A confidence interval estimates the interval within which the real coefficient will fall with a certain probability
the confidence intervals are “tight” (not wide) and do not include zero, suggesting the parameters are significant.
For a large repetitive experiments, 95% chance of the true value lies in the region defined by CI.
Or:
If you perform nonlinear regression many times (on different data sets), you expect the confidence interval to include the true value 95% of the time, but to exclude the true value the other 5% of the time (but you won't know when this happens).
In most of cases, you will use the CI to get a sense of if your results are any good. If CI are narrow, you know the parameter precisely; otherwise, you know that you have not determined the parameters very precisely.
Normally, that 95% confidence ranges over a factor of about two (e.g. 20~40 or 80~160) is very satisfactory.
By Harvey Motulsky & Arthur Christopoulos, "Fitting Models to Biological Data Using Linear and Nonlinear Regression"
http://matlab.cheme.cmu.edu/2011/08/29/nonlinear-curve-fitting-with-parameter-confidence-intervals/
http://books.google.com/books?id=g1FO9pquF3kC&pg=PA100&lpg=PA100&dq=confidence+interval+curve+fitting&source=bl&ots=m-T6aj4y4q&sig=T1-Q37uBPAxPL5M9Li395iAlr88&hl=en&sa=X&ei=9h00UeOOIKvJ0AGf04HYBA&ved=0CFoQ6AEwBTgU#v=onepage&q=confidence%20interval%20curve%20fitting&f=false
https://www.zoology.ubc.ca/~schluter/R/fit-model/
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