Looking for a Test of Quotients So I have roughly data from 8.000 persons: each person's working time and the time of absence from work due to illness.  What test should I use to determine whether there is a difference of the proportions
$$\text{absence quote}:=\frac{\text{absence time}}{\text{working time}}$$
in different groups, say men/women?
Edit: To clarify, I may be allowed to present the problem the I think I'm facing.  I'm just interested in the described quotient. I'm given 8000 person's data.
Assume in one group ($n_0 = 100$) the mean of absence time for a period, say a month, is 8h, in the other group ($n_1 = 300$) the mean is 17h.  The mean of  working time in the first group is 110h, in the second 150h.  Hence the quotient in the first group is $8/110$ whereas it's in the second $17/150$.  Now how to perform a test for the quotients?  (Sorry in case the question ist silly.)
First pic shows a box plot of working time ("Arbeitszeit"), the second on a box plot of absence time ("Ausfall").
Edit_2:  Following the advice I've plotted absence vs. work time an got something to think over:

But why should I take a square root scale?
Edit_3: I did a prtest for the proportion of people which absence time is more than 30% of their working time and got significant difference: 5.9% for group 0 and 12.8% for group 1.  Seems to me a much better strategy since the individual absence quotes are hardly comparable.
Thanks for your help again -- and any suggestions are welcome.



In case you are interested whether a typical absence quote (i.e. one that represents your data the best) significantly differs by gender, you can use the Yuen's t-test on 20% trimmed means (provided that no more than 20% of your dataset is comprised of outliers). In R this test can be conducted easily with the yuen function from the WRS2 package. Given your large sample size, the explanatory measure of effect size $\xi$ reported by this function will be also of particular importance. $\xi = 0.15, 0.30, 0.50$ translates roughly to small, medium respectively large effect size.
library(WRS2)

yuen(ratio ~ gender, yourdf)

