I am a materials engineer so please take my answer with a grain of salt.  But here is what I would do:

1. Rearrange the equation so that

1-F=exp(-at^b)

2. Take double natural logarithms so that

ln(-ln(1-F))=b(ln(t))-ln(a)

3. Conduct regression analysis between as "ln(-ln(1-F))" as your Y variable and "ln(t)" as your x variable.  Slope will be b and intercept will be "-ln(a)".

Good luck!

Thank you very much @Murat Tiryakioglu

Excel has a powerful nonlilnear regression function built in called Solver. The 30 second overview is to 1) can load your experimental data into two columns-- time and your amounts or % dissolved, for instance; 2) Creat cells for the variables to be fit. (For instance, if F = A * exp(-bt^n), create cells for A, B and n); 3) create a third column (nice to put it next to the experimental data) to calculate a simulated amount or % dissolved using the corresponding time and the value for A, b  and n (these are the same for all time points); 4) create fourth column with squared residuals or weighted squared residuals-- for instance, (exp value - simulated value)^2 , Alternatively, you can weight the squared residuals; create a cell (convenient if right below the cells for A, b and n) with the sum of the squares (i.e., add all of the squared residuals from the 4th column); 5) on the Excel ribbon, go to Data >> Solver (Solver is all the way to the right on the ribbon). In Solver, for the Set Objective cell, click on the cell denoting the sum of the residuals squared. Next, click on Minimize. Next, in the By Changing cells, click on the cells containing values of A, b and n.  Once this is done, click on Solve, and you will have your answers for the "best fit: of A, b and n.(Excel has enough information on using Solver to clear this up if it you need more help on that function.)

A couple of notes. First, you may have to activate Solver before using it. When you go to data, if you do not see Solver anywhere on the ribbon (should be to the far right, below "Data Analysis"), then you can activate it by going to the add-ins. If you do that, it's also useful to activate the Analysis Toolpak, which contains special functions such as Bessel, etc.)

Second, it is handy while doing this to create a plot with the experimental data (preferably as discrete points or symbols) as well as the simulated data (as a smooth line). This visual comparison will help you... it is also useful because you can guess values for A, b and n that roughly fit the experimental data. (When performing nonlinear regressions, the initial guesses are important... bad guesses will not converge. Good guesses will result in the simulated line being similar (not necessarily identical) to the experimental data. 

Hope this help!

Thanks alot Prof. Robert for detailed explanation.

You are very welcome. Good luck!

Nauman, I am familiar with Murat advised, Only, in the estimation of ln(-ln(1-F(t))), since F(t) is estimated by the median rank given by the Benard's approximation given by F(t)=(i-0.3)/(n+0.4), then the sample size n has big impact in the estimated b and a parameters. Thus, in order to select the right n, you has to determine first the desired or expected reliability percentile R(t), and based on R(t), determine n as n=-1/ln[R(t)]. (see eq.(35) of the reference paper given below).

In particular, since b is completely determined by the standard deviation of the lifetime data [eq.(41)], and a is completely determined by the mean of the lifetime data [eq.(46)], then, selecting the right n is determinant to accurately estimate the b and a parameters.

My paper where these issues are addressed and solved is "Weibull and lognormal Taguchi analysis using multiple linear regression". which is uploaded to RG. Please observe that in addition to estimate n correctly, the paper also gives the method to determine b and a for a desired combination of the factors that determine your life times. Thus, if you have the factors that determine the lifetimes you can use them in the estimation of the related Weibull family. Any question related to the paper content or how it could be implemented by using your data, please let me know.

Good luck.