What method is recommended after outliers removal?
based on this information I would recommend to use


Wilcoxon signed-rank test


or


paired Student's t test


This depends on your sample size and distribution. To test if your data set is normally distributed you can use the Jarque-Bera test.

I didn't work with Matlab yet, but  I guess all the tests should be implemented in Matlab. 

From that point you could evaluate the impact of the condition on response time. 
Visualize your data (X on Y) - Scatter plot / Heatmap (if multivariate data). 

From here you could start building a model


starting with linear regression
going into more advanced models/approaches -- random forests, bootstrapping etc.


Added 

If you only have binary variables (condition / no condition) - you need to set up dummy variables to perform a regression model (1 / 0) . Data then could look like:

 obs  response-time   any_condition  condition_1   condition_2   condition_3
  1   12.54           1              1             0             0
  2   19.34           0              0             0             0
  3   13.32           1              1             1             0
  4   14.7            0              0             0             0


If you have multiple different "state" conditions, I would recommend to use a control variable ("any_condition") - to see if conditions (not specifying which, have an impact) on the response time. 

I hope that helps.