P-value for T-statistics

Hey Jamie!

This is a very interesting question (if I understood it correctly).

I believe your question is: 'it is clear with the z-table, but it doesn't work with the t-table'.

Correct?

If so, think about this:

The z-table is very detailed - most values are there. The rows and columns work together to describe all possible cells. Note that it is not REALLY needed to be a table. You can just have one very, very long row (or column) of values.

As ONE column (300 rows):

z = 0.01 -> value
z = 0.02 -> value
z = 0.03 -> value
...
z = 2.99 -> value
z = 3.00 -> value

As ONE row (300 columns):

z =         0.01 ; 0.02 ; 0.03 ; ... ; 2.99 ; 3.00
value =    a    ;    b   ;    c   ; ...  ;   d    ;    e

Alternatively, we can reverse those (it won't make a difference). For my purposes I need this representation:

value =    a    ;    b   ;    c   ; ...  ;   d    ;    e
z =         0.01 ; 0.02 ; 0.03 ; ... ; 2.99 ; 3.00

(remember that!)

We have a z-TABLE, which is usually (30x10) so it takes less space and can be 'carried arround' (before computer were good enough).

***

The t-table is not as detailed.

Why?

Because there is another dimension - the degrees of freedom. But a table can only have two dimensions. For the t-table, they are: degrees of freedom and level significance.

Looking at the (first 5 rows of the) t-table:

We realize that it is actually (from above) ...

value =    a    ;    b   ;    c   ; ...  ;   d    ;    e
t  =         0.01 ; 0.02 ; 0.03 ; ... ; 2.99 ; 3.00

Right?

But this is done for EACH degree of freedom.

So, We don't have one row, but have as many rows as there are degrees of freedom. And the T values are different for each degree of freedom. We have 30 degrees of freedom in most tables so we have 30 rows.

Usually we also include the z-row, because after df=30 for t, we can assume normality:

With this last row, we are basically summarizing t>30 (infinitely many rows).

***

Okay, but what about the columns?

Didn't we say that when there is a single row, we need 300 columns?

Well, we already have 30 rows for the degrees of freedom. And we can't pull of the 'trick' from the z-table to make it 30x10, instead it will be 30x300. Which is a table which we cannot really carry around or investigate manually (doesn't make sense when we have p-value calculators).

There are several solutions to this problem:

1) Make 30 t-tables. T-table for 1 degree of freedom, T-table for 2 df, 3df, etc., where we pull of the z-table trick, so we get 30 t-tables, each 30x10.
2) Make a 3-dimensional table. 30x30x10.

That's cool, but not very useful per se.

3) Cut most of the table and keep only the most improtant values. The result?

The t-table you know.

Obviously that's the best decision if we want a paper copy.

***

After this long explanation we can answer your question:

The information you are looking for is NOT in the t-table, because it was intentionally cut out.

The solution we've opted for is an online p-value calculator. You can find a pdf with detailed instructions with this lecture: https://www.udemy.com/the-data-science-course-complete-data-science-bootcamp/learn/v4/t/lecture/10764560?start=15

In practice, usually, you'd have a software calculate it for you!

Hope this helps!
Best,
The 365 Team