Find the degrees of freedom for the F statistic and then use Table E to approximate the P-value

Question

a. seven groups are being compared with 5 observations per group. the value of the f statistic is 2.69

b. five groups are being compared with 11 observations per group. the value for the f statistic is 2.43

c. six groups are being compared using 34 total observations. the value of the f statistic is 3.06

Answer 1

This seems to be an old question, but in case someone ends up finding it while searching for something similar... here goes.

 

F-Tests have two degrees of freedom associated with them: df-between groups and df-within groups.  

 

Between groups is simply the number of groups - 1.  So in a) with seven groups, df-between is 6!  b) 5-1 groups = 4 df-between!  c) 6 groups means 5 df-between!  The actual formula is k - 1, where "k" = number of groups.

 

Within groups df is a tiny bit trickier.  It's the number of people in each group - 1, or (group 1 - 1) + (group 2 - 1) + ... (group N - 1).  The actual formula is N - k, where N = "total people in the study" and "k" still = "number of groups."

 

So, for a) 7 groups with 5 people each in them makes N = 35.  N - k = 35 - 7 = 28 df-within.  Or, you could say that for group 1: (5 people - 1) = 4; group 2 (5 - 1) = 4; group 3 (5 - 1) = 4; group 4 (5 - 1) = 4; group 5 (5 - 1) = 4, group 6 (5 - 1) = 4, and finally, group 7 (5 - 1) = 4... That's 4  x 7 = 28!  The same answer I got using N - k.

 

So b) df-within = N - k = 55 - 5 = 50.

c) N - k = 34 - 6 = 28.

 

To summarize so far:

 

a) df = (6, 28); F = 2.69.

b) df = (4, 50); F = 2.43.

c) df = (5, 28); F = 3.06.

 

Now all you have to do is look up the critical values in an F-Table such as this one: http://www.uncert.com/uncert/grid/t-1a.gif (all I did was search "table of F critical values").  If you go across the table using df-between, and then down the table using df-within, you'll find a number at that juncture.  If you compare the number given in the problem for F to the critical value on the table, you'll know whether p is greater than or less than 0.05 (you can be more specific with tables that also list alpha levels at 0.01, but let's save that for another time!).

 

For a) we go over 6 and down 28, and the critical value on the table is 2.45; the F we were given is 2.69, which exceeds the value in the table.  So p < 0.05!  I understand that this answer might include a discussion on what p is and why we care that it's less than or greater than 0.05, but we only have so much room here - feel free to ask if you want to know!

 

For b) the critical value on the table is between the 2.61 and 2.53.  Well, gee, that makes it harder.  So let's use this: http://www.danielsoper.com/statcalc3/calc.aspx?id=4, a free online critical values calculator!  Sticking in (4,50) gets us 2.56 (which is between 2.61 and 2.53, above!).  Since the F we were given is 2.43, it is NOT greater than the critical value, so p > 0.05.

 

Finally (whew!) for c) the critical value for (5,28) = 2.56 as well.  But the F we were given is 3.06, which IS greater than the critical value, so p < 0.05!

 

I hope this helped clarify for anyone who is looking at this kind of problem.  However, I do feel that a quick lecture on what critical values, F statisics, and p values really are would be a lot more helpful than just giving these answers!  Hit me up if you have any questions. =)

 

Seth