#This is a quick look at the relation within the F-distribution
#   of values and areas looking to the left and to the right.

# As pointed out in the web page, textbooks often present 
#    tables of the F-distribution that only give critical
#    values for a few specified areas and then only give
#    those values for the specified areas to the right.
#    How then are textbook readers to find critical values
#    for areas to the left?

#    First, observe the following specific situation.
#      Consider the situation where our first
#      sample is of size 20 and has standard deviation
#      of s_1 = 6.249 while the second sample is of size 27
#      and has a standard deviation of s_2=4.475.  Now, look
#      at the ratio (s_1^2)/(s_2^2).
s_1 <- 6.249
s_2 <- 4.475

ratio  <- s_1^2 / s_2^2
ratio

#     What is the area to the right of 
#     1.95 for the situation where we have 19 degrees of
#     freedom in the numerator and 26 degrees of freedom
#     in the denominator?

pf( 1.95, 19, 26, lower.tail=FALSE)

# now, look at the area to the left of the reciprocal of
#      1.95 with 26 degrees of freedom in the numerator and
#      19 degrees of freedom in the denominator.

pf( 1/1.95, 26, 19, lower.tail=TRUE)

# This last computation reflects looking at the area to the
# left of (s_2^2) / (s_1^2).  Note the reversal of the degrees
# of freedom.

# Stated in general terms, the area to the right of a value
#     that we may call x  with n_1 degrees of freedom in the 
#     numerator and n_2 degrees of freedom in the denominator
#  is equal to the area to the left of 1/x for n_2 degrees of
#     freedom in the numerator and n_1 degrees of freedom in 
#     denominator.
#
# That relationship allows textbooks to just publish tables 
#     for critical values for given areas to the right.
# Thus, if you need the critical value that has 5% to the left
#     of it for 31 and 43 degrees of freedom (numerator and
#     denominator) you first look up, in the table, the value 
#     that has 5% to the right for 43 and 31 degrees of freedom.
#     Then, the critical value that you seek is just the
#     reciprocal of the value that you found in the table.
#
#  we can do this in R
c_val_right <- qf( 0.05, 43, 31, lower.tail=FALSE)
c_val_right  # this is the value we would get from the table
c_val_left <- 1 / c_val_right
c_val_left   # this is the critical value we want

#  all of this is silly for us since we could get that
#  critical value by just using qf()
qf(0.05, 31, 43 )