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P-value for T-statistics

P-value for T-statistics

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How does one find the p-value for the T-statistic? Got kinda lost there…

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365 Team
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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 (30×10) 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 30×10, instead it will be 30×300. 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 30×10. 
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