I am a beginner in statistics. My problem is as follows: I have a set of $N$ observations $\{(x_i,y_i,e_i)\}$, where $e_i$ is the error in the $i^\mathrm{th}$ observation. The individual observations were calculated as averages within bins of raw data, and the errors as square-roots of the bin variances.
I also have a function $f(x)$. Now, I would like to quantify how good a fit the function $f$ is to the observations. Also using $f(x) = 0$, I would like to quantify how likely it is that the observations are just random, 0 mean noise.
In my data, I see, among a lot of noise-looking, equally positive and negative values, a long sequence of positive values. I would like to quantify how unlikely its occurrence is and whether it's real or not.
Could you please help me with that or point me in the right direction?
Thanks! -SSF