You have proof that $(\Delta W/A)^3 \propto t$. The best option at this point is to do a true regression fit to the equation $(\Delta W/A) = k_r t^{1/3} + C$ rather than continuing with the linearized equation. You will find $k_r ± \Delta k_r$ in this manner. The "offset" $C$ may or may not be appropriately forced to zero as needed. FWIW, the $R^2$ value is otherwise meaningless fluff.

 Thank you for your help.  Do you mean that I need to substitute each value in \Delta W/A = k_r  t^{1/3}  (assuming C=0) and find \Delta k_r? What do you mean by k_r+- \Delta k_r?

Thanks. 

Use a regression-analysis program (my favorite is Igor Pro, though this is also supported in Origin, MatLab, Mathematica, Maple, MathCad ...). Create a plot of (\Delta W/A) versus t. Fit that plot to the equation (\Delta W/A) = k_r t^3 + C. You can (again certainly in Igor Pro if not elsewhere) allow C to be free or fixed at a defined value. The output of the regression fit is k_r ± \Delta k_r (as well as C ± \Delta C as appropriate). The \Delta k_r (and \Delta C) are the standard uncertainties of the regression coefficients.

It looks like your plot only has 6 actual data points. I'd start with an exact fit with a fifth order polynomial to see how close it really is to a 3rd order polynomial (are the X^4 and X^5 terms really near zero).  As Dr Weimer points out there are plenty of great statistical based regression programs out there.  I'd prefer using real kinetic rate expressions to fit the data but  if a polynomial or other regression based fit suits your purpose there is nothing wrong with using it.

@R Manner: A point or two of serious contention. The data contain 7 points since (0, 0) is demanded too. A polynomial fit is a lazy man's way of saying "I don't know the fundamentals, so I'll punt this problem". A fifth order polynomial is ... (mildly put) abusive, especially based on the quality of fit to the 3rd-order linearized data set.

Jeffrey,

 I'm not saying that the fifth order polynomial is the best way to go but there is something to be learned from the coefficient in polynomial fits. But Ziad has already proposed a non traditional sort of polynomial fit(cubic term only).  You should at least check how it compares to a more complete polynomial fit.  If all of the coefficient except for the cubic term are near zero I would consider that to be good evidence that there really is a reason why a cubic fit has some merit.  

I don't know the purpose of this particular data fit so I do not know if 0,0 is "demanded".  In its current state it is useful data fit not a rigorous kinetic expression.  It is a correlation.  It looks to me like the data does not actually go  through (0,0) because the first 3 points are above the line. There may be a good reason why it doesn't go through (0,0) but if the correlation will not be used near 0 than fitting the 0,0 point is not necessary.  If the area near 0,0 is important than take more data in that region to see if something unexpected is going on.

I would prefer a more rigorous Kinetics based fit. But if this correlation based technique is fit for purpose there is no reason to overwork the problem. 

My purpose of all this is to compare different set of data for different temperatures. I need to find the difference in kinetics constants. To be clearer in my question, how the following cubic rate Kc were obtained in the reference paper since they are not directly the slope of the line? Thanks.

I'll leave detailed debates about the validity of different methods to fit data to other (private) discussions. Suffice to say that I disagree entirely with the school of thought that polynomial trend-line analysis with R^2 values gives anything useful for any level of further reasoned discussion except in narrowly-defined circumstances.

(amended): The OP's question requires that he take the same approach as the literature to retain the "best possible" comparison. That does NOT validate the approach. Indeed, the approach is sloppy because it looses all statistical or physical information of merit, and it likely has far greater inaccuracy than truly appreciated.

@Z Smoqi - Check the units. The slopes are per hour. The values are per second. My second guess is to check whether a cube or cube-root transformation of values was (mistakenly) applied going from the graph to the table.

I agree with Jeffrey that something looks wrong with your data compared with the reference data you sited.  The time unit could be a problem your data appears to be over a much shorter time scale than the reference and your fit at 1000 C is much better. This is also a very curious "kinetic" relationship.  3rd or 4th order kinetics is extremely unusual.  I am usually skeptical of 2nd order kinetics. 

It would probably be useful to us if you could describe your procedure in more detail?  Are you oxidizing with? Air? Oxygen? Something else?    What substance are you oxidizing.  Are you approaching a state where it is fully oxidized?