Let's look into using a t table to
find the p value in a one sample t-test.
Suppose we have a situation where
we are testing the null hypothesis that mu is equal to mu_0,
against the alternative hypothesis that
mu is greater than mu_0,
and we find that the observed value of
the test statistic is 2.59, with sample size of 7.
We need to find the p-value in order to reach an appropriate conclusion.
 
And first we should draw in the
appropriate t distribution here.
For one sample t-tests the degrees of
freedom are n-1,
so here we would have 6 degrees of freedom.
Here I've drawn in a t distribution with
6 degrees of freedom,
and our observed value of our test statistic, 2.59,
falls somewhere around there on that distribution.
With the alternative hypothesis that mu is greater than mu_0,
the p-value is the probability, under the null hypothesis,
of getting this observed value of the
test statistic or something even greater.
In other words, the p-value is the area to the right of the observed value of our test statistic.
That is the p value in this scenario.
If we used software,
we could find that the p value is approximately 0.021,
but if we have to use a t table we're
not going to be able to find the exact value
and we're only going to be able to come up with a range of values.
So let's go to the t table and see what I mean by that.
We need to find the area to the right of 2.59
under a t distribution with 6 degrees of freedom,
so 2.59 is somewhere around there.
This is the area we need with 6 degrees of freedom.
Values in the table here, these are t values with these areas to the right.
And if we focus in here on 6 degrees of freedom
and we look along this line, we're not
going to be able to come up with 2.59 exactly,
 
but we are going to be able to find that 2.59 
falls in between these two values.
And the table tells me that the area to the right of 2.447 is 0.025,
so I could draw that out and at 2.447 the area out here is 0.025,
 
 
 
 
and similarly the area to the right of 3.143 is 0.01.
 
Our value of 2.59 falls in between these two values,
so the area to the right of our value
must fall in between these two values.
So the table tells us that this area falls in between 0.01 and 0.025.
 
But for this alternative hypothesis that area is the p-value,
so we would simply say that the p-value 
falls in between 0.01 and 0.025.
What if we had an alternative hypothesis in the other direction?
 
 
but the alternative hypothesis was in
the other direction
and H_a was that the population mean mu is actually less than mu_0.
For that alternative hypothesis the p-value is
the area to the left of the observed
value the test statistic,
which is simply 1 minus the area to the right of that value which we found up top.
The area to the left of 2.59 must lie between 0.975 and 0.99,
 
since the area to the right was between 0.01 and 0.025.
And for this alternative hypothesis that area to the left is the p-value,
so for this alternative the p-value must lie between 0.975 and 0.99.
 
What about a two-sided alternative hypothesis?
Suppose we had this situation where
we're testing a two-sided alternative hypothesis,
 
and we find an observed value of the test statistic of -1.31.
We have a sample size 10 so we have 9 degrees of freedom.
Here I'm just going to roughly draw in a t distribution
with 9 degrees of freedom, it looks something like this.
We don't have to get it exactly correct here.
And -1.31 is going to lie somewhere out there.
As we've discussed previously for a two-sided alternative hypothesis,
 
the p-value is double the area in the tail
beyond the observed value of the test statistic.
So here the p value is going to be double
this area in the tail to the left of the test statistic.
But before we go to the table we need to remember
that negative values of t are not given in the table,
and that's because the t distribution is symmetric about zero,
so we can find areas in the left tail by
using areas in the right tail and a little logic.
So here we first must recognize that the area to the left of -1.31
is exactly equal to the area to the right of 1.31.
 
Those two areas are exactly the same.
We need to find the area to the right of 1.31
under a t distribution with 9 degrees of freedom.
And if we focus in here on 9 degrees of freedom,
we'd see that 1.31 lies between 1.100 and 1.383,
and the table tells me that the area to the right 
of 1.100 is up here at 0.15,
so this area is 0.15,
 
 
 
and our value of 1.31 lies in between these two values,
 
 
 
So the table tells me that the area to the right of 1.31 
lies between 0.10 and 0.15,
which implies that the area to the left of -1.31 
also lies between 0.10 and 0.15.
 
 
 
But since the alternative hypothesis is two-sided
we must double that area to get our p-value.
So the p-value must lie between 0.20 (0.1 doubled) 
and 0.30 (0.15 doubled).
If we have access to software it's much better to use software
 
to find the precise value of 0.223.
If we were to do so we'd see that the
real p value from the software
falls into the range of values we found from the t table.
