location along the x-axis specific to its value.
From the calculation of the first sample to the nth sample, a bin is dropped each and
every time there's a new measurement. As you can see, the sampling distribution is quite
vividly formed in this way.
Now the means of these samples tend to have an approximately bell-shaped distribution.
This is called the central limit theorem.
Theorem, theorem, theorem, complex, pedantic theorem, theorem, theorem, blah, blah, blah,
blah. Yes.
Question, question, confusion, bewilderment, question?
Response. Theorem, theorem, theorem. Yes.
Combative response.
Disgusted reply. Theorem, theorem, theorem. Yes.
My pen blew up.
This theorem states that when n number of measurements are drawn from a population with
a mean of mu and a standard deviation of sigma, the sampling distribution of the means of
the samples, x-bar, is approximately normally distributed, provided that n is very large.
Now this is very similar to what we saw with our distributions of continuous random variables.
Basically, what we have is a population, and we take samples with equal number of measurements
from that population. Then we find the mean of each of the samples.
The mean of the samples form their own distribution, which is a sampling distribution.
According to the central limit theorem, this type of sampling distribution should be normally
distributed, or curve-shaped like a bell when the sample size is large.
Here's a couple other important points.
The mean of the distribution of the sample means should be equal to the mean of the original
population, and the standard deviation of the distribution of the sample mean should equal
the standard deviation of the population, divided by the square root of the number of
measurements in the sample, or n.
These approximations become more accurate as n, or our sample size, becomes larger.
Also realize that the standard deviation of the mean sampling distribution is the best
measure of reliability of a sample. It tells us how good of an estimation the sample mean
is of the population mean. This is our goal, to make estimations about our population
from our samples.
According to the central limit theorem, the mean of the sampling distribution is just about
equal to the mean of the population. Therefore, the standard deviation of the sampling
distribution tells us how different the mean for each sample is from the mean of the
population. For this reason, the standard deviation of a sampling distribution of the
means is also called the standard error of the mean, which basically gives us an idea
of how far away we are from the population.
Statisticians use what are called estimators to make inferences from the sample about the
population itself. In other words, we use the mean, or any other measure from a sample,
to make inferences about the greater population. Now, what exactly is an estimator?
Well, an estimator is simply the rule that tells us how to calculate an estimate. It's
used to draw conclusions about the population from the sample in the form of this
estimate. So an estimator walks into a bar. All right, bartender says, Hey, estimator,
what time is it? Estimator says, I don't know about three o'clock. Thank you. Thank
you. I wrote that one myself. There are two types of estimators, point and interval.
Both of these types use some statistic from the sample to calculate a value which helps
us make inferences about the population parameter. Remember from the beginning of the
video, population, parameter. Let's take the first one. A point estimator of a
population parameter is a statistic which indicates how to calculate a single number
using sample data. Now, this number is called a point estimate. A point estimate
allows us to draw a conclusion about a parameter of the population. There's another
type called interval estimators of a population parameter. Interval estimators
allow us to calculate two numbers forming an interval within which the population
parameter is believed to lie. This pair of numbers is called the interval estimate,
or also known as the confidence interval. Unlike the point estimator which estimates
that the population parameter is a single value, the interval estimator indicates
that the population parameter lies within a certain range. Also, statisticians use
measures of accuracy in order to determine trueness of these estimators. Such measures
of accuracy are the error of estimation and the confidence coefficient. Let's start
with the first one. The error of estimation is the distance from or the difference
between the calculated estimate and the parameter we're estimating. The confidence
coefficient, on the other hand, is the probability that the calculated confidence
interval will enclose this parameter that we're estimating. Thus, if we're trying to
estimate the mean of a population, we would use the mean of the sample as our
estimator. Alright, thank you, thank you. Now I want to show you a little dance that was
invented by my cousin Shecky. It's called the interval dance. Watch.
See, I was dancing at intervals. Thank you, thank you cousin Shecky. You helped out a
lot for my act. It's the last time I ever give you that rent check. Alright, thank
you, thank you. Remember that the standard deviation of the sampling distribution is
calculated by dividing the standard deviation of the population by the square root
of the number of measurements in the sample, n. This new calculated value is useful
in determining the error of estimation only if the number of measurements in each
sample is greater than or equal to 30. n equals 30 is a standard in statistics for
using the central limit theorem. Let's go back to five-card Charlie as an example.
Remember that Charlie needed information about his population of fuzzy dice cola
drinkers. Suppose he takes a random sample of 50 measurements. It produced the mean
of 871 ounces of cola consumed and a standard deviation of 21. Charlie wants to
estimate the mean of the population, mu, using the mean of the sample, x-bar. He
does this by defining a range called the error bound. For this example, the error
bound is defined by plus or minus 1.96 multiplied by the quantity 21 divided by
the square root of 50. This value is 5.82 ounces. If you recall, the central limit
theorem says that x-bar is normally distributed. Here, we use the z-score of 1.96
to identify a 95% interval because the z-score of 1.96 contains 47.5% of the
measurements on either side of the mean. We are also using the standard error of
x-bar. Therefore, we're 95% confident that the mean of the population is between
plus or minus 5.82 of our estimate, 871 ounces of cola consumed. In other words,
we are 95% confident that the actual mean of the population will be between
876.82 and 865.18. These two values are the upper confidence limit and the lower
confidence limit, respectively.
Excuse me. I was wondering if you'd like to come back with me to my cabin in
Saskatchewan. Please. I've got a wood stove. Please.
How you doing, baby? How about you and me take off to my cabin in Rio?
Yes. Let's do it.
As we said earlier, statisticians make inferences from the sample about the
population. Hypothesis testing is one way in which to make inferences. Using this
method, statisticians make a hypothesis and then need to test this hypothesis.
Using a statistical test, this hypothetical statement is given the name of
the alternate hypothesis. In calculations like this, the alternate hypothesis is
represented as H sub A.
In order to test this hypothesis, there must be something to test it against.
Therefore, the exact opposite of the alternate hypothesis is created, and it's
called the null hypothesis. It's shown in a similar way to the alternate hypothesis
in calculations as H sub 0. When set up properly, if the alternate hypothesis is
true, then the null hypothesis is false. Basically, the null hypothesis and the
alternate hypothesis must be written to support one of them as acceptable and
reject the other as unacceptable.
If we test the null hypothesis, the result of the test will determine whether to
accept or reject it.
In order to go ahead and test the null hypothesis, there must be a decision-making
process. This process is based on what's called a test statistic.
Remember that we're testing the null hypothesis. Therefore, if the test statistic
lies in the rejection region, then we're accepting the alternate hypothesis,
thereby rejecting the null hypothesis. If the test statistic lies in the acceptance
region, we're accepting the null hypothesis, thereby rejecting the alternate
hypothesis.
There are two types of errors that can be made in the testing process, a Type I
error and a Type II error. A Type I error is made by rejecting the null hypothesis
when it's true. The probability of making a Type I error is denoted as lowercase
alpha. A Type II error is made by accepting the null hypothesis when it's false.
The probability of making a Type II error is denoted as lowercase beta.
This table represents the decisions a statistician can make from a statistical
test. If he or she rejects the null hypothesis and the null hypothesis is true,
then a Type I error has been made. However, if the null hypothesis is false, then the
correct decision has been made. If he or she accepts the null hypothesis and the
null hypothesis is true, then a correct decision has been made. However, if the
null hypothesis is false, a Type II error has been made.
In these types of tests, the value which separates the rejection from the
acceptance region is called the critical value. In our case, the critical values
would have z-scores of 1.96 and negative 1.96 because they are the boundaries
between the acceptance region and the rejection region.
Because we're still dealing with normal distributions, keep in mind we still have
a symmetrical bell curve about the mean. Therefore, the rejection region can lie
on either or both sides of the curve. Why is this important? Take a look at the
graph.
A statistical test that locates the rejection region on only one side of the
curve is called a one-tailed test, while one that locates the rejection region on
both sides of the curve is called a two-tailed test. It's easy to see because
they really do look like tails.
Let's go back to our example with five-card Charlie. Suppose we want to test
the null hypothesis that mu is equal to 880 against the alternate hypothesis that
mu is not 880. Remember n equals 50, the mean is 871, and the standard deviation
is 21. Also, we're 95% confident that the mean, mu, is between 865.18 and
876.82. Since this interval does not include the value 880 given by the null
hypothesis, we're 95% confident that the null hypothesis is not true and should
therefore reject it. Usually, statisticians test a hypothesis using a test
statistic, which is a z-score in our case.
Oh, how many standard deviations in z-scores pick up? Oh, I think it's six
standard deviations or maybe five.
Here's how to calculate the test statistic for this test. Since we are
testing our hypothesis using the z-score distribution, our test statistic is a
z-score which corresponds to the null hypothesis in our distribution of the
means of the samples.
Remember that the point estimate for mu is x-bar. Therefore, our test statistic
is z equals the quantity x-bar minus mu sub-zero divided by the quantity sigma
over the square root of n. Mu sub-zero indicates the value of our null
hypothesis. Here's what we get when we insert the values. z equals the quantity
271 minus 880 divided by 21 over the square root of 50. This is our test
statistic and calculates to be z equals negative 3.03. Therefore, we should
reject the null hypothesis that mu equals 880.
All right. With the foundation of large samples established, you should do just
fine in the next section.
Section C, small samples.
Statistical tests for smaller samples and for larger samples are basically the
same procedure with one variation. The statistical test for smaller samples
uses the t-statistic, which is similar to the z-score when using larger samples.
The t-statistic is calculated by subtracting the mean of the population
from the mean of the sample and dividing that by the quantity of the standard
deviation of the sample divided by the square root of the sample size.
In order to use the t-statistic, we have to also determine what's referred to as
degrees of freedom. Degrees of freedom is equal to the sample size minus 1 or
n minus 1. The degrees of freedom is used to adjust the t-statistic to the
specific size of the small sample. The critical values for a test based on the
t-statistic come from a t distribution. For a given t distribution and a given
value alpha, we use t sub alpha as the critical value in our test. T sub alpha
is such that the area to the right of the value on the distribution is equal to
alpha, meaning the probability of a t-statistic taking on a value larger than
t sub alpha is equal to alpha.
There's a table in your statistics book which helps you easily determine certain
critical values of the t distribution. The degrees of freedom are down the first
column and the values of the percentage of the area to the right of t or alpha is
along the top row. For example, if we wanted to find the critical value for t
such that the degrees of freedom is 6 and 5% of the area under the curve is to
the t's right, then we look down the degrees of freedom column to 6 and along
the top row for t sub.050 since we're looking for 5% of the area to be to the
right of t. The corresponding value for t in this case is 1.943.
We should note that for a one-tailed test, we find the value for t using t sub
alpha along the top row. However, for a two-tailed test, the value for t is
determined by using alpha divided by 2 because the critical region is split
between both ends of the distribution on the curve.
As an example, let's say for a sampling distribution, we have n equal to 6,
x-bar equal to.53, and s equal to.0559. We may test that the null hypothesis is
mu equal to.5, and the alternate hypothesis is mu greater than.5, making
this a one-tailed test. Using the formula for the t statistic of the null
hypothesis, we find that t equals 1.32. Now we need to find the rejection region
such that alpha equals.05. Looking on the table, we find degrees of freedom of
origin, n minus 1, or 5, and t sub.050. This value is 2.015. Since the t
statistic of the null hypothesis does not fall in the rejection region, we must
accept it, and thus we cannot accept the alternate hypothesis that mu is greater
than.5.
Guys, wake up. That's a test for smaller samples. As you can see, it's pretty much
like a statistical test for large samples, except that you use the t statistic
instead of the z score. You've come a long way. You know now how to test
inferences about a population using sample statistics and sampling
distributions, which are based on probability distributions, which are based
on the models for discrete and continuous random variables, which of course come
from our data sets, which is where this whole thing started. You've done a great
job. Pat yourself on the back.
Sometimes when you're studying your statistics, it's really helpful to sing a
song or two or just take a little bit of a study break. I like to take a study
break with a good cup of Joe, and I've got a song about it. Coffee, coffee, the
wonderful drink. You can make it from the water in the sink. You drink it down and
it wakes you up. It's made from beans and it comes in a cup. It's coffee, coffee,
coffee, coffee. Thank you. Well, we hope you enjoyed watching this as much as we
enjoyed making it. In this video, we covered everything from statistical
problems and data sets to probability and distributions to sampling and
sampling distributions. That's right. And make sure you look in your bookstore for
standard deviants video reviews with the standard deviants. Bye. Bye mom. Bye bye.
The stems of the data set in ascending numerical order. Sorry, did I say data or
data? And to represent the data we found. Oh, but I said data. The stems of the
data set. Data, data. Burp Charlie, burp. Burp Charlie. Why are you going towards me?
This was a really good video. How long have we been studying? A long time. Hey Steve,
what did you think of that? Steve? Steve? Oh my God. Oh my God. I am Z-score. In N
trials. In N trials. In M trials. In N trials. In M trials. In N. Action. How you
How you doing, baby?
How about you and me go to my cabin in Rio?
Yes!
What was it doing?
You
Thank you.
Thank you.