I am completing a stats project in R and so far I have created a function that computes the sample kurtosis of a data set. I have then used bootstrapping to compute a 95% confidence interval for the kurtosis. Now I am wanting to perform a hypothesis test where the null hypothesis is that the kurtosis is zero and the alternative hypothesis is that the kurtosis is not zero. What test should I be using for this?
I know that a zero kurtosis value suggests the data is normally distributed so should I somehow be using the student t test where I assume the data is normally distributed?