How to extrapolate my sample results to population We have tested randomly selected bacteria ($k = 198$) for antibiotic resistance over a period of 3 years (total isolates $n = 444$) and observed $117$ resistant strains.
I would like to extrapolate my results to total isolates. I presume,
$$\hat{q} = \frac{117 \times 444}{198} = 262.$$ 
Is there any specific statistical method for this analysis?
 A: First of all, the selected bacteria must have been selected randomly to do any inference.
262 is indeed a good estimate of the resistant strains you would have in your population. 
But what you would like to get - I guess - is to know how accurate is this estimation. The simplest way to do that is to compute confidence interval.
The 95% confidence interval for your proportion would be (if your population was infinite) :
$$\hat{p}\pm z_{\alpha/2}\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n} }$$
In your case, this is different, as you study almost half of your population, you can pretend to have a better estimate of your proportion :
$$\hat{p}\pm z_{\alpha/2}\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n} \cdot \frac{N-n}{N-1}}$$
In your case you have $z_{\alpha/2}=1.96$, $N=444$, $n=198$, $\hat{p}=\frac{117}{198}$
If I typed it right wolfram alpha gives $0.066$
This means that in your population $0.591-0.066<p<0.591+0.066$ which in turns mean that you are 95% confident that $444*(0.591-0.066)<Nresistant<444*(0.591+0.066)$ 
To wrap it up :
$$P(233.1<Nresistant<291.7)=95\%$$
