how to statistically analyse accuracy data (psycholinguistics) My experiment is a naming task, in which participants name pictures and their vocal responses are recorded and the onset of vocalization. The data I am dealing with are reaction times in ms and accuracy ( correct answers in percentages. I did the analysis of reaction times and what is left is the accuracy data in which I need your valued help.

Now, just to explain what I have done so far:
Each subject has an average score results for four conditions: Condition A, scored 10/15 , B 9/15, C 6/15 , and D 7/15. I transferred the discrete numbers into percentages and intended to use a mixed anova as I have One independent variable with two levels and two dependent variables with two level each. 

My questions are:

1- Do I have to run a normality test for these data? the data size is over 30?
2- what if they were not normally distributed? what should I do?
3- Is transferring data into percentage right or wrong? 

Many Thanks.
Sounds interesting!
Transferring counts into percentages definitely loses some information: 1/2 is not the same evidence as 100/200, but the difference is lost as a percentage.

ANOVA is pretty popular, but as you note in the question it can have problems.
Because you have the classic linguistics/psych use-case of non-independent data (ie grouped by participant), you probably want to jump on the mixed effects model bandwagon if you possibly can. I hear it's all the rage.

Some justification/motivation here:

Jaeger, T. F. (2008). Categorical data analysis: Away from ANOVAs (transformation or not) and towards logit mixed models. Journal of memory and language, 59(4), 434-446.

And R package to do the heavy lifting for you here:

https://cran.r-project.org/web/packages/glmm/vignettes/intro.pdf

Sounds like you have the binomial-input case.