Hello Hajime,

Thank you! Great question. You are correct; the traditional approach assumes that that sampling distribution of the mean is normal once there is a sufficiently large sample and observations are sampled appropriately. The potential issue comes with if the supporting assumptions, such as sample size, cannot be met. If they are not met, then it becomes more difficult to make reliable conclusions about the estimates because you are not certain about the theoretical sampling distribution.

With the bootstrapping approach, however, you can plot the sampling distribution using the estimates from the resampled set (and then run the various tests) to determine whether the distribution is normal. This way, you do not have to make any theoretical assumptions to make conclusions about the estimates.