Can chi-square test be used on non-integer observed frequencies? I am using a model to calculate observed frequencies, which some times gives non-integer values. I can round these frequencies but that seems like artificially distorting the information I have. 
For Example:
Example Data
          Yes        No
Male       11        19
Female     16        17

Assume my model just divides everything by 3, so model data becomes:
           Yes        No
Male       3.67        6.33
Female     5.33        5.67

This data has to be  used as "observed frequencies". Doing a chi-square test gives p value of 0.58. However, if I round this data to integers, chi-square test will give a p value of 0.8, which is very much different. My question is: is chi-square test theoretically valid on non-integer observed frequencies?
Edit: Please note that data and model specified in the question are not real, just to make you understand the problem I am facing. Real data is of this type. 
            Male     Female
Source1     10.8      18.2
Source2     16        17

The real data is the prediction of males and females according to the job roles and City from Bureau of Labor Statistics.
I have no control over the data coming from source1, which (surprisingly) contains decimal point numbers. All I can do is round the data from source1.
 A: 
observed count have decimal points. 

If you have fractions, you don't have observed counts but something else. Counts actually count things, 0, 1, 2... 

. The real data is the prediction of males and females according to the job roles and City from Bureau of Labor Statistics

Predictions don't have the same properties (including the same uncertainty) as count data. 
The chi-squared test relies on the data being actual observed counts, not predictions of counts or any other manipulation of counts. This is needed to obtain the correct scaling of $O_i-E_i$ (the denominator is based on particular assumptions that in general won't hold for things that are not counts).
As a result, your test won't work - you can't just treat predictions as observed counts. It's irrelevant whether they were rounded integers or not (the only difference of any consequence is that the non-integer values made it obvious you didn't have actual observed counts; if the predictions had been rounded you might never have known there was a problem). 
A: Even before rounding, parts of your question describes things which seem to me problematic. IMHO, it pays to consider them, as they relate to the cause of rounding.

Scaling

Say my model just divides everything by 3

The rationale behind this test involves the multinomial distribution, and contains combinatorical terms of the form
$${n \choose n_1 \cdot n_k}$$
These terms are not invariant to scaling. I.e., you cannot replace this with 
$${\alpha n \choose \alpha^k n_1 \cdot n_k} = {\alpha n \choose \alpha n_1 \cdot \alpha n_k}$$
and expect to get the same results. 

Test Assumptions
It's possible your division of 3 is caused by this being an average of three observations. In this case, though, there's a problem with assuming that the test is relevant here:

A common rule is 5 or more in all cells of a 2-by-2 table

Post the division by 3, this does not hold, and the numbers are not in the range where it can be assumed that this test is applicable.
A: What you are doing here is in a manner of speaking called weighting of cases. Let's assume that you are doing a study where you want to find out the prevalence of child abuse in high school population. In the population you have 50% boys and 50% girls, however your sample is 60% boys and 40% girls due to the sampling error.
If you assume that the prevalence is higher among girls than boys then your sample estimate would be deflated because the sample does not represent the population well. What you do here is that you employ weights to cases and you will declare that each male participant has the weight of 0.83, and each female participant has the weight of 1.25 so you wil get e.g. 60 male participants * 0,83=50 and 40 female * 1,25=50.
When you do this your frequencies can become fractional values and this is quite common in survey based studies.
Softwares have issues with this and e.g. SPSS when calculating nonparametric statistics ignores the decimal parts (imagine calculating ranks when each entity has its own weight), and SPSS even ignores the weights when performing CHI Square test.
However the logic of chi square is that you are first calculating the difference between the observed and expected value, then you square it to remove the direction and to give more importance to bigger differences, and the you divide this by expected value to return it to original metric and to put it in terms of expected value. From this perspective, if you use the fractional counts, the logic is still preserved. If you want to calculate this, you should check out how your software deals with weighted data, or do it by hand.
Be aware that meddling with weights means meddling with statistical power so use caution :)
