Approximate confidence interval
Variables a and b may be measured in different units, so there is no way to directly combine the standard errors as they may also be in different units. The most complete discussion of this is given by Fieller (1954). [1]
Fieller showed that if a and b are (possibly correlated) means of two samples with expectations 
and 
, and variances 
and 
and covariance 
, and if 
are all known, then a (1 − α) confidence interval (mL, mU) for 
is given by
where
Here 
is an unbiased estimator of 
based on r degrees of freedom, and 
is the 
-level deviate from the Student's t-distribution based on r degrees of freedom.
Three features of this formula are important in this context:
a) The expression inside the square root has to be positive, or else the resulting interval will be imaginary.
b) When g is very close to 1, the confidence interval is infinite.
c) When g is greater than 1, the overall divisor outside the square brackets is negative and the confidence interval is exclusive.
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