>>We met the variance during
the very first part of this course.
It was the odd ball parameter we described
as spreadness and skrinchness.
It told us how uniform the data were,
how alike they were.
It said that if we were looking at ages of people
we would expect the ages in the first grade class
to have a small variance.
The ages in the whole city would have a larger variance
because it included all the children,
and all the retiries and everybody else in between.
The variance was found to be the foundation
of all of our hypothesis tests
and our confidence intervals.
Our confidence intervals were always plus or minus
something and the standard deviation, read variance,
was always determining that something.
The standard deviation, square root of the variance,
was the bottom term in all of our hypothesis test formulas.
We have been, perhaps, taking the variance a bit
by implaction, the standard deviation for granted.
In order to test a hypothesis using the F Distribution,
we need to learn how the F Table works.
We need to know this because we need the critical values
to determine if the calculated F values fall in the tail.
In this way, hypothesis tests using the F Distribution
work exactly like any other hypothesis test.
The F Fistribution is created as the ratio
of two Chi Square Distributions.
The F Distribution looks like a Chi Square Distribution.
It is scewed and it changes shape .
as the degrees of freedom change.
Also, like the Chi Square Distribution.
Remembering that the Chi Square Distribution
has degrees of freedom and now knowing
that the F Distribution is a ratio
of two Chi Square Distributions,
we see that the F Distribution must
have two set of degrees of freedom.
One for the Chi Square Distribution in the numerator
and one for the Chi Square Distribution in the denominator.
To determine the critical values for our statistical test,
we find F sub alpha, D-F meaning degrees of freedom one
and D-F degrees of freedom two
in the F Distribution table.
The two different degrees of freedom,
one associated with the numerator, D-F one
and associated with the denominator, D-F two.
The degrees of freedom in the numerator
is N one minus one, where N one
is the sample size for group one.
The degrees of degrees of freedom in the denominator
is N two minus 1, where N two
is the sample size for group two.
F alpha D-F one, D-F two, will give the critical value
in the upper end of the F Distribution.
As an example, imagine that we have sampled two groups,
and we want to know if they have different variances.
We want to have a 90% level of confidence.
If group one has a sample size of six,
N one equals six, and group two
has a sample size of 13, N two equals 13,
we can calculate the degrees of freedom
by subtracting one from each sample size.
The gives us the degrees of freedom for the numerator
and the degrees of freedom for the denominator.
We draw the F Distribution and mark
the alpha level in each tail.
For the right hand tail, the upper tail,
we look up F critical value by
going to .05 F Distribution Table
and looking for five degrees of freedom across the top row
D-F one and then going down
until 12 degrees of freedom at D-F two.
We find 3.11 and mark this on the graph.
This gives us the critical value for the right hand tail
at the upper end of the distribution.
Like the Chi Square Distribution,
the F Distribution is skewed so we have to take
a second step to get the critical value for the lower end
of the distribution or the left hand tail.
To find the critical value for
the lower end of the distribution,
reverse the degrees and divide
the F value we find in the table into one.
By reversing the degrees of freedom,
this means to change the order of which one
is in the numerator and which one is in the denominator.
In this example, we look up 13 degrees of freedom
across the top row now, and go down to five and read 4.68.
We then divide this into one and get .214.
This is the critical value for the lower tail.
The two formulas for finding the critical values
for the F Distribution are these.
I know this seems a bit crazy,
but this is how the F Distribution is constructed.
More on this later in the next unit.