How to interpret the confidence interval of a variance F-test using R? I'm trying to understand the confidence interval returned by the function var.test() in R. More specifically, the confidence interval returned by var.test() is not the one I find when doing the calculation of the F-test by myself.
For example :
> s1 <- 10:12 ; s2 <- 13:16
> var(s1)
[1] 1
> var(s2)
[1] 1.666667
> var.test(s1,s2)

    F test to compare two variances

data:  s1 and s2 
F = 0.6, num df = 2, denom df = 3, p-value = 0.7926
alternative hypothesis: true ratio of variances is not equal to 1 
95 percent confidence interval:
  0.03739691 23.49929674 
sample estimates:
ratio of variances 
               0.6

The 95% confidence interval here is [0.037,23.499]. I interpret "confidence interval" as "rejection region", i.e. if the test statistic F is inside this interval, the null hypothesis should be accepted, for a given statistical level (95% here). However, when I try to calculate this, I find :
> qf(c(0.025,0.975),length(s1)-1,length(s2)-1)
[1]  0.02553268 16.04410643

So I guess I'm wrong when I interpret var.test()'s "confidence interval" as the "rejection region". 
So my question is : what does this confidence interval represent?
 A: Your method using qf computes the rejection region that you would compare the ratio of the 2 variance to.  Using the central F is the correct thing to do because we calculate the rejection region assuming the null hypothesis is true and if the null (that the variances are equal) is true then we have a central F distribution.
The derivation for the formula for the confidence interval goes along these lines (I only show the lower limit, some slight modifications give the upper limit) 
$ \frac{v1}{\sigma_1^2} / \frac{v2}{\sigma_2^2} \sim f(df_1,df_2) $
$ \frac{v1}{v2} \times \frac{\sigma_1^2}{\sigma_2^2} \sim f(df_1,df_2)$
$prob( \frac{v1}{v2} \times \frac{\sigma_2^2}{\sigma_1^2} > f_{0.975} ) = 0.025 $
$prob( \frac{\sigma_1^2}{\sigma_2^2} < \frac{v1}{v2}/f_{0.975} )= 0.025$
So the confidence interval is also based on the central F since it is estimating the ratio of the 2 true variances.
And in R to get the value "manually" you do
vr <- var(s1)/var(s2)
vr/qf(.975,2,3)

which matches the result of var.test()
A: The rejection region for a test and a confidence interval are different things.  The rejection region is the region where you reject the null hypothesis that the ratio of variances equals $1$.  A $100(1-α)\%$ confidence interval is an interval that has the property that in repeated sampling would include the true parameter value in $100(1-α)\%$ of the cases.  There is a 1-1 correspondence between a confidence interval and the related hypothesis test but that does not mean that the rejection region equals the confidence region even when the confidence level is the same as the significance level for the test.
