>>We have developed the formula for calculating
a confidence interval for a population mean from a sample
when we do not have the population standard deviation
and we have a small sample size.
This formula naturally comes from the formula
for cases when we had a sample size greater than 30.
For that case we used the normal distribution
but for sample sizes less than 30
we need the Student's t distribution.
Here is the example of what we can do with our new formula.
From a sample of 10 industrial stocks
randomly selected from those listed on the Dow Jones
the average earnings per share, or EPS,
was found to be $1.85 with a standard
deviation of 0.395, or about $0.39.
Calculate the 99% confidence interval
for the average earnings per share
of all industrials listed on the Dow Jones Average.
The appropriate formula for this is
our confidence interval where x-bar
is plus or minus t sub nu,
alpha over two times s over the square root of n.
Typically we would have z in here
but because the sample size is less than 30,
then we use the t distribution.
We can visualize the process by including
the confidence interval and we draw
the appropriate distribution so up here
we have the random variable x,
which happens to be the earnings per share.
This is the sampling distribution
so we have, looks something like this,
these are actually x-bars and what we have now
is the particular x-bar
that we found from our sample.
This value here is equal to $1.85
where s, the standard deviation
we found from our sample is equal to about $0.39.
Now what we want to do is find
the confidence interval around this value
so down below here we have
the normal distribution except
we've changed it to the t distribution.
Normally this would be my z value
but in this case it's going to be a t value.
What we wanted to have was a confidence interval of .99
so I have up here .99
and I drop this down to here
and this is going to be .99, so that's what I'm looking for.
What I'm actually after, when we've put
it on a graph like this, is I want to know
that number and that number.
I want to know the range of values
within which, with 99% confidence,
the true population mean can be found.
So I have to have my t value down here,
this is zero of course, and my minus t value over here
so I can put it into my formula to get my range.
To find the appropriate t requires two pieces
of information, the level of confidence,
in this case it's .99
and the degrees of freedom.
The degrees of freedom is equal to n minus one.
In this case here our degrees of freedom,
which Greek letter nu often written df,
degrees of freedom, is n minus one
so in this case here we have nine degrees of freedom.
We divide our confidence level by half
so I've got 01 here, this is going to be alpha over two,
which is going to be .005
and this is going to be .005 over here.
This is equal to alpha over two.
A confidence interval is two-sided, meaning
you have probabilities on both sides.
Now we need to look that up and when
we do that and we find it in the t table.
The way you do that is you go to
the .005 column and you go down to
nine degrees of freedom and that's
the t value that we're going to place right here.
In this case here it's 3.2498,
carried out the four digits for reasons I can't fathom.
We take this number here, we put it in this formula.
When we do that we end up with a confidence interval
of 1.44 on this end
and 2.26 on this end.
We state the formal conclusion as
with 99% confidence level the earnings per share
of all industries listed on the Dow Jones Average
is between $1.44 and $2.26.
From a small sample we are able to,
with very high confidence, say what
the average earnings per share of all stocks
must be, plus or minus some number
but perhaps the most important feature
of this tool is the ability to say
the level of confidence we have in our estimates.
After all, people are always giving estimates
of this and of that but when pressed,
how confident are you in what you're saying?
Then we see nothing but hesitation,
not required with us here.